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Calculus

Calculus

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Table of Contents

1. Calculus.................................................. ........................25Precalculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Basics of Differentiation

See Picture License Information Here

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7

Applications of Derivatives


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 8

Important Theorems


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 8

Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Basics of Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Integration techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Applications of Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Parametric and Polar Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Parametric Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Polar Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Sequences and Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Series and calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Multivariable and Differential Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Further Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Formal Theory of Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Acknowledgements and Further Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2. Contributing................................................ ......................34Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Templates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

{{Calculus/Top Nav}}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

{{Calculus/TOC}}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

{{Calculus/Def}}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

{{Calculus/Stub}}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

On Inclusiveness vs. Exclusiveness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

TODO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3. Introduction................................................. .....................37What is calculus?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Why learn calculus?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

What is involved in learning calculus?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

What you should know before using this text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Scope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4. Precalculus.................................................. ....................41Exercises.................................................. ....................42

Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Convert to interval notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

State the following intervals using inequalities. . . . . . . . . . . . . . . . . . . . . . . 42

Which one of the following is a true statement?. . . . . . . . . . . . . . . . . . . . . . 43

Evaluate the following expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Simplify the following. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Factor the following expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

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Simplify the following. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Decomposition of functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Graphing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Solutions.................................................. .....................49Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5. Algebra................................................. ..........................50Rules of arithmetic and algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Interval notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Exponents and radicals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Factoring and roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Simplifying rational expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6. Functions.................................................. .......................57Classical understanding of functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Modern understanding of functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

The vertical line test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Important functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Example functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Manipulating functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Composition of functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Domain and Range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

One-to-one Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Inverse functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7. Graphing linear functions............................................... .......72

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Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Slope-intercept form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Point-slope form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Calculating slope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Two-point form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

8. Limits.................................................. ............................75An Introduction to Limits............................................... .....76

Intuitive Look. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Informal definition of a limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Limit rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

The Squeeze Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Finding limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Using limit notation to describe asymptotes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Key application of limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

External links. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Exercises.................................................. ....................92Limits with Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Basic Limit Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

One Sided Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Two Sided Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Limits to Infinity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Limits of Piece Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Contents.................................................. .....................97Solutions.................................................. .....................98

Basic Limit Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Harder Limit Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

9. Finite Limits................................................. ...................100Informal Finite Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

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One-Sided Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

10. Infinite Limits................................................ .................103Informal infinite limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Limits at infinity of rational functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Infinity is not a number.............................................. .......107Addition breaks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Reinterpret formulas that use

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 8

Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

11. Continuity................................................ .....................112Defining Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Discontinuities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Removable Discontinuities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Jump Discontinuities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

One-Sided Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Intermediate Value Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Application: bisection method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

12. Formal Definition of the Limit............................................ ..117Formal Definition of the Limit at Infinity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Formal Definition of a Limit Being Infinity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

13. Differentiation................................................ .................127Basics of Differentiation


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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Important Theorems


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 8

Differentiation Defined................................................. ....129What is differentiation?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

The Definition of Slope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Of a Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Of a Graph of a Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

The Rate of Change of a Function at a Point. . . . . . . . . . . . . . . . . . . . . . . . . . 135

The Definition of the Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Understanding the Derivative Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Differentiation rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Derivative of a Constant Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Derivative of a Linear Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Constant multiple and addition rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

The Power Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Derivatives of polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Exercises................................................. ....................148Find The Derivative By Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Prove Differentiation Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Find The Derivative By Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

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Power Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Product Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Quotient Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Chain Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Exponentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Logarithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Trig Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

More Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Implicit Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Logarithmic Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Equation of Tangent Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Higher Order Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Relative Extrema. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Range of Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Absolute Extrema. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Determine Intervals of Change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Determine Intervals of Concavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Word Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Graphing Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Contents.................................................. ....................158Basics of Differentiation


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5 8



. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5 8

Calculus

- 9 -




Important Theorems


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5 9

Solutions.................................................. ...................160Find The Derivative By Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Prove Differentiation Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Proof of the Derivative of a Constant Function. . . . . . . . . . . . . . . . . . . . . . 162

Proof of the Derivative of a Linear Function. . . . . . . . . . . . . . . . . . . . . . . . . 163

Proof of the Constant Multiple Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Proof of the Addition and Subtraction Rules. . . . . . . . . . . . . . . . . . . . . . . . 164

Find The Derivative By Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Implicit Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Higher Order Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

14. Product and Quotient Rules.............................................. .167Product Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Application, proof of the power rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Quotient rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

15. Derivatives of Trigonometric Functions..................................17116. Chain Rule................................................. ...................17517. More Differentiation Rules............................................... ..177

External links. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

18. Higher Order Derivatives............................................. ......178Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

19. Implicit differentiation................................................ ........181Explicit differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

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Implicit differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Uses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Implicit Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

Inverse Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

20. Derivatives of Exponential and Logarithm Functions..................187Exponential Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Logarithm Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

Logarithmic differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

21. Extrema and Points of Inflection..........................................192The Extremum Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

22. Newton's Method................................................ ............196Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

See Also. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

23. Related Rates................................................. ...............200Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

Related Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

Common Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Filling Tank. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Problem Set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Solution Set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

24. Kinematics.................................................. ..................207Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Average Velocity and Acceleration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Instantaneous Velocity and Acceleration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

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Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

25. Optimization.................................................. ................211Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Volume Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Sales Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

26. Euler's Method................................................ ...............215Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

27. Extreme Value Theorem............................................... .....216First Derivative Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Second Derivative Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

28. Rolle's Theorem................................................. ............224Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

29. Mean Value Theorem for Functions......................................226Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

Differentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

Definition of Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

Cauchy's Mean Value Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

30. Integration................................................ .....................231Basics of Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

Integration techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

Applications of Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

Exercises................................................. ....................233Set One: Sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

Set Two: Integration of Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

Indefinite Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

Integration by parts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

Contents................................................... ...................235Basics of Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Calculus

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Integration techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

Applications of Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

Solutions.................................................. ...................237Solutions to Set One. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Solutions to Set Two. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Solutions to Set Three. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

31. Indefinite integral.............................................. ..............239Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

Indefinite integral identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

Basic Properties of Indefinite Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

Indefinite integrals of Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

Integral of the Inverse function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Integral of the Exponential function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Integral of Sine and Cosine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

The Substitution Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Preliminary Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Generalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

General Substitution Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

Integration by Parts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

Preliminary Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

General Integration by Parts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

32. Definite integral.............................................. ................251Definition of the Definite Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Independence of Variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

Left and Right Handed Riemann Sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

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Basic Properties of the Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

The Constant Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

The addition and subtraction rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

The Comparison Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

Linearity with respect to endpoints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

Even and odd functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

33. Fundamental Theorem of Calculus.......................................266Fundamental Theorem of Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

Statement of the Fundamental Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Integration of Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

34. Infinite Sums................................................. .................273Exact Integrals as Limits of Sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

35. Recognizing Derivatives and the Substitution Rule....................276Recognizing Derivatives and Reversing Derivative Rules. . . . . . . . . . . . . . . . . . . 276

Integration by Substitution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

Integrating with the derivative present. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

Proof of the substitution rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

36. Integration by Parts........................................... ...............281Integration by Parts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

With definite integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

37. Integration by Complexifying............................................ ...28538. Partial Fraction Decomposition........................................... .286

Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

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39. Trigonometric Substitution............................................. ....290Sine substitution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

Tangent substitution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

Secant substitution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

40. Tangent Half Angle............................................... ...........295Alternate Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

41. Trigonometric Integrals............................................ .........300Powers of Sine and Cosine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

Powers of Tan and Secant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

More trigonometric combinations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

42. Reduction Formula.............................................. ............30643. Irrational Functions................................................ ..........308

Type 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

Type 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

Type 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

Type 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

Type 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

44. Numerical Approximations............................................... ..311Riemann Sum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

Right Rectangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

Left Rectangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

Trapezoidal Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

Simpson's Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

Further reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

45. Improper integrals............................................... ............313L'Hopital's Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

Improper Integrals with Infinite Limits of Integration. . . . . . . . . . . . . . . . . . . . . . . . . 316

Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

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Improper Integrals with a Finite Number Discontinuities. . . . . . . . . . . . . . . . . . . . 320

Definition of improper integrals with a single discontinuity. . . . . . . . . . . . . . . . 320

Definition: Improper integrals with finite number of discontinuities. . . . . . . 322

Comparison Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

An extension of the comparison theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

46. Area.................................................. ..........................327Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

Area between two curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

47. Volume................................................. .......................330Formal Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

Example 1: A right cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

Example 2: A right circular cone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

Example 3: A sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

Extension to Non-trivial Solids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

48. Volume of solids of revolution........................................... ...336Revolution solids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

Calculating the volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

49. Arc length................................................. ....................338Definition (Length of a Curve). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

The Arclength Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

Arclength of a parametric curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

50. Surface area................................................ ..................342Definition (Surface of Revolution). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

The Surface Area Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

51. Work................................................ ...........................34552. Centre of mass................................................ ...............34653. Parametric and Polar Equations..........................................347

Parametric Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

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Polar Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

54. Parametric Introduction.............................................. .......348Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

Forms of Parametric Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

Parametric Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

Vector Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

Equalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

Converting Parametric Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

55. Parametric Differentiation............................................ ......351Taking Derivatives of Parametric Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

Slope of Parametric Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

Concavity of Parametric Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

56. Parametric Integration............................................ ..........354Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

57. Polar Introduction................................................ ............355Plotting points with polar coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

Converting between polar and Cartesian coordinates. . . . . . . . . . . . . . . . . . . . 357

Polar equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

Circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

Polar rose. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

Archimedean spiral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

Conic sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

58. Polar Differentiation.............................................. ...........362Differential calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

59. Polar Integration.............................................. ...............363Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

Proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

Integral calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

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Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

An interesting example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

60. Sequences and Series............................................... .......367Basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

Series and calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

Exercises................................................. ....................369Hints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

Answers only. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

Full solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

61. Sequences.................................................. ..................376Examples and notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

Types and properties of sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

Sequences in analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

62. Series................................................. .........................379Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

Absolute convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

Ratio test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

Integral test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

Limit comparison test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

Alternating series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

Geometric series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

Telescoping series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

63. Taylor series.............................................. ....................388Taylor Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

Derivation/why this works. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

List of Taylor series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

Multiple dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

History. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

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Constructing a Taylor Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

Generalized Mean Value Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

64. Power series............................................. .....................396Motivations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396

An example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396

Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396

Radius of convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

An example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

Another example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

Abstraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

Differentiation and Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

Further reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

65. Multivariable and differential calculus....................................40066. Vectors................................................. .......................402

Two-Dimensional Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

Component Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

Magnitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

Direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

Vector Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

Applications of Scalar Multiplication and Dot Product. . . . . . . . . . . . . . . . . . . . 406

Polar coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

Three-Dimensional Coordinates and Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

Basic definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

Three dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

Cylindrical coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

Spherical coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

Cross Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

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Triple Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

Three-Dimensional Lines and Planes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

Vector-Valued Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

Limits, Derivatives, and Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

Velocity, Acceleration, Curvature, and a brief mention of the Binormal. . . 418

67. Lines and Planes in Space.............................................. ...420Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

Parametric Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

Line in Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

Plane in Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

Vector Equation (of a Plane in Space, or of a Line in a Plane). . . . . . . . . . . . . . 421

Scalar Equation (of a Plane in Space, or of a Line in a Plane). . . . . . . . . . . . . . 422

68. Multivariable calculus.............................................. .........423

Topology in Rn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

Lengths and distances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

Open and closed balls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

Boundary points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

Curves and parameterizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

Collision and intersection points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

Continuity and differentiability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

Tangent vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

Different parameterizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

Limits and continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430

Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

Special note about limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

Differentiable functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

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Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

Rules of taking Jacobians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

Alternate notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

Directional derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

Gradient vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

Divergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

Curl. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

Product and chain rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

Second order differentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

Riemann sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

Iterated integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

Order of integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

Parametric integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

Line integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

Surface and Volume Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

Gauss's divergence theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

Stokes' curl theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

69. Ordinary differential equations............................................453Notations and terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

Terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

Some simple differential equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

Basic first order DEs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

Separable equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

Homogeneous equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

Linear equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460

Exact equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

Basic second and higher order ODE's. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

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Reducible ODE's. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

Linear ODEs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

70. Partial differential equations............................................. ..472Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

First order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

Special cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

Quasilinear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

Fully non-linear PDEs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

Second order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488

Elliptic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

Hyperbolic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

Parabolic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

71. Multivariable and differential calculus:Exercises.......................49672. Extensions.................................................. ..................497

Further Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

Formal Theory of Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

73. Systems of ordinary differential equations..............................498First order systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498

74. Real numbers................................................ ................501Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

Constructing the Real Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

Properties of Real Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

75. Complex numbers.............................................. .............506Equality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

Notation and operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

The field of complex numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

The complex plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508

Polar form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508

Conversion from the polar form to the Cartesian form. . . . . . . . . . . . . . . . . . . . 509

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Conversion from the Cartesian form to the polar form. . . . . . . . . . . . . . . . . . . . 509

Notation of the polar form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

Multiplication, division, exponentiation, and root extraction in the polarform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

Absolute value, conjugation and distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

Complex fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

Matrix representation of complex numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

76. References................................................. ...................51577. Table of Trigonometry........................................... ............516

Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

Pythagorean Identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

Double Angle Identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

Angle Sum Identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

Product-to-sum identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518

78. Summation notation................................................. ........519Common summations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

79. Tables of Derivatives.......................................... ..............521General Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

Powers and Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522

Exponential and Logarithmic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522


Hyperbolic and Inverse Hyperbolic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523

80. Tables of Integrals........................................... ................524Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

Powers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

Basic Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

Reciprocal Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526

Inverse Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526

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Exponential and Logarithmic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527


81. Acknowledgements................................................ ..........528Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528

Contributors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528

Further Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528

Other Calculus Textbooks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

82. Choosing delta................................................. ..............53083. More Differentation Rules............................................... ...533

External links. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

84. Mean Value Theorem............................................... ........534Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534

Differentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536

Definition of Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

Cauchy's Mean Value Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

85. Infinite series................................................ .................539Basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

Series and calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

GNU Free Documentation License. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .540

List of Contributors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .548

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Precalculus

• Algebra


• Functions


• Graphing linear functions


• Exercises

Limits


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http://en.wikibooks.org/w/index.php?title=File:Limit-at-infinity-graph.png&filetimestamp=20060319153541

• Calculus/Limits/An Introduction to Limits


• Finite Limits


• Infinite Limits


• Continuity


• Formal Definition of the Limit


• Calculus/Limits/Exercises

Differentiation




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• Differentiation Defined• Product and Quotient Rules• Derivatives of Trigonometric Functions• Chain Rule• More differentiation rules- More rules for differentiation• Higher Order derivatives- An introduction to second power derivatives• Implicit Differentiation• Derivatives of Exponential and Logarithm Functions• Exercises



• Extrema and Points of Inflection• Newton's Method• Related Rates• Kinematics• Optimization• Euler's Method• Exercises

Important Theorems


• Extreme Value Theorem• Rolle's Theorem• Mean Value Theorem

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http://en.wikibooks.org/wiki/Calculus/Differentiation/Exercises#Applications


Integration


Basics of Integration

• Indefinite integral


• Definite integral


• Fundamental Theorem of Calculus


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Integration techniques

• Infinite Sums• Derivative Rules and the Substitution Rule• Integration by Parts• Complexifying• Rational Functions by Partial Fraction Decomposition• Trigonometric Substitutions• Tangent Half Angle Substitution• Trigonometric Integrals• Reduction Formula• Irrational Functions• Numerical Approximations• Integration techniques


• Improper integrals


• Exercises

Applications of Integration

• Area• Volume• Volume of solids of revolution• Arc length• Surface area• Work• Centre of mass• Pressure and force• Probability

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http://en.wikibooks.org/w/index.php?title=Calculus/Probability&action=edit&redlink=1

Parametric and Polar Equations


Parametric Equations

• Introduction to Parametric Equations• Differentiation and Parametric Equations• Integration and Parametric Equations

Polar Equations

• Introduction to Polar Equations• Differentiation and Polar Equations• Integration and Polar Equations

Sequences and Series

Basics

• Sequences• Series

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Series and calculus

• Taylor series• Power series• Exercises

Multivariable and Differential Calculus


• Vectors


• Lines and Planes in Space


• Multivariable Calculus


• Ordinary Differential Equations


• Partial Differential Equations


• Exercises

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Extensions

Further Analysis

• Systems of Ordinary Differential Equations


Formal Theory of Calculus

• Real numbers


• Complex numbers

References

• Table of Trigonometry• Summation notation• Tables of Derivatives• Tables of Integrals

Acknowledgements and Further Reading


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Contributing

Precalculus→Calculus← Introduction

Contributing

Notes on contributing to theCalculustextbook. This is theWikibooks:Local manuals ofstylefor theCalculustextbook.

Structure

The structure currently being implemented is as follows:

• The book is divided into major sections (these sections may incorporate material thatmight comprise two or more chapters in a standard text).

• These sections and their articles are listed on themain page. Some articles actually spanmore than one page and if a user clicks on the link to them they will be taken to thesection's contents page where they can select which page of the article they would liketo view.

• Each section has a page with its complete, detailed contents listed plus a link to thatsection's page(s) of Exercises and Problems. Section pages should incorporate moreactual material such as descriptions of articles and discussion of the chapter's importantpoints or important/unusual prerequisites.

• Pages of Exercises are divided into sets (by type of problem) of sequentially numberedexercises or problems. Each page of exercises links to a corresponding page of solu-tions.NamingConvention:Calculus/[Section-Name]/Exercises(ex.Calculus/Differenti-ation/Exercises)

• Pages of solutions are organized based on their corresponding page of exercises (exercis-es set one has a corresponding set one of solutions, etc.) Solutions should be completeand logical. Particularly important steps should be noted and decribed in words as wellas being shown symbolicly.Naming Convention: Calculus/[Section-Name]/Solutions(ex. Calculus/Differentiation/Solutions)

• All content pages and section pages should include at the end {{Calculus/TOC}} whichis a quick-navigation template.

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http://en.wikibooks.org/wiki/Wikibooks:Local_manuals_of_style

http://en.wikibooks.org/wiki/Wikibooks:Local_manuals_of_style

Templates

SeeCategory:Calculus (book)/Templatesfor a list of allCalculustextbook templates.

{{Calculus/Top Nav}}

{{ Calculus/Top Nav |Limits|Infinite Limits}} produces this navigationbox:

Infinite Limits →Calculus← Limits

Contributing

All Calculuscontent pages should include this at the top of the page.

{{Calculus/TOC}}

{{ Calculus/TOC }} produces this navigation box:

All Calculuscontent pages should include this at the bottom of the page. This also adds thepage toCategory:Calculus (book).

{{Calculus/Def}}

{{ Calculus/Def |text=My definition here.}} produces a box for importanttext:

My definition here.Use this to introduce significant new definitions and concepts. SeeCalculus/Limits#Informal

definition of a limit, where the informal definition of a limit is inside such a box.

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http://en.wikibooks.org/wiki/Category:Calculus_%28book%29/Templates

http://en.wikibooks.org/wiki/Template:Calculus/Top_Nav

http://en.wikibooks.org/wiki/Template:Calculus/TOC

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http://en.wikibooks.org/wiki/Calculus/Limits#Informal_definition_of_a_limit

{{Calculus/Stub}}

{{ Calculus/Stub }} produces a stub notice, signifying that the given page or sectionis too short.

On Inclusiveness vs. Exclusiveness

An Extensions section is included for further topics. Any study beyond fairly basic calculusshould be in this section. We should aim to include more than is necessary and ensure that ourreaders are aware of this. The book should be structured so that J. Random Student can find arigorous course in the essentials of Calculus but also include further study of the topic that readerscan pick and choose topics from, as their interests warrant.

TODO

• ADD MORE, ORGANIZE MORE.• Provide answer for Differentation exercise on the absolute value function.• Provide a more in-depth Precalculus section.• Create problem sets (preferrably original, rigorous and realistic).• Create answer sets.• Move existing problems from within the articles to the appropriate Exercises page.

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http://en.wikibooks.org/wiki/Template:Calculus/Stub

Introduction

Contributing→Calculus

Introduction

Wikipediahas related information at

SeePic-

Calculus

tureLi-censeInfor-ma-tionHere

What is calculus?

Calculus is the branch of mathematics dealing with instantaneous rates of change of continu-ously varying quantities. For example, consider a moving car. It is possible to create a functiondescribing thedisplacementof the car (where it is located in relation to a reference point) at anypoint in time as well as a function describing thevelocity(speed and direction of movement) ofthe car at any point in time. If the car were traveling at a constant velocity, then algebra wouldbe sufficient to determine the position of the car at any time; if the velocity is unknown but stillconstant, the position of the car could be used (along with the time) to find the velocity.

However, the velocity of a car cannot jump from zero to 35 miles per hour at the beginningof a trip, stay constant throughout, and then jump back to zero at the end. As the accelerator ispressed down, the velocity rises gradually, and usually not at a constantrate (i.e., the driver maypush on the gas pedal harder at the beginning, in order to speed up). Describing such motion andfinding velocities and distances at particular times cannot be done using methods taught in pre-calculus, but it is not only possible but straightforward with calculus.

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http://en.wikipedia.org/wiki/Main_Page

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Calculus has two basic applications:differential calculusandintegral calculus. The simplestintroduction to differential calculus involves an explicit series of numbers. Given the series (42,43, 3, 18, 34), the differential of this series would be (1, -40, 15, 16). The new series is derivedfrom the difference of successive numbers which gives rise to its name "differential". Rarely, ifever, are differentials used on an explicit series of numbers as done here. Instead, they are derivedfrom a series of numbers defined by a continuous function which are described later.

Integral calculus, like differential calculus, can also be introduced via series of numbers.Notice that in the previous example, the original series can almost be derived solely from itsdifferential. Instead of taking the difference, however, integration involves taking the sum. Giventhe first number of the original series, 42 in this case, the rest of the original series can be derivedby adding each successive number in its differential (42+1, 43-40, 3+15, 18+16). Note thatknowledge of the first number in the original series is crucial in deriving the integral. As withdifferentials, integration is performed on continuous functions rather than explicit series ofnumbers, but the concept is still the same. Integral calculus allows us to calculate the area undera curve of almost any shape; in the car example, this enables you to find the displacement of thecar based on the velocity curve. This is because the area under the curve is the total distancemoved, as we will soon see.

Why learn calculus?

Calculus is essential for many areas of science and engineering. Both make heavy use ofmathematical functions to describe and predict physical phenomena that are subject to continualchange, and this requires the use of calculus. Take our car example: if you want to design cars,you need to know how to calculate forces, velocities, accelerations, and positions. All requirecalculus. Calculus is also necessary to study the motion of gases and particles, the interaction offorces, and the transfer of energy. It is also useful in business whenever rates are involved. Forexample, equations involving interest or supply and demand curves are grounded in the languageof calculus.

Calculus also provides important tools in understanding functions and has led to the develop-ment of new areas of mathematics including real and complex analysis, topology, and non-eu-clidean geometry.

Notwithstanding calculus'functionalutility (pun intended), many non-scientists and non-engineers have chosen to study calculus just for the challenge of doing so. A smaller number ofpersons undertake such a challenge and then discover that calculus is beautiful in and of itself.Is calculus then perhaps a manifestation of some divine ordering of the universe?

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What is involved in learning calculus?

Learning calculus, like much of mathematics, involves two parts:

• Understanding the concepts: You must be able to explain what it means when you takeaderivativerather than merely apply the formulas for finding a derivative. Otherwise,you will have no idea whether or not your solution is correct. Drawing diagrams, forexample, can help clarify abstract concepts.

• Symbolic manipulation: Like other branches of mathematics, calculus is written insymbols that represent concepts. You will learn what these symbols mean and how touse them. A good working knowledge oftrigonometryandalgebrais a must, especiallyin integral calculus. Sometimes you will need to manipulate expressions into a usableform before it is possible to perform operations in calculus.

What you should know before using this text

There are some basic skills that you need before you can use this text. Continuing with ourexample of a moving car:

• You will need to describe the motion of the car in symbols. This involves understandingfunctions.

• You need to manipulate these functions. This involves algebra.• You need to translate symbols into graphs and vice verse. This involves understanding

the graphing of functions.• It also helps (although it isn't necessarily essential) if you understand the functions used

in trigonometry since these functions appear frequently in science.

Scope

The first four chapters of this textbook cover the topics taught in a typical high school orfirst year college course. The first chapter,Precalculus, reviews those aspects of functions mostessential to the mastery of Calculus, the secondLimits, introduces the concept of the limit pro-cess. It also discusses some applications of limits and proposes using limits to examine slope andarea of functions. The next two chapters,DifferentiationandIntegration, apply limits to calculate

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http://en.wikipedia.org/wiki/derivative

http://en.wikibooks.org/wiki/Trigonometry

http://en.wikibooks.org/wiki/Algebra

derivatives and integrals. The Fundamental Theorem of Calculus is used, as are the essentialformulae for computation of derivatives and integrals without resorting to the limit process. Thethird and fourth chapters include articles that apply the concepts previously learned to calculatingvolumes, and so on as well as other important formulae.

The remainder of the central Calculus chapters cover topics taught in higher level Calculustopics: multivariable calculus, vectors, and series (Taylor, convergent, divergent).

Finally, the other chapters cover the same material, using formal notation. They introducethe material at a much faster pace, and cover many more theorems than the other two sections.They assume knowledge of some set theory and set notation.

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Precalculus

Algebra→Calculus← Contributing

Precalculus

• Algebra


• Functions


• Graphing linear functions


• Exercises

Calculus

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Exercises

Limits/An Introduction toLimits →

Calculus← Graphing linear functions

Precalculus/Exercises

Algebra

Convert to interval notation

1.2.3.4.5.6.7.8.9.10.

State the following intervals using inequalities

1.2.3.4.5.6.

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Which one of the following is a true statement?

Hint: the true statement is often referred to as thetriangle inequality.Give examples wherethe other two are false.

1.2.3.

Evaluate the following expressions

1.2.

3.

4.

5.

6.

7.

8.

9.

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Simplify the following

1.2.

3.4.

5.6.

7.

8.

Factor the following expressions

For 1-8, determine what values ofx make the expression 0 (i.e. determine the roots).

1.2.3.4.5.6.7.8.9.10.11.

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Simplify the following

1.

2.

3.

4.

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Functions

1. Let f(x) = x2.1. Computef(0) andf(2).2. What are the domain and range off?3. Doesf have an inverse? If so, find a formula for it.

2. Let f(x) = x + 2, g(x) = 1 / x.Give formulae for1.

f + g,1.2. f − g,3. g − f,4.

,5. f / g,6. g / f,7.

and8.

.2. Computef(g(2)) andg(f(2)).3. Do f andg have inverses? If so, find formulae for them.

3. Does this graph represent a function?


4. Consider the following function

1. What is the domain?2. What is the range?

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http://en.wikibooks.org/wiki/File:Sinx_over_x.svg

3. Where isf continuous?5. Consider the following function

1. What is the domain?2. What is the range?3. Where isf continuous?







Decomposition of functions

For each of the following functions,h, find functionsf andg such that (f(g(x)) = h(x)

1.2.

3.

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Graphing

1. Find the equation of the line that passes through the point (1,-1) and has slope 3.2. Find the equation of the line that passes through the origin and the point (2,3).

Solutions

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Solutions

Functions

1. 1. f(0) = 0, f(2) = 42. The domain is

; the range is

,3. No, sincef isn't one-to-one; for example,f( − 1) = f(1) = 1.

2. 1. 1. (f + g)(x) = x + 2 + 1 /x = (x2 + 2x + 1) / x.2. (f − g)(x) = x + 2 − 1 /x = (x2 + 2x − 1) / x.3. (g − f)(x) = 1 / x − x − 2 = (1 −x2 − 2x) / x.4.

.5. (f / g)(x) = x(x + 2) provided

. Note that 0 is not in the domain off / g, since it's not in the domain ofg,and you can't divide by something that doesn't exist!

6. (g / f)(x) = 1 / [x(x + 2)]. Although 0 is still not in the domain, we don'tneed to state it now, since 0 isn't in the domain of the expression 1 / [x(x +2)] either.

7..

8..

2. f(g(2)) = 5 / 2;g(f(2)) = 1 / 4.3. Yes;f − 1(x) = x − 2 andg − 1(x) = 1 / x. Note thatg and its inverse are the same.

3. As pictured, by the Vertical Line test, this graph represents a function.

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Algebra

Functions→Calculus← Precalculus

Algebra

This section is intended to review algebraic manipulation. It is important to understand alge-bra in order to do calculus. If you have a good knowledge of algebra, you should probably justskim this section to be sure you are familiar with the ideas.

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Rules of arithmetic and algebra

The following rules are always true.

• AdditionCommutative Law:•

.• Associative Law:

.• Additive Identity:

.• Additive Inverse:

.• Subtraction

Definition:•

.• Multiplication

Commutative law:•

.• Associative law:

.• Multiplicative Identity:

.• Multiplicative Inverse:

, whenever

• Distributive law:

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.• Division

• Definition:

, whenever

.

The above laws are true for alla, b, andc, whethera, b, andc are numbers, variables, func-tions, or other expressions. For instance,

=

=

=

=

Of course, the above is much longer than simply cancellingx + 3 out in both the numeratorand denominator. But, when you are cancelling, you are really just doing the above steps, so itis important to know what the rules are so as to know when you are allowed to cancel. Occasional-ly people do the following, for instance, which is incorrect:

.

The correct simplification is

,

where the number 2 cancels out in both the numerator and the denominator.

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Interval notation

There are a few different ways that one can express with symbols a specific interval (all thenumbers between two numbers). One way is with inequalities. If we wanted to denote the set ofall numbers between, say, 2 and 4, we could write "allx satisfying 2<x<4." This excludes theendpoints 2 and 4 because we use < instead of

. If we wanted in include the endpoints, we would write "allx satisfying

." This includes the endpoints.

Another way to write these intervals would be with interval notation. If we wished to convey"all x satisfying 2<x<4" we would write (2,4). This doesnot include the endpoints 2 and 4. If wewanted to include the endpoints we would write [2,4]. If we wanted to include 2 and not 4 wewould write [2,4); if we wanted to exclude 2 and include 4, we would write (2,4].

Thus, we have the following table:

Interval notationInequality notationEndpoint conditions

all x satisfyingNot including 2 nor 4

all x satisfyingIncluding 2 not 4

all x satisfyingIncluding 4 not 2

all x satisfyingIncluding both 2 and 4

In general, we have the following table:

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Set NotationInterval Nota-tion

Meaning

All values greater than or equal toa and less than or equal tob

All values greater thana and less thanb

All values greater than or equal toa and less thanb

All values greater thana and less than or equal tob

All values greater than or equal toa.

All values greater thana.

All values less than or equal toa.

All values less thana.

All values.

Note that

and

must always have an exclusive parenthesis rather than an inclusive bracket. This is because

is not a number, and therefore cannot be in our set.

is really just a symbol that makes things easier to write, like the intervals above.

Exponents and radicals

There are a few rules and properties involving exponents and radicals that you'd do well to

remember. As a definition we have that ifn is a positive integer thenan denotesn factors ofa.That is,

If

then we say that

. If n is a positive integer we say that

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If we have an exponent that's a fraction we say that

These definitions yield the following table of properties:

ExampleRule

Factoring and roots

Given the expressionx2 + 3x+ 2, one may ask "what are the values ofx that make this expres-sion 0?" If we factor we obtain

If x=-1 or -2, then one of the factors on the right becomes zero. Therefore, the whole mustbe zero. So, by factoring we have discovered the values ofx that render the expression zero.

These values are termed "roots." In general, given a quadratic polynomialpx2 + qx+ r that factorsas

then we have thatx = -c/a andx = -d/b are roots of the original polynomial.

A special case to be on the look out for is the difference of two squares,a2 − b2. In this case,we are always able to factor as

For example, consider 4x2 − 9. On initial inspection we would see that both 4x2 and 9 are

squares ((2x)2 = 4x2 and 32 = 9). Applying the previous rule we have

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Simplifying rational expressions

Consider the two polynomials

and

When we take the quotient of the two we obtain

The ratio of two polynomials is called arational expression. Many times we would like tosimplify such a beast. For example, say we are given

We may simplify this in the following way:

This is nice because we have obtained something we understand quite well,x − 1, fromsomething we didn't.

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Functions

Graphing linear functions→Calculus← Algebra

Functions

Classical understanding of functions

To provide the classical understanding of functions, think of afunctionas a kind of machine.You feed the machine raw materials, and the machine changes the raw materials into a finishedproduct based on a specific set of instructions. The kinds of functions we consider here, for themost part, take in a real number, change it in a formulaic way, and give out a real number (possi-bly the same as the one it took in). Think of this as aninput-output machine; you give the functionan input, and it gives you an output. For example, the squaring function takes the input 4 andgives the output value 16. The same squaring function takes the input − 1 and gives the outputvalue 1.

A function is usually written asf, g, or something similar - although it doesn't have to be. Afunction is always defined as "of a variable" which tells us what to replace in the formula for thefunction.

For example,

tells us:

• The functionf is a function ofx.• To evaluate the function at a certain number, replace thex with that number.• Replacingxwith that number in the right side of the function will produce the function's

output for that certain input.• In English, the definition of

is interpreted, "Given a number,f will return two more than the triple of that number."

Thus, if we want to know the value (or output) of the function at 3:

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We evaluate the function atx = 3.

The value of

at 3 is 11.

See? It's easy!

Note that

means the value of the dependent variable when

takes on the value of 3. So we see that the number11 is the output of the function when wegive the number3 as the input. We refer to the input as theargumentof the function (or theinde-pendent variable), and to the output as thevalue of the function at the given argument (or thedependent variable). A good way to think of it is the dependent variable

'depends' on the value of the independent variable

. This is read as "the value off at three is eleven", or simply "f of three equals eleven".

Notation

Functions are used so much that there is a special notation for them. The notation is some-what ambiguous, so familiarity with it is important in order to understand the intention of anequation or formula.

Though there are no strict rules for naming a function, it is standard practice to use the lettersf, g, andh to denote functions, and the variablex to denote an independent variable.y is used forboth dependent and independent variables.

When discussing or working with a functionf, it's important to know not only the function,but also its independent variablex. Thus, when referring to a functionf, you usually do not writef, but insteadf(x). The function is now referred to as "f of x". The name of the function is adjacent

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to the independent variable (in parentheses). This is useful for indicating the value of the functionat a particular value of the independent variable. For instance, if

,

and if we want to use the value off for x equal to 2, then we would substitute 2 forx on bothsides of the definition above and write

This notation is more informative than leaving off the independent variable and writingsimply 'f', but can be ambiguous since the parentheses can be misinterpreted as multiplication.

Modern understanding of functions

The formal definition of a function states that a function is actually arule that associates ele-ments of one set called thedomainof the function, with the elements of another set called therangeof the function. For each value we select from the domain of the function, there exists exact-ly one corresponding element in the range of the function. The definition of the function tells uswhich element in the range corresponds to the element we picked from the domain. Classically,the element picked from the domain is pictured as something that is fed into the function and thecorresponding element in the range is pictured as the output. Since we "pick" the element in thedomain whose corresponding element in the range we want to find, we have control over whatelement we pick and hence this element is also known as the "independent variable". The elementmapped in the range is beyond our control and is "mapped to" by the function. This element ishence also known as the "dependent variable", for it depends on which independent variable wepick. Since the elementary idea of functions is better understood from the classical viewpoint,we shall use it hereafter. However, it is still important to remember the correct definition offunctions at all times.

To make it simple, for the functionf(x), all of the possiblex values constitute the domain,and all of the valuesf(x) (y on the x-y plane) constitute the range.

Remarks

The following arise as a direct consequence of the definition of functions:

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1. By definition, for each "input" a function returns only one "output", corresponding tothat input. While the same output may correspond to more than one input, one inputcannot correspond to more than one output. This is expressed graphically as theverticalline test: a line drawn parallel to the axis of the dependent variable (normally vertical)will intersect the graph of a function only once. However, a line drawn parallel to theaxis of the independent variable (normally horizontal) may intersect the graph of afunction as many times as it likes. Equivalently, this has an algebraic (or formula-based) interpretation. We can always say ifa = b, thenf(a) = f(b), but if we only knowthatf(a) = f(b) then we can't be sure thata = b.

2. Each function has a set of values, the function'sdomain, which it can accept as input.Perhaps this set is all positive real numbers; perhaps it is the set {pork, mutton, beef}.This set must be implicitly/explicitly defined in the definition of the function. Youcannot feed the function an element that isn't in the domain, as the function is not de-fined for that input element.

3. Each function has a set of values, the function'srange, which it can output. This maybe the set of real numbers. It may be the set of positive integers or even the set {0,1}.This set, too, must be implicitly/explicitly defined in the definition of the function.


This is an example of an expression which fails the vertical line test.

The vertical line test

The vertical line test, mentioned in the preceding paragraph, is a systematic test to find outif an equation involvingx andy can serve as a function (withx the independent variable andythe dependent variable). Simply graph the equation and draw a vertical line through each pointof thex-axis. If any vertical line ever touches the graph at more than one point, then the equationis not a function; if the line always touches at most one point of the graph, then the equation isa function.

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http://en.wikibooks.org/wiki/File:Vertlinetest.png

(There are a lot of useful curves, like circles, that aren't functions (see picture). Some peoplecall these graphs with multiple intercepts, like our circle, "multi-valued functions"; they wouldrefer to our "functions" as "single-valued functions".)

Important functions

Constant functionIt disregards the input and always outputs the constantc, and is a polyno-

mial of thezerothdegree wheref(x) = cx0= c(1) = c. Its graph is a hori-zontal line.

Linear functionTakes an input, multiplies bymand addsc. It is a polynomial of thefirstdegree. Its graph is a line (slanted, exceptm = 0).

Identity functionTakes an input and outputs it unchanged. A polynomial of thefirst de-

gree,f(x) = x1 = x. Special case of a linear function.

Quadratic functionA polynomial of theseconddegree. Its graph is a parabola, unlessa =0. (Don't worry if you don't know what this is.)

Polynomial functionThe numbern is called thedegree.

Signum function

Determines the sign of the argumentx.

Example functions

Some more simple examples of functions have been listed below.

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Gives 1 if input is positive, -1 if input is negative. Note that the function only acceptsnegative and positive numbers, not 0. Mathematics describes this condition by saying 0is not in the domain of the function.

Takes an input and squares it.

Exactly the same function, rewritten with a different independent variable. This is perfect-ly legal and sometimes done to prevent confusion (e.g. when there are already too manyuses ofx or y in the same paragraph.)

Note that we can define a function by a totally arbitrary rule.

It is possible to replace the independent variable with any mathematical expression, not justa number. For instance, if the independent variable is itself a function of another variable, thenit could be replaced with that function. This is called composition, and is discussed later.

Manipulating functions

Functions can be manipulated in the same ways as variables; they can be added, multiplied,raised to powers, etc. For instance, let

and

.

Then

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,

,

,

.

Composition of functions

However, there is one particular way to combine functions which cannot be done with vari-ables. The value of a functionf depends upon the value of another variablex; however, thatvariable could be equal to another functiong, so its value depends on the value of a third variable.If this is the case, then the first variable is a functionh of the third variable; this function (h) iscalled thecompositionof the other two functions (f andg). Composition is denoted by

.

This can be read as either "f composed with g" or "f of g of x."

For instance, let

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and

.

Then

.

Here,h is the composition off andg and we write

. Note that composition is not commutative:

, and

so

.

Composition of functions is very common, mainly because functions themselves are com-mon. For instance, squaring and sine are both functions:

,

Thus, the expression sin2x is a composition of functions:

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=sin2x

=

.

(Note that this isnot the same as

.) Since the function sine equals 1 / 2 ifx = π / 6,

.

Since the function square equals 1 / 4 ifx = 1 / 2,

.

Transformations

Transformations are a type of function manipulation that are very common. They consist ofmultiplying, dividing, adding or subtracting constants to either the input or the output. Multiply-ing by a constant is calleddilation and adding a constant is calledtranslation. Here are a fewexamples:

Dilation

Translation

Dilation

Translation

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Examples of horizontal and vertical translations


Examples of horizontal and vertical dilationsTranslations and dilations can be either horizontal or vertical. Examples of both vertical and

horizontal translations can be seen at right. The red graphs represent functions in their 'original'state, the solid blue graphs have been translated (shifted) horizontally, and the dashed graphshave been translated vertically.

Dilations are demonstrated in a similar fashion. The function

has had its input doubled. One way to think about this is that now any change in the inputwill be doubled. If I add one tox, I add two to the input off, so it will now change twice asquickly. Thus, this is a horizontal dilation by

because the distance to they-axis has beenhalved. A vertical dilation, such as

is slightly more straightforward. In this case, you double the output of the function. Theoutput represents the distance from thex-axis, so in effect, you have made the graph of thefunction 'taller'. Here are a few basic examples wherea is any positive constant:

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http://en.wikibooks.org/wiki/File:Translations.jpg

http://en.wikibooks.org/wiki/File:Dilations.jpg

Rotation about originOriginal graph

Horizontal translation bya unitsrightHorizontal translation bya units left

Vertical dilation by a factor ofaHorizontal dilation by a factor ofa

Vertical translation bya unitsupVertical translation bya unitsdown

Reflection abouty-axisReflection aboutx-axis

Domain and Range

Domain


The domain of the function is the interval from -1 to 1Thedomain of a function is the set of all points over which it is defined. More simply, it

represents the set of x-values which the function can accept as input. For instance, if

thenf(x) is only defined for values ofx between − 1 and 1, because the square root functionis not defined (in real numbers) for negative values. Thus, the domain, in interval notation, is

. In other words,

.

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http://en.wikibooks.org/wiki/File:Domainofsemicirclefunction.png


The range of the function is the interval from 0 to 1

Range

The range of a function is the set of all values which it attains (i.e. the y-values). For in-stance, if:

,

thenf(x) can only equal values in the interval from 0 to 1. Thus, the range off is

.

One-to-one Functions

A function f(x) is one-to-one(or less commonlyinjective) if, for every value off, there isonly one value ofx that corresponds to that value off. For instance, the function

is not one-to-one, because bothx = 1 andx = - 1 result inf(x) = 0. However, the functionf(x) = x + 2 is one-to-one, because, for every possible value off(x), there is exactly one corre-

sponding value ofx. Other examples of one-to-one functions aref(x) = x3 + ax, where

. Note that if you have a one-to-one function and translate or dilate it, it remains one-to-one.(Of course you can't multiplyx or f by a zero factor).

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http://en.wikibooks.org/wiki/File:Rangeofsemicirclefunction.png

Horizontal Line Test

If you know what the graph of a function looks like, it is easy to determine whether or notthe function is one-to-one. If every horizontal line intersects the graph in at most one point, thenthe function is one-to-one. This is known as the Horizontal Line Test.

Algebraic 1-1 Test

If you dont know what the graph of the function looks like, it is also easy to determinewhether or not the function is one-to-one. The rule

applies.eg. Is

a 1-1 function?

Therefore by the algebraic 1-1 test, the function

is 1-1

Inverse functions

We callg(x) the inverse function off(x) if, for all x:

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.

A function f(x) has an inverse function if and only iff(x) is one-to-one. For example, the in-verse off(x) = x + 2 isg(x) = x - 2. The function

has no inverse.

Notation

The inverse function off is denoted asf - 1(x). Thus,f - 1(x) is defined as the function thatfollows this rule

f(f − 1(x)) = f − 1(f(x)) = x:

To determinef - 1(x) when given a functionf, substitutef - 1(x) for x and substitutex for f(x).

Then solve forf - 1(x), provided that it is also a function.

Example: Givenf(x) = 2x − 7, find f - 1(x).

Substitutef - 1(x) for x and substitutex for f(x). Then solve forf - 1(x):

To check your work, confirm thatf − 1(f(x)) = x:

f − 1(f(x)) =

f − 1(2x − 7) =

If f isn't one-to-one, then, as we said before, it doesn't have an inverse. Then this methodwill fail.

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Example: Givenf(x) = x2, find f - 1(x).

Substitutef - 1(x) for x and substitutex for f(x). Then solve forf - 1(x):

Since there are two possibilities forf - 1(x), it's not a function. Thusf(x) = x2 doesn't have aninverse. Of course, we could also have found this out from the graph by applying the HorizontalLine Test. It's useful, though, to have lots of ways to solve a problem, since in a specific casesome of them might be very difficult while others might be easy. For example, we might onlyknow an algebraic expression forf(x) but not a graph.

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Graphing linear functions

Precalculus/Exercises→Calculus← Functions

Graphing linear functions


Graph of y=2xIt is sometimes difficult to understand the behavior of a function given only its definition; a

visual representation or graph can be very helpful. Agraph is a set of points in the Cartesianplane, where each point (x,y) indicates thatf(x) = y. In other words, a graph uses the position ofa point in one direction (thevertical-axisor y-axis) to indicate the value off for a position of thepoint in the other direction (thehorizontal-axisor x-axis).

Functions may be graphed by finding the value off for variousx and plotting the points (x,f(x)) in a Cartesian plane. For the functions that you will deal with, the parts of the function be-tween the points can generally be approximated by drawing a line or curve between the points.Extending the function beyond the set of points is also possible, but becomes increasingly inaccu-rate.

Example

Plotting points like this is laborious. Fortunately, many functions' graphs fall into generalpatterns. For a simple case, consider functions of the form

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http://en.wikibooks.org/wiki/File:Graphofyequals2x.png

The graph off is a single line, passing through the point (0,2) with slope 3. Thus, after plot-ting the point, a straightedge may be used to draw the graph. This type of function is calledlinearand there are a few different ways to present a function of this type.

Slope-intercept form

When we see a function presented as

we call this presentation theslope-intercept form. This is because, not surprisingly, thisway of writing a linear function involves the slope,m, and they-intercept,b.

x+y=7 y-x=7

Point-slope form

If someone walks up to you and gives you one point and a slope, you can draw one line andonly one line that goes through that point and has that slope. Said differently, a point and a slopeuniquely determine a line. So, if given a point (x0,y0) and a slopem, we present the graph as

We call this presentation thepoint-slope form. The point-slope and slope-intercept formare essentially the same. In the point-slope form we can use any point the graph passes through.Where as, in the slope-intercept form, we use they-intercept, that is the point (0,b).

Calculating slope

If given two points, (x1,y1) and (x2,y2), we may then compute the slope of the line that passes

through these two points. Remember, the slope is determined as "rise over run." That is, the slopeis the change iny-values divided by the change inx-values. In symbols,

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So now the question is, "what'sΔy andΔx?" We have thatΔy = y2 − y1 andΔx = x2 - x1.

Thus,

Two-point form

Two points also uniquely determine a line. Given points (x1,y1) and (x2,y2), we have the

equation

This presentation is in thetwo-point form . It is essentially the same as the point-slope formexcept we substitute the expression

for m.

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Limits


Calculus← Precalculus/Exercises

Limits




• Finite Limits


• Infinite Limits


• Continuity





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http://en.wikibooks.org/wiki/File:Limit-at-infinity-graph.png






An Introduction to Limits

Finite Limits→Calculus← Limits/Contents

Limits/An Introduction toLimits

Intuitive Look

A limit looks at what happens to a function when the input approaches a certain value. Thegeneral notation for a limit is as follows:

This is read as "The limit off(x) asx approachesa". We'll take up later the question of howwe can determine whether a limit exists forf(x) at a and, if so, what it is. For now, we'll look atit from an intuitive standpoint.

Let's say that the function that we're interested in isf(x) = x2, and that we're interested in itslimit asx approaches 2. Using the above notation, we can write the limit that we're interested inas follows:

One way to try to evaluate what this limit is would be to choose values near 2, computef(x)for each, and see what happens as they get closer to 2. This is implemented as follows:

1.9991.991.951.91.81.7x

3.9960013.96013.80253.613.242.89f(x) = x2

Here we chose numbers smaller than 2, and approached 2 from below. We can also choosenumbers larger than 2, and approach 2 from above:

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2.0012.012.052.12.22.3x

4.0040014.04014.20254.414.845.29f(x) = x2

We can see from the tables that asx grows closer and closer to 2,f(x) seems to get closerand closer to 4, regardless of whetherx approaches 2 from above or from below. For this reason,

we feel reasonably confident that the limit ofx2 asx approaches 2 is 4, or, written in limit nota-tion,

Now let's look at another example. Suppose we're interested in the behavior of the function

asx approaches 2. Here's the limit in limit notation:

Just as before, we can compute function values asx approaches 2 from below and fromabove. Here's a table, approaching from below:

1.9991.991.951.91.81.7x

-1000-100-20-10-5-3.333

And here from above:

2.0012.012.052.12.22.3x

1000100201053.333

In this case, the function doesn't seem to be approaching any value asx approaches 2. In thiscase we would say that the limit doesn't exist.

Both of these examples may seem trivial, but consider the following function:

This function is the same as

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Note that these functions are really completely identical; not just "almost the same," but actu-ally, in terms of the definition of a function, completely the same; they give exactly the sameoutput for every input.

In algebra, we would simply say that we can cancel the term (x − 2), and then we have the

functionf(x) = x2. This, however, would be a bit dishonest; the function that we have now is notreally the same as the one we started with, because it is defined whenx = 2, and our originalfunction was specifically not defined whenx = 2. In algebra we were willing to ignore this diffi-culty because we had no better way of dealing with this type of function. Now, however, in calcu-lus, we can introduce a better, more correct way of looking at this type of function. What wewant is to be able to say that, although the function doesn't exist whenx = 2, it works almost asthough it does. It may not get there, but it gets really, really close. That is,f(1.99999) = 3.99996.The only question that we have is: what do we mean by "close"?

Informal definition of a limit

As the precise definition of a limit is a bit technical, it is easier to start with an informaldefinition; we'll explain the formal definition later.

We suppose that a functionf is defined forx nearc (but we do not require that it be definedwhenx = c).

Definition: (Informal definition of a limit)We callL the limit of f(x) asx approachesc if f(x) becomes close toL whenx is close (but

not equal) toc.

When this holds we write

or

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Notice that the definition of a limit is not concerned with the value off(x) whenx = c (whichmay exist or may not). All we care about are the values off(x) whenx is close toc, on either theleft or the right (i.e. less or greater).

Limit rules

Now that we have defined, informally, what a limit is, we will list some rules that are usefulfor working with and computing limits. You will be able to prove all these once we formallydefine the fundamental concept of the limit of a function.

First, theconstant rulestates that iff(x) = b (that is,f is constant for allx) then the limit asx approachesc must be equal tob. In other words

Constant Rule for Limits

If b andc are constants then

.

Second, theidentity rule states that iff(x) = x (that is,f just gives back whatever numberyou put in) then the limit off asx approachesc is equal toc. That is,

Identity Rule for Limits

If c is a constant then

.

The next few rules tell us how, given the values of some limits, to compute others.

Operational Identities for LimitsSuppose that

and

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and thatk is constant. Then

•

•

•

•

•

Notice that in the last rule we need to require thatM is not equal to zero (otherwise we wouldbe dividing by zero which is an undefined operation).

These rules are known asidentities; they are the scalar product, sum, difference, product,and quotient rules for limits. (A scalar is a constant, and, when you multiply a function by aconstant, we say that you are performingscalar multiplication .)

Using these rules we can deduce another. Namely, using the rule for products many timeswe get that

for a positive integern.

This is called thepower rule.

Examples

Example 1

Find the limit

.

We need to simplify the problem, since we have no rules about this expression by itself. Weknow from the identity rule above that

. By the power rule,

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. Lastly, by the scalar multiplication rule, we get

.

Example 2

Find the limit

.

To do this informally, we split up the expression, once again, into its components. As above,

.

Also

and

. Adding these together gives

.

Example 3

Find the limit

.

From the previous example the limit of the numerator is

. The limit of the denominator is

As the limit of the denominator is not equal to zero we can divide. This gives

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.

Example 4

Find the limit

.

We apply the same process here as we did in the previous set of examples;

.

We can evaluate each of these;

and

Thus, the answer is

.

Example 5

Find the limit

.

To evaluate this seemingly complex limit, we will need to recall some sine and cosineidentities. We will also have to use two new facts. First, iff(x) is a trigonometric function (thatis, one of sine, cosine, tangent, cotangent, secant or cosecant) and is defined ata, then

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. Second,

.

To evaluate the limit, recognize that 1 − cosx can be multiplied by 1 + cosx to obtain (1 −

cos2x) which, by our trig identities, is sin2x. So, multiply the top and bottom by 1 + cosx. (Thisis allowed because it is identical to multiplying by one.) This is a standard trick for evaluatinglimits of fractions; multiply the numerator and the denominator by a carefully chosen expressionwhich will make the expression simplify somehow. In this case, we should end up with:

.

Our next step should be to break this up into

by the product rule. As mentioned above,

.

Next,

.

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Thus, by multiplying these two results, we obtain 0.

We will now present an amazingly useful result, even though we cannot prove it yet. Wecan find the limit atc of any polynomial or rational function, as long as that rational function isdefined atc (so we are not dividing by zero). That is,c must be in the domain of the function.

Limits of Polynomials and Rational functionsIf f is a polynomial or rational function that is defined atc then

We already learned this for trigonometric functions, so we see that it is easy to find limitsof polynomial, rational or trigonometric functions wherever they are defined. In fact, this is trueeven for combinations of these functions; thus, for example,

.

The Squeeze Theorem


Graph showingf being squeezed betweeng andhThe Squeeze Theorem is very important in calculus, where it is typically used to find the

limit of a function by comparison with two other functions whose limits are known.

It is called the Squeeze Theorem because it refers to a functionf whose values are squeezedbetween the values of two other functionsg andh, both of which have the same limitL. If thevalue off is trapped between the values of the two functionsg andh, the values off must alsoapproachL.

Expressed more precisely:

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http://en.wikibooks.org/w/index.php?title=File:Squeeze_theorem_example.svg&filetimestamp=20070621004114

Theorem: (Squeeze Theorem)Suppose that

holds for allx in some open interval containinga, except possibly atx = a itself. Suppose alsothat

. Then

also.


Plot of x*sin(1/x) for -0.5 < x <0.5Example: Compute

. Note that the sine of anything is in the interval [ − 1,1]. That is,

for all x. If x is positive, we can multiply these inequalities byx and get

. If x is negative, we can similarly multiply the inequalities by the positive number -x andget

. Putting these together, we can see that, for all nonzerox,

. But it's easy to see that

. So, by the Squeeze Theorem,

.

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http://en.wikibooks.org/w/index.php?title=File:Xsin1_x.PNG&filetimestamp=20060611070806

Finding limits

Now, we will discuss how, in practice, to find limits. First, if the function can be built outof rational, trigonometric, logarithmic and exponential functions, then if a numberc is in thedomain of the function, then the limit atc is simply the value of the function atc.

If c is not in the domain of the function, then in many cases (as with rational functions) thedomain of the function includes all the points nearc, but notc itself. An example would be if wewanted to find

, where the domain includes all numbers besides 0.

In that case, in order to find

we want to find a functiong(x) similar to f(x), except with the hole atc filled in. The limitsof f andg will be the same, as can be seen from the definition of a limit. By definition, the limitdepends onf(x) only at the points wherex is close toc but not equal to it, so the limit atc doesnot depend on the value of the function atc. Therefore, if

,

also. And since the domain of our new functiong includesc, we can now (assumingg is stillbuilt out of rational, trigonometric, logarithmic and exponential functions) just evaluate it atc asbefore. Thus we have

.

In our example, this is easy; canceling thex's givesg(x) = 1, which equalsf(x) = x / x at allpoints except 0. Thus, we have

. In general, when computing limits of rational functions, it's a good idea to look for commonfactors in the numerator and denominator.

Lastly, note that the limit might not exist at all. There are a number of ways in which thiscan occur:

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"Gap"There is a gap (not just a single point) where the function is not defined. As an example,in

does not exist when

. There is no way to "approach" the middle of the graph. Note that the function also has nolimit at the endpoints of the two curves generated (atc = − 4 andc = 4). For the limit toexist, the point must be approachable fromboththe left and the right. Note also that thereis no limit at a totally isolated point on the graph.

"Jump"If the graph suddenly jumps to a different level, there is no limit. For example, letf(x) bethe greatest integer

. Then, ifc is an integer, whenx approachesc from the rightf(x) = c, while whenx ap-proachesc from the leftf(x) = c − 1. Thus

will not exist.


A graph of 1/(x2) on the interval [-2,2].Vertical asymptote

In

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http://en.wikibooks.org/w/index.php?title=File:Graph_of_x-2.svg&filetimestamp=20090221212127

the graph gets arbitrarily high as it approaches 0, so there is no limit. (In this case wesometimes say the limit is infinite; see the next section.)


A graph of sin(1/x) on the interval (0,1/π].Infinite oscillation

These next two can be tricky to visualize. In this one, we mean that a graph continuallyrises above and falls below a horizontal line. In fact, it does this infinitely often as you ap-proach a certainx-value. This often means that there is no limit, as the graph never ap-proaches a particular value. However, if the height (and depth) of each oscillation diminish-es as the graph approaches thex-value, so that the oscillations get arbitrarily smaller, thenthere might actually be a limit.

The use of oscillation naturally calls to mind the trigonometric functions. An example ofa trigonometric function that does not have a limit asx approaches 0 is

As x gets closer to 0 the function keeps oscillating between − 1 and 1. In fact, sin(1 /x)oscillates an infinite number of times on the interval between 0 and any positive value ofx. The sine function is equal to zero wheneverx = kπ, wherek is a positive integer. Betweenevery two integersk, sinx goes back and forth between 0 and − 1 or 0 and 1. Hence, sin(1/ x) = 0 for everyx = 1 / (kπ). In between consecutive pairs of these values, 1 / (kπ) and 1/ [(k + 1)π], sin(1 /x) goes back and forth from 0, to either − 1 or 1 and back to 0. We mayalso observe that there are an infinite number of such pairs, and they are all between 0 and1 / π. There are a finite number of such pairs between any positive value ofx and 1 /π, sothere must be infinitely many between any positive value ofx and 0. From our reasoning

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http://en.wikibooks.org/w/index.php?title=File:Graph_of_sin%28x-1%29.svg&filetimestamp=20090221205432

we may conclude that, asx approaches 0 from the right, the function sin(1 /x) does notapproach any specific value. Thus,

does not exist.

Using limit notation to describe asymptotes

Now consider the function

What is the limit asx approaches zero? The value ofg(0) does not exist; it is not defined.

Notice, also, that we can makeg(x) as large as we like, by choosing a smallx, as long as

. For example, to makeg(x) equal to one trillion, we choosex to be 10- 6. Thus,

does not exist.

However, wedoknow something about what happens tog(x) whenx gets close to 0 withoutreaching it. We want to say we can makeg(x) arbitrarily large (as large as we like) by takingxto be sufficiently close to zero, but not equal to zero. We express this symbolically as follows:

Note that the limit does not exist at 0; for a limit, being

is a special kind of not existing. In general, we make the following definition.

Definition: Informal definition of a limit being

We say thelimit of f(x) asx approachesc is infinity if f(x) becomes very big (as big as welike) whenx is close (but not equal) toc.

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In this case we write

or

.

Similarly, we say thelimit of f(x) as x approachesc is negative infinity if f(x) becomesvery negative whenx is close (but not equal) toc.

In this case we write

or

.

An example of the second half of the definition would be that

.

Key application of limits

To see the power of the concept of the limit, let's consider a moving car. Suppose we havea car whose position is linear with respect to time (that is, a graph plotting the position with re-spect to time will show a straight line). We want to find the velocity. This is easy to do from alge-bra; we just take the slope, and that's our velocity.

But unfortunately, things in the real world don't always travel in nice straight lines. Carsspeed up, slow down, and generally behave in ways that make it difficult to calculate their veloci-ties.

Now what we really want to do is to find the velocity at a given moment (the instantaneousvelocity). The trouble is that in order to find the velocity we need two points, while at any given

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time, we only have one point. We can, of course, always find the average speed of the car, giventwo points in time, but we want to find the speed of the car at one precise moment.

This is the basic trick of differential calculus, the first of the two main subjects of this book.We take the average speed at two moments in time, and then make those two moments in timecloser and closer together. We then see what the limit of the slope is as these two moments intime are closer and closer, and say that this limit is the slope at a single instant.

We will study this process in much greater depth later in the book. First, however, we willneed to study limits more carefully.

External links

• Online interactive exercises on limits• Tutorials for the calculus phobe

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http://wims.unice.fr/wims/wims.cgi?module=home&search_keywords=limit&search_category=X

http://www.calculus-help.com/funstuff/phobe.html

Exercises

Differentiation→Calculus← Formal Definition of theLimit

Limits/Exercises

Limits with Graphs

Given the following graph, evaluate the succeeding limits

Basic Limit Exercises

1.

2.

Solutions

One Sided Limits

Evaluate the following limits or state that the limit does not exist.

1.

2.

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http://en.wikibooks.org/wiki/Calculus/Limits/Solutions#Basic_Limit_Exercises

Two Sided Limits


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1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

Solutions

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http://en.wikibooks.org/wiki/Calculus/Limits/Solutions#Harder_Limit_Exercises

Limits to Infinity


1.

2.

3.

4.

5.

6.

7.

Limits of Piece Functions


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1. Consider the function

1.

2.

3.


1.

2.

3.

4.

5.

6.


1.

2.

3.

4.

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Contents


Calculus← Precalculus/Exercises

Limits




• Finite Limits


• Infinite Limits


• Continuity





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http://en.wikibooks.org/wiki/File:Limit-at-infinity-graph.png






Solutions

Basic Limit Exercises

(1)

Since this is a polynomial, two can simply be plugged in. This results in4(4)-2(3)+1=16-6+1=11

11

(2)

25

Harder Limit Exercises


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(3)

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(4)D.N.E.(5)-6(6)6(7)3(8)13/8(9)10(10)D.N.E.(11)+infinity

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Finite Limits

Infinite Limits →Calculus← Limits/An Introduction toLimits

Finite Limits

Informal Finite Limits

Now, we will try to more carefully restate the ideas of the last chapter. We said then that theequation

meant that, whenx gets close to 2,f(x) gets close to 4. What exactly does this mean? Howclose is "close"? The first way we can approach the problem is to say that, atx = 1.99,f(x) =3.9601, which is pretty close to 4.

Sometimes however, the function might do something completely different. For instance,

supposef(x) = x4 − 2x2 − 3.77, sof(1.99) = 3.99219201. Next, if you take a value even closer to2, f(1.999) = 4.20602, in this case you actually move further from 4. As you can see here, theproblem with some functions is that, no matter how close we get, we can never be sure what theydo.

The solution is to find out what happensarbitrarily close to the point. In particular, we wantto say that, no matter how close we want the function to get to 4, if we makex close enough to2 then it will get there. In this case, we will write

and say "The limit off(x), asx approaches 2, equals 4" or "Asx approaches 2,f(x) approach-es 4." In general:

Definition: (New definition of a limit)

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We callL thelimit of f(x) asx approachesc if f(x) becomesarbitrarily close to L whenev-er x is sufficiently close(and not equal) toc.


or

One-Sided Limits

Sometimes, it is necessary to consider what happens when we approach anx value from oneparticular direction. To account for this, we have one-sided limits. In a left-handed limit,x ap-proachesa from the left-hand side. Likewise, in a right-handed limit,x approachesa from theright-hand side.

For example, if we consider

, there is a problem because there is no way forx to approach 2 from the left hand side (thefunction is undefined here). But, ifx approaches 2 only from the right-hand side, we want to saythat

approaches 0.

Definition: (Informal definition of a one-sided limit)We callL the limit of f(x) as x approachesc from the right if f(x) becomesarbitrarily

closeto L wheneverx is sufficiently closeto andgreater than c.


Similarly, we callL the limit of f(x) asx approachesc from the left if f(x) becomesarbi-trarily close to L wheneverx is sufficiently closeto andless thanc.


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In our example, the left-handed limit

does not exist.

The right-handed limit, however,

.

It is a fact that

exists if and only if

and

exist and are equal to each other. In this case,

will be equal to the same number.

In our example, one limit does not even exist. Thus

does not exist either.

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Infinite Limits

Continuity→Calculus← Finite Limits

Infinite Limits

Informal infinite limits

Another kind of limit involves looking at what happens tof(x) asx gets very big. For exam-ple, consider the functionf(x) = 1 / x. As x gets very big, 1 /x gets very small. In fact, 1 /x getscloser and closer to zero the biggerx gets. Without limits it is very difficult to talk about this fact,becausex can keep getting bigger and bigger and 1 /x never actually gets to zero; but the lan-guage of limits exists precisely to let us talk about the behavior of a function as it approachessomething - without caring about the fact that it will never get there. In this case, however, wehave the same problem as before: how big doesx have to be to be sure thatf(x) is really goingtowards 0?

In this case, we want to say that, however close we wantf(x) to get to 0, forx big enoughf(x) is guaranteed to get that close. So we have yet another definition.

Definition: (Definition of a limit at infinity)We callL the limit of f(x) asx approaches infinity if f(x) becomesarbitrarily close to L

wheneverx is sufficiently large.


or

Similarly, we callL thelimit of f(x) asx approaches negative infinityif f(x) becomesarbi-trarily close to L wheneverx is sufficiently negative.

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or

So, in this case, we write:

and say "The limit, as x approaches infinity, equals 0," or "as x approaches infinity, thefunction approaches 0".

We can also write:

because makingx very negative also forces 1 /x to be close to 0.

Notice, however, thatinfinity is not a number; it's just shorthand for saying "no matter howbig." Thus, this is not the same as the regular limits we learned about in the last two chapters.

Limits at infinity of rational functions

One special case that comes up frequently is when we want to find the limit at

(or

) of a rational function. A rational function is just one made by dividing two polynomials by

each other. For example,f(x) = (x3 + x − 6) / (x2 − 4x + 3) is a rational function. Also, any polyno-mial is a rational function, since 1 is just a (very simple) polynomial, so we can write the function

f(x) = x2 − 3 asf(x) = (x2 − 3) / 1, the quotient of two polynomials.

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There is a simple rule for determining a limit of a rational function as the variable approachesinfinity. Look for the term with the highest exponent in the numerator. Look for the same in thedenominator. This rule is based on that information.

• If the exponent of the highest term in the numerator matches the exponent of the highestterm in the denominator, the limit (at both

and

) is the ratio of the coefficients of the highest terms.

• If thenumeratorhas the highest term, then the fraction is called "top-heavy" and neitherlimit (at

or at

) exists.

• If the denominatorhas the highest term, then the fraction is called "bottom-heavy" andthe limit (at both

and

) is zero.

Note that, if the numerator or denominator is a constant (including 1, as above), then this is

the same asx0. Also, a straight power ofx, like x3, has coefficient 1, since it is the same as 1x3.

Example

Find

.

The functionf(x) = (x − 5) / (x − 3) is the quotient of two polynomials,x − 5 andx − 3. Byour rule we look for the term with highest exponent in the numerator; it'sx. The term with highestexponent in the denominator is alsox. So, the limit is the ratio of their coefficients. Sincex = 1x,both coefficients are 1, so

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.

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Infinity is not a number

Most people seem to struggle with this fact when first introduced to calculus, and in particu-lar limits.

But

is different.

is not a number.

Mathematics is based on formal rules that govern the subject. When a list of formal rulesapplies to a type of object (e.g., "a number") those rules mustalwaysapply — no exceptions!

What makes

different is this: "there is no number greater than infinity". You can write down the formulain a lot of different ways, but here's one way:

. If you add one to infinity, you still have infinity; you don't have a bigger number. If youbelieve that, then infinity is not a number.

Since

does not follow the rules laid down for numbers, it cannot be a number. Every time you usethe symbol

in a formula where you would normally use a number, you have to interpret the formuladifferently. Let's look at how

does not follow the rules that every actual number does:

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Addition breaks

Every number has a negative, and addition is associative. For

we could write

and note that

. This is a good thing, since it means we can prove if you take one away from infinity, youstill have infinity:

But it also means we can prove 1 = 0, which is not so good.

.

Reinterpret formulas that use

We started off with a formula that does "mean" something, even though it used

and

is not a number.

What does this mean, compared to what it means when we have a regular number insteadof an infinity symbol:

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This formula says that I can make sure the values of

don't differ very much from

, so long I can control how muchx varies away from 2. I don't have to make

exactly equal to

, but I also can't controlx too tightly. I have to give you a range to varyx within. It's justgoing to be very, very small (probably) if you want to make

very very close to

. And by the way, it doesn't matter at all what happens whenx = 2.

If we could use the same paragraph as a template for my original formula, we'll see someproblems. Let's substitute 0 for 2, and

for

.


don't differ very much from

, so long I can control how muchx varies away from 0. I don't have to make

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exactly equal to

, but I also can't controlx too tightly. I have to give you a range to varyx within. It's just goingto be very, very small (probably) if you want to see that

gets very, veryclose to

. And by the way, it doesn't matter at all what happens whenx = 0.

It's close to making sense, but it isn't quite there. It doesn't make sense to say that some realnumber is really "close" to

. For example, whenx = .001 and

does it really makes sense to say 1000 is closer to

than 1 is? Solve the following equations forδ:

No real number is very close to

; that's what makes

so special! So we have to rephrase the paragraph:


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get as big as any number you pick, so long I can control how muchx varies away from 0. Idon't have to make

bigger than every number, but I also can't controlx too tightly. I have to give you a range tovaryx within. It's just going to be very, very small (probably) if you want to see that

gets very, verylarge. And by the way, it doesn't matter at all what happens whenx = 0.

You can see that the essential nature of the formula hasn't changed, but the exact details re-quire some human interpretation. While rigorous definitions and clear distinctions are essentialto the study of mathematics, sometimes a bit of casual rewording is okay. You just have to makesure you understand what a formula really means so you can draw conclusions correctly.

Exercises

Write out an explanatory paragraph for the following limits that include

. Remember that you will have to change any comparison of magnitude between a realnumber and

to a different phrase. In the second case, you will have to work out for yourself what theformula means.

1.

2.

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Continuity

Formal Definition of the Limit→

Calculus← Infinite Limits

Continuity

Defining Continuity

We are now ready to define the concept of a function beingcontinuous. The idea is that wewant to say that a function is continuous if you can draw its graph without taking your pencil offthe page. But sometimes this will be true for some parts of a graph but not for others. Therefore,we want to start by defining what it means for a function to be continuous atone point. The defini-tion is simple, now that we have the concept of limits:

Definition: (continuity at a point)If f(x) is defined on an open interval containingc, thenf(x) is said to becontinuous atc if

and only if

.Note that forf to be continuous atc, the definition in effect requires three conditions:

1. thatf is defined atc, sof(c) exists,2. the limit asx approachesc exists, and3. the limit andf(c) are equal.

If any of these do not hold thenf is not continuous atc.

The idea of the definition is that the point of the graph corresponding toc will be close tothe points of the graph corresponding to nearbyx-values. Now we can define what it means fora function to be continuous in general, not just at one point.

Definition: (continuity)A function is said to becontinuous on (a,b) if it is continuous at every point of the interval (a,b).

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We often use the phrase "the function is continuous" to mean that the function is continuousat every real number. This would be the same as saying the function was continuous on (−∞, ∞),but it is a bit more convenient to simply say "continuous".

Note that, by what we already know, the limit of a rational, exponential, trigonometric orlogarithmic function at a point is just its value at that point, so long as it's defined there. So, allsuch functions are continuous wherever they're defined. (Of course, they can't be continuouswhere they'renotdefined!)

Discontinuities

A discontinuity is a point where a function is not continuous. There are lots of possibleways this could happen, of course. Here we'll just discuss two simple ways.

Removable Discontinuities

The function

is not continuous atx = 3. It is discontinuous at that point because the fraction then becomes

, which is undefined. Therefore the function fails the first of our three conditions for continu-ity at the point 3; 3 is just not in its domain.

However, we say that this discontinuity isremovable. This is because, if we modify thefunction at that point, we can eliminate the discontinuity and make the function continuous. Tosee how to make the functionf(x) continuous, we have to simplifyf(x), getting

. We can define a new functiong(x) whereg(x) = x + 3. Note that the functiong(x) is notthe same as the original functionf(x), becauseg(x) is defined atx = 3, while f(x) is not. Thus,g(x) is continuous atx = 3, since

. However, whenever

, f(x) = g(x); all we did tof to getg was to make it defined atx = 3.

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In fact, this kind of simplification is often possible with a discontinuity in a rational function.We can divide the numerator and the denominator by a common factor (in our examplex − 3)to get a function which is the same except where that common factor was 0 (in our example atx= 3). This new function will be identical to the old except for being defined at new points wherepreviously we had division by 0.

Unfortunately this is not possible in every case. For example, the function

has a common factor ofx − 3 in both the numerator and denominator, but when you simplifyyou are left with

. Which is still not defined atx = 3. In this case the domain off(x) andg(x) are the same, andthey are equal everywhere they are defined, so they are in fact the same function. The reason thatg(x) differed fromf(x) in the first example was because we could take it to have a larger domainand not simply that the formulas definingf(x) andg(x) were different.

Jump Discontinuities

Unfortunately, not all discontinuities can be removed from a function. Consider this function:

Since

does not exist, there is no way to redefinek at one point so that it will be continuous at 0.These sorts of discontinuities are callednonremovablediscontinuities.

Note, however, that both one-sided limits exist;

and

. The problem is that they are not equal, so the graph "jumps" from one side of 0 to the other.In such a case, we say the function has ajumpdiscontinuity. (Note that a jump discontinuity isa kind of nonremovable discontinuity.)

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One-Sided Continuity

Just as a function can have a one-sided limit, a function can be continuous from a particularside. For a function to be continuous at a point from a given side, we need the following threeconditions:

1. the function is defined at the point,2. the function has a limit from that side at that point and3. the one-sided limit equals the value of the function at the point.

A function will be continuous at a point if and only if it is continuous from both sides at thatpoint. Now we can define what it means for a function to be continuous on a closed interval.

Definition: (continuity on a closed interval)A function is said to becontinuous on [a,b] if and only if

1. it is continuous on (a,b),2. it is continuous from the right ata and3. it is continuous from the left atb.

Notice that, if a function is continuous, then it is continuous on every closed interval con-tained in its domain.

Intermediate Value Theorem

The definition of continuity we've given might not seem to have much to do with the intu-itive notion we started with of being able to draw the graph without lifting one's pencil. Fortunate-ly, there is a connection, given by the so-called intermediate value theorem, which says, informal-ly, that if a function is continuous then its graph can be drawn without ever picking up one'spencil. More precisely:

Intermediate Value TheoremIf a functionf is continuous on a closed interval [a,b], then for every valuek betweenf(a) andf(b) there is a valuec betweena andb such thatf(c) = k.

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Application: bisection method


A few steps of the bisection method applied over the starting range [a1;b1]. The bigger red dotis the root of the function.

The bisection method is the simplest and most reliable algorithm to find zeros of a continu-ous function.

Suppose we want to solve the equationf(x) = 0. Given two pointsa andb such thatf(a) andf(b) have opposite signs, the intermediate value theorem tells us thatf must have at least one rootbetweena andb as long asf is continuous on the interval [a,b]. If we know f is continuous ingeneral (say, because it's made out of rational, trigonometric, exponential and logarithmic func-tions), then this will work so long asf is defined at all points betweena andb. So, let's dividethe interval [a,b] in two by computingc = (a + b) / 2. There are now three possibilities:

1. f(c) = 0,2. f(a) andf(c) have opposite signs, or3. f(c) andf(b) have opposite signs.

In the first case, we're done. In the second and third cases, we can repeat the process on thesub-interval where the sign change occurs. In this way we home in to a small sub-interval contain-ing the zero. The midpoint of that small sub-interval is usually taken as a good approximation tothe zero.

Note that, unlike the methods you may have learned in algebra, this works foranycontinuousfunction that you (or your calculator) know how to compute.

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http://en.wikibooks.org/wiki/File:Bisection_method.png

Formal Definition of the Limit

Limits/Exercises→Calculus← Continuity

Formal Definition of theLimit

In preliminary calculus, the concept of a limit is probably the most difficult one to grasp (ifnothing else, it took some of the most brilliant mathematicians 150 years to arrive at it); it is alsothe most important and most useful.

The intuitive definition of a limit is inadequate to prove anything rigorously about it. Theproblem lies in the vague term "arbitrarily close". We discussed earlier that the meaning of thisterm is that the closerx gets to the specified value, the closer the function must get to the limit,so that however close we want the function to the limit, we can accomplish this by makingxsufficiently close to our value. We can express this requirement technically as follows:

Definition: (Formal definition of a limit)Let f(x) be a function defined on an open intervalD that containsc, except possibly atx =

c. Let L be a number. Then we say that

if, for every

, there exists aδ > 0 such that for all

with

we have

.

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To further explain, earlier we said that "however close we want the function to the limit, wecan find a correspondingx close to our value." Using our new notation of epsilon (

) and delta (δ), we mean that if we want to makef(x) within

of L, the limit, then we know that makingx within δ of c puts it there.

Again, since this is tricky, let's resume our example from before:f(x) = x2, atx = 2. To start,let's say we wantf(x) to be within .01 of the limit. We know by now that the limit should be 4,so we say: for

, there is someδ so that as long as

, then

To show this, we can pickanyδ that is bigger than 0, so long as it works. For example, youmight pick .00000000000001, because you are absolutely sure that ifx is within.00000000000001 of 2, thenf(x) will be within .01 of 4. Thisδ works for

. But we can't just pick a specific value for

, like .01, because we said in our definition "forevery

." This means that we need to be able to show an infinite number ofδs, one for each

. We can't list an infinite number ofδs!

Of course, we know of a very good way to do this; we simply create a function, so that forevery

, it can give us aδ. In this case, it's a rather simple function; all we need is

.

So, in general, how do you show thatf(x) tends toL asx tends toc? Well imagine somebodygave you a small number

(e.g., say

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). Then you have to find aδ > 0 and show that whenever

we have |f(x) − L | < 0.03. Now if that person gave you a smaller

(say

) then you would have to find anotherδ, but this time with 0.03 replaced by 0.002. If you can dothis foranychoice of

then you have shown thatf(x) tends toL asx tends toc. Of course, the way you would do this ingeneral would be to create a function giving you aδ for every

, just as in the example above.

Formal Definition of the Limit at Infinity

Definition: (Limit of a function at infinity)We callL the limit of f(x) asx approaches

if for every number

there exists aδ such that wheneverx > δ we have


or

as

Similarly, we callL the limit of f(x) asx approaches

if for every number

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, there exists a numberδ such that wheneverx < δ we have


or

as

Notice the difference in these two definitions. For the limit off(x) asx approaches

we are interested in thosex such thatx > δ. For the limit off(x) asx approaches

we are interested in thosex such thatx < δ.

Examples

Here are some examples of the formal definition.

Example 1

We know from earlier in the chapter that

.

What isδ when

for this limit?

We start with the desired conclusion and substitute the given values forf(x) and

:

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.

Then we solve the inequality forx:

7.96 <x < 8.04

This is the same as saying

- 0.04 <x - 8 < 0.04.

(We want the thing in the middle of the inequality to bex − 8 because that's where we'retaking the limit.) We normally choose the smaller of

and 0.04 forδ, soδ = 0.04, but any smaller number will also work.

Example 2

What is the limit off(x) = x + 7 asx approaches 4?

There are two steps to answering such a question; first we must determine the answer —this is where intuition and guessing is useful, as well as the informal definition of a limit — andthen we must prove that the answer is right.

In this case, 11 is the limit because we knowf(x) = x + 7 is a continuous function whosedomain is all real numbers. Thus, we can find the limit by just substituting 4 in forx, so the an-swer is 4 + 7 = 11.

We're not done, though, because we never proved any of the limit laws rigorously; we juststated them. In fact, we couldn't have proved them, because we didn't have the formal definitionof the limit yet, Therefore, in order to be sure that 11 is the right answer, we need to prove thatno matter what value of

is given to us, we can find a value ofδ such that

whenever

For this particular problem, letting

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works (seechoosing deltafor help in determining the value ofδ to use in other problems). Now,we have to prove

given that

.

Since

, we know

which is what we wished to prove.

Example 3

What is the limit off(x) = x2 asx approaches 4?

As before, we use what we learned earlier in this chapter to guess that the limit is 42 = 16.Also as before, we pull out of thin air that

.

Note that, since

is always positive, so isδ, as required. Now, we have to prove

given that

.

We know that

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(because of the triangle inequality), so

Example 4

Show that the limit of sin(1 /x) asx approaches 0 does not exist.

We will proceed by contradiction. Suppose the limit exists; call itL. For simplicity, we'llassume that

; the case forL = 1 is similar. Choose

. Then if the limit wereL there would be someδ > 0 such that

for everyx with

. But, for everyδ > 0, there exists some (possibly very large)n such that

, but | sin(1 /x0) − L | = | 1 −L | , a contradiction.

Example 5

What is the limit ofxsin(1 /x) asx approaches 0?

By the Squeeze Theorem, we know the answer should be 0. To prove this, we let

. Then for allx, if 0 < | x | < δ, then

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as required.

Example 6

Suppose that

and

. What is

?

Of course, we know the answer should beL + M, but now we can prove this rigorously.Given some

, we know there's aδ1 such that, for anyx with

,

(since the definition of limit says "for any

", so it must be true for

as well). Similarly, there's aδ2 such that, for anyx with

,

. We can setδ to be the lesser ofδ1 andδ2. Then, for anyx with

,

, as required.

If you like, you can prove the other limit rules too using the new definition. Mathematicianshave already done this, which is how we know the rules work. Therefore, when computing alimit from now on, we can go back to just using the rules and still be confident that our limit iscorrect according to the rigorous definition.

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Formal Definition of a Limit Being Infinity

Definition: (Formal definition of a limit being infinity)Let f(x) be a function defined on an open intervalD that containsc, except possibly atx =

c. Then we say that

if, for every


with

we have

.


or

as

.

Similarly, we say that

if, for every


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with

we have

.


or

as

.

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Differentiation

Differentiation/DifferentiationDefined→

Calculus← Limits/Exercises

Differentiation






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http://en.wikibooks.org/wiki/File:Derivative1.png



Important Theorems



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Differentiation Defined

Product and Quotient Rules→Calculus← Limits/Exercises

Differentiation/Differentia-tion Defined

What is differentiation?

Differentiation is an operation that allows us to find a function that outputs therate ofchangeof one variable with respect to another variable when given that second variable.

Informally, we may suppose that we're tracking the position of a car on a two-lane road withno passing lanes. Assuming the car never pulls off the road, we can abstractly study the car'sposition by assigning it a variable,x. Since the car's position changes as the time changes, wesay thatx is dependent on time, orx = x(t). This tells where the car is at each specific time. Differ-entiation gives us a functiondx / dt which represents the car's speed, that is the rate of change ofits position with respect to time.

Equivalently, differentiation gives us the slope at any point of the graph of a non-linearfunction. For a linear function, of formf(x) = ax+ b, a is the slope. For non-linear functions, such

asf(x) = 3x2, the slope can depend onx; differentiation gives us a function which represents thisslope.

The Definition of Slope

Historically, the primary motivation for the study ofdifferentiation was the tangent lineproblem: for a given curve, find the slope of the straight line that is tangent to the curve at agiven point. The word tangent comes from the Latin wordtangens, which means touching. Thus,to solve the tangent line problem, we need to find the slope of a line that is "touching" a given

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curve at a given point, or, in modern language, that has the same slope. But what exactly do wemean by "slope" for a curve?

The solution is obvious in some cases: for example, a liney = mx+ c is its own tangent; the

slope at any point ism. For the parabolay = x2, the slope at the point (0,0) is 0; the tangent lineis horizontal.

But how can you find the slope of, say,y = sinx + x2 atx = 1.5? This is in general a nontrivialquestion, but first we will deal carefully with the slope of lines.

Of a Line


Three lines with different slopesTheslopeof a line, also called thegradient of the line, is a measure of its inclination. A line

that is horizontal has slope 0, a line from the bottom left to the top right has a positive slope anda line from the top left to the bottom right has a negative slope.

The slope can be defined in two (equivalent) ways. The first way is to express it as howmuch the line climbs for a given motion horizontally. We denote a change in a quantity usingthe symbolΔ (pronounced "delta"). Thus, a change inx is written asΔx. We can therefore writethis definition of slope as:

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http://en.wikibooks.org/w/index.php?title=File:Lines_with_Gradients.svg&filetimestamp=20071024001446

An example may make this definition clearer. If we have two points on a line,

and

, the change inx from P to Q is given by:

Likewise, the change iny from P to Q is given by:

This leads to the very important result below.

The slope of the line between the points (x1,y1) and (x2,y2) is

.

Alternatively, we can define slope trigonometrically, using the tangent function:

whereα is the angle from the line to the rightward-pointing horizontal (measured clockwise).If you recall that the tangent of an angle in a right triangle is defined as the length of the sideopposite the angle over the length of the leg adjacent to the angle, you should be able to spot theequivalence here.

Of a Graph of a Function

The graphs of most functions we are interested in are not straight lines (although they canbe) but rather curves. We cannot define the slope of a curve in the same way as we can for a line.

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In order for us to understand how to find the slope of a curve at a point, we will first have tocover the idea oftangency. Intuitively, atangent is a line whichjust touches a curve at a point,such that the angle between them at that point is zero. Consider the following four curves andlines:

(ii)(i)

See Picture License InformationHereSee Picture License InformationHere

(iv)(iii)

See Picture License InformationHereSee Picture License InformationHere

1. The lineL crosses, but is not tangent toC atP.2. The lineL crosses, and is tangent toC atP.3. The lineL crossesC at two points, but is tangent toC only atP.4. There are many lines that crossC atP, but none are tangent. In fact, this curve has no

tangent atP.

A secantis a line drawn through two points on a curve. We can construct a definition of atangent as the limit of a secant of the curve taken as the separation between the points tends tozero. Consider the diagram below.

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http://en.wikibooks.org/w/index.php?title=File:Tangency_Example_2.svg&filetimestamp=20071030170706





As the distanceh tends to zero, the secant line becomes the tangent at the pointx0. The two

points we draw our line through are:

and

As a secant line is simply a line and we know two points on it, we can find its slope,mh,

using the formula from before:

(We will refer to the slope asmh because it may, and generally will, depend onh.) Substitut-

ing in the points on the line,

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http://en.wikibooks.org/w/index.php?title=File:Tangent_as_Secant_Limit.svg&filetimestamp=20071030165651

This simplifies to

This expression is called thedifference quotient. Note thath can be positive or negative —it is perfectly valid to take a secant through any two points on the curve — but cannot be 0.

The definition of the tangent line we gave was not rigorous, since we've only defined limitsof numbers— or, more precisely, of functions that output numbers — not oflines. But wecandefine theslopeof the tangent line at a point rigorously, by taking the limit of the slopes of thesecant lines from the last paragraph. Having done so, we canthendefine the tangent line as well.Note that we cannot simply seth to zero as this would imply division of zero by zero whichwould yield an undefined result. Instead we must find thelimit of the above expression ashtendsto zero:

Definition: (Slope of the graph of a function)Theslopeof the graph off(x) at the point (x0,f(x0)) is

If this limit does not exist, then we say the slope isundefined.

If the slope is defined, saym, then thetangent line to the graph off(x) at the point (x0,f(x0))

is the line with equation

This last equation is just the point-slope form for the line through (x0,f(x0)) with slopem.

Exercises

1. Find the slope of the tangent to the curvey = x2 at (1,1).

Answer: The definition of the slop off atx0 is

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Substituting inf(x) = x2 andx0 = 1 gives:

The Rate of Change of a Function at a Point

Consider the formula for average velocity in thex-direction,

, whereΔx is the change inx over the time intervalΔt. This formula gives the average veloci-ty over a period of time, but suppose we want to define the instantaneous velocity. To this endwe look at thechange in position as the change in time approaches 0. Mathematically this iswritten as:

, which we abbreviate by the symbol

. (The idea of this notation is that the letterd denotes change.) Compare the symbold withΔ. The (entirely non-rigorous) idea is that both indicate a difference between two numbers, butΔ denotes a finite difference whiled denotes an infinitesimal difference. Please note that thesymbolsdxanddt have no rigorous meaning on their own, since

, and we can't divide by 0.

(Note that the letters is often used to denote distance, which would yield

. The letterd is often avoided in denoting distance due to the potential confusion resultingfrom the expression

.)

The Definition of the Derivative

You may have noticed that the two operations we've discussed — computing the slope ofthe tangent to the graph of a function and computing the instantaneous rate of change of the

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function — involved exactly the same limit. That is, the slope of the tangent to the graph ofy =f(x) is

. Of course,

can, and generally will, depend onx, so we should really think of it as afunctionof x. We callthis process (of computing

) differentiation . Differentiation results in another function whose value for any valuex is theslope of the original function atx. This function is known as thederivative of the original func-tion.

Since lots of different sorts of people use derivatives, there are lots of different mathematicalnotations for them. Here are some:

•(read "f prime of x") for the derivative off(x),

• Dx[f(x)],• Df(x),•

for the derivative ofy as a function ofx or•

, which is more useful in some cases.

Most of the time the brackets are not needed, but are useful for clarity if we are dealing withsomething likeD(fg), where we want to differentiate the product of two functions,f andg.

The first notation has the advantage that it makes clear that the derivative is a function. Thatis, if we want to talk about the derivative off(x) atx = 2, we can just writef'(2).

In any event, here is the formal definition:

Definition: (derivative)Let f(x) be a function. Then

wherever this limit exists. In this case we say thatf is differentiable atx and itsderivative atxis f'(x).

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Examples

Example 1

The derivative off(x) = x / 2 is

,

no matter whatx is. This is consistent with the definition of the derivative as the slope of afunction.

Example 2

What is the slope of the graph ofy = 3x2 at (4,48)? We can do it "the hard (and imprecise)way", without using differentiation, as follows, using a calculator and using small differencesbelow and above the given point:

Whenx = 3.999,y = 47.976003.

Whenx = 4.001,y = 48.024003.

Then the difference between the two values ofx is Δx = 0.002.

Then the difference between the two values ofy is Δy = 0.048.

Thus, the slope

at the point of the graph at whichx = 4.

But, to solve the problem precisely, we compute

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=

=

=

=

=3(8)

=24.

We were lucky this time; the approximation we got above turned out to be exactly right. Butthis won't always be so, and, anyway, this way we didn't need a calculator.

In general, the derivative off(x) = 3x2 is

=

=

=

=

=

=3(2x)

=6x.

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Example 3

If

(the absolute value function) then

Here,f(x) is not smooth (though it is continuous) atx = 0 and so the limits

and

(the limits as 0 is approached from the right and left respectively) are not equal. From thedefinition,

, which does not exist. Thus,f'(0)is undefined, and sof'(x) has a discontinuity at 0. This sortof point of non-differentiability is called a cusp. Functions may also not be differentiable becausethey go to infinity at a point, or oscillate infinitely frequently.

Understanding the Derivative Notation


SeePic-

Notation for differentiation


The derivative notation is special and unique in mathematics. The most common notationfor derivatives you'll run into when first starting out with differentiating is the Leibniz notation,expressed as

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http://en.wikipedia.org/wiki/Special:Search/Notation_for_differentiation?go=Go



. You may think of this as "rate of change iny with respect tox". You may also think of it as"infinitesimal value ofy divided by infinitesimal value ofx". Either way is a good way of think-ing, although you should remember that the precise definition is the one we gave above. Often,in an equation, you will see just

, which literally means "derivative with respect to x". This means we should take the derivativeof whatever is written to the right; that is,

means

wherey = x + 2.

As you advance through your studies, you will see that we sometimes pretend thatdy anddxare separate entities that can be multiplied and divided, by writing things like

. Eventually you will see derivatives such as

, which just means that the input variable of our function is calledy and our output variableis calledx; sometimes, we will write

, to mean the derivative with respect toy of whatever is written on the right. In general, thevariables could be anything, say

.

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All of the following are equivalent for expressing the derivative ofy = x2

•

•

•••

Exercises

1. Using the definition of the derivative find the derivative of the functionf(x) = 2x + 3.

2. Using the definition of the derivative find the derivative of the functionf(x) = x3. Now try

f(x) = x4. Can you see a pattern? In the next section we will find the derivative off(x) = xn for alln.

3. The text states that the derivative of

is not defined atx = 0. Use the definition of the derivative to show this.

4. Graph the derivative toy = 4x2 on a piece of graph paper without solving fordy / dx. Then,solve fordy / dxand graph that; compare the two graphs.

5. Use the definition of the derivative to show that the derivative of sinx is cosx. Hint: Usea suitable sum to product formula and the fact that

.

Differentiation rules

The process of differentiation is tedious for complicated functions. Therefore, rules for differ-entiating general functions have been developed, and can be proved with a little effort. Oncesufficient rules have been proved, it will be fairly easy to differentiate a wide variety of functions.Some of the simplest rules involve the derivative of linear functions.

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Derivative of a Constant Function

For any fixed real numberc,

Intuition

The graph of the functionf(x) = c is a horizontal line, which has a constant slope of zero.Therefore, it should be expected that the derivative of this function is zero, regardless of thevalues ofx andc.

Proof

The definition of a derivative is

Let f(x) = c for all x. (That is,f is a constant function.) Thenf(x + Δx) = c. Therefore

Examples

1.

2.

Note that, in the second example,z is just a constant.

Derivative of a Linear Function

For any fixed real numbersm andc,

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The special case

shows the advantage of the

notation -- rules are intuitive by basic algebra, though this does not constitute a proof, andcan lead to misconceptions to what exactlydxanddyactually are.

Intuition

The graph ofy = mx+ c is a line with constant slopem.

Proof

If f(x) = mx+ c, thenf(x + Δx) = m(x + Δx) + c. So,

==

==

=m.

Constant multiple and addition rules

Since we already know the rules for some very basic functions, we would like to be able totake the derivative of more complex functions by breaking them up into simpler functions. Twotools that let us do this are the constant multiple rule and the addition rule.

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The Constant Rule

For any fixed real numberc,

The reason, of course, is that one can factorc out of the numerator, and then of the entirelimit, in the definition.

Example

We already know that

.

Suppose we want to find the derivative of 3x2

=

=

=

Another simple rule for breaking up functions is the addition rule.

The Addition and Subtraction Rules

Proof

From the definition:

By definition then, this last term is

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Example

What is:

=

=

=

=

The fact that both of these rules work is extremely significant mathematically because itmeans that differentiation islinear. You can take an equation, break it up into terms, figure outthe derivative individually and build the answer back up, and nothing odd will happen.

We now need only one more piece of information before we can take the derivatives of anypolynomial.

The Power Rule

For example, in the case ofx2 the derivative is 2x1 = 2x as was established earlier. A special

case of this rule is thatdx / dx= dx1 / dx= 1x0 = 1.

Since polynomials are sums of monomials, using this rule and the addition rule lets youdifferentiate any polynomial. A relatively simple proof for this can be derived from the binomialexpansion theorem.

This rule also applies to fractional and negative powers. Therefore

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=

=

=

Derivatives of polynomials

With these rules in hand, you can now find the derivative of any polynomial you comeacross. Rather than write the general formula, let's go step by step through the process.

The first thing we can do is to use the addition rule to split the equation up into terms:

We can immediately use the linear and constant rules to get rid of some terms:

Now you may use the constant multiplier rule to move the constants outside the derivatives:

Then use the power rule to work with the individual monomials:

And then do some algebra to get the final answer:

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These are not the only differentiation rules. There are other, more advanced,differentiationrules, which will be described in a later chapter.

Exercises

• Find the derivatives of the following equations:

f(x) = 42

f(x) = 6x + 10

f(x) = 2x2 + 12x + 3

• Use the definition of a derivative to prove the derivative of a constant function, of alinear function, and the constant rule and addition or subtraction rules.

• Answers:

f'(x) = 0

f'(x) = 6

f'(x) = 4x + 12

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Exercises

Extrema and Points of Inflec-tion →

Calculus← Implicit differentiation

Differentiation/Exercises

Find The Derivative By Definition

Find the derivative of the following functions using the limit definition of the derivative.

1.2.3.

4.5.6.

7.

8.

9.

Solutions

Prove Differentiation Rules

Use the definition of the derivative to prove the following general rules:

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1. For any fixed real numberc,

2. For any fixed real numbersm andc,

3. For any fixed real numberc,

4.

Solutions

Find The Derivative By Rules

Find the derivative of the following functions:

Power Rule

1.2.3.4.5.

6.

7.

8.

9.

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http://en.wikibooks.org/wiki/Calculus/Differentiation/Solutions#Prove_Differentiation_Rules

Product Rule

1.2.3.4.

Quotient Rule

1.

2.

3.

4.

5.

6.

7.

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Chain Rule

1.2.3.4.

5.6.

7.8.

9.10.11.

Exponentials

1.2.

3.

4.

Logarithms

1.2.3.4.5.

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Trig Functions

1.

2.

Solutions

More Differentiation

Given the above rules, practice differentiation on the following.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Answers

Click arrow to show answers

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http://en.wikibooks.org/wiki/Calculus/Differentiation/Solutions#Find_The_Derivative_By_Rules

1. 30x2(x3 + 5)9

2. 3x2 + 33. (x − 3)(x + 2) + (x + 4)(x + 2) + (x − 3)(x + 4)4.

5. 9x2

6.7.8.9.10.

Implicit Differentiation

Use implicit differentiation to find y'

1.2.3.

Solutions

Logarithmic Differentiation

Use logarithmic differentiation to find

1.2.

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http://en.wikibooks.org/wiki/Calculus/Differentiation/Solutions#Implicit_Differentiation

Equation of Tangent Line

For each function,f, (a) determine for what values ofx the tangent line tof is horizontal and(b) find an equation of the tangent line tof at the given point.

1.

2.3.

4.

5.6.

7. Find an equation of the tangent line to the graph defined by

at the point (1,-1).8. Find an equation of the tangent line to the graph defined by

at the point (1,0).

Higher Order Derivatives

1. What is the second derivative of 3x4 + 3x2 + 2x?

Solutions

Relative Extrema

Find the relative maximum(s) and minimum(s), if any, of the following functions.

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http://en.wikibooks.org/wiki/Calculus/Differentiation/Solutions#Higher_Order_Derivatives

1.

2.

3.

4.

5.6.

Range of Function

1. Show that the expressionx + 1 / x cannot take on any value strictly between 2 and -2.

Absolute Extrema

Determine the absolute maximum and minimum of the following functions on the givendomain

1.

on [0,3]2.

on [-1,1]3.

on [-1/2,2]

Determine Intervals of Change

Note: There are currently no answers given for these exercises.

Find the intervals where the following functions are increasing or decreasing

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1. 10-6x-2x2

2. 2x3-12x2+18x+153. 5+36x+3x2-2x3

4. 8+36x+3x2-2x3

5. 5x3-15x2-120x+36. x3-6x2-36x+2

Determine Intervals of Concavity

Find the intervals where the following functions are concave up or concave down

1. 10-6x-2x2

2. 2x3-12x2+18x+153. 5+36x+3x2-2x3

4. 8+36x+3x2-2x3

5. 5x3-15x2-120x+36. x3-6x2-36x+2

Word Problems

1. You peer around a corner. A velociraptor 64 meters away spots you. You run away ata speed of 6 meters per second. The raptor chases, running towards the corner you justleft at a speed of 4t meters per second (timet measured in seconds after spotting). Afteryou have run 4 seconds the raptor is 32 meters from the corner. At this time, how fastis death approaching your soon to be mangled flesh? That is, what is the rate of changein the distance between you and the raptor?

2. Two goldcarts leave an intersection at the same time. One heads north going 12 mphand the other heads east going 5 mph. How fast are the cars getting away from eachother after one hour?

3. You're making a can of volume 200 m3 with a gold side and silver top/bottom. Say

gold costs 10 dollars per m2 and silver costs 1 dollar per m2. What's the minimum costsof such a can?

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Graphing Functions

For each of the following, graph a function that abides by the provided characteristics

1.

2.

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Contents

Differentiation/DifferentiationDefined→

Calculus← Limits/Exercises

Differentiation






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http://en.wikibooks.org/wiki/File:Derivative1.png



Important Theorems



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Solutions

Find The Derivative By Definition

1. 2x

=

=

=

=

=

=

2. 2

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=

=

=

=

=

3. x

=

=

=

=

=

=

=

4. 4x + 4

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=

=

=

=

=

=

=

Prove Differentiation Rules

Proof of the Derivative of a Constant Function

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Proof of the Derivative of a Linear Function

Proof of the Constant Multiple Rule

=

=

=

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Proof of the Addition and Subtraction Rules

=

=

=

=

=

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Find The Derivative By Rules

1.2.

3.

4.

5.6.7.

8.

9.10.

11.

12.

13. 10x4 + 16x + 114. 49x6 + 40x4 + 3x2 + 2x − 1


Recall that

is the same asy'.

1.

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solve for

2.

solve for


1. 36x2 + 6

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Product and Quotient Rules

Derivatives of TrigonometricFunctions→

Calculus← Differentiation/Differentia-tion Defined

Product and Quotient Rules

Product Rule

When we wish to differentiate a more complicated expression such as:

our only way (up to this point) to differentiate the expression is to expand it and get a polyno-mial, and then differentiate that polynomial. This method becomes very complicated and is partic-ularly error prone when doing calculations by hand. A beginner might guess that the derivativeof a product is the product of the derivatives, similar to the sum and difference rules, but this isnot true. To take the derivative of a product, we use the product rule.

Derivatives of products (Product rule)

Proof

Proving this rule is relatively straightforward, first let us state the equation for the derivative:

We will then apply one of the oldest tricks in the book—adding a term that cancels itself outto the middle:

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Notice that those terms sum to zero, and so all we have done is add 0 to the equation. Nowwe can split the equation up into forms that we already know how to solve:

Looking at this, we see that we can separate the common terms out of the numerators to get:

Which, when we take the limit, becomes:

, or the mnemonic "one D-two plus two D-one"

This can be extended to 3 functions:

For any number of functions, the derivative of their product is the sum, for each function,of its derivative times each other function.

Back to our original example of a product,

, we find the derivative by the product rule is

Note, its derivative wouldnot be

which is what you would get if you assumed the derivative of a product is the product of thederivatives.

To apply the product rule we multiply the first function by the derivative of the second andadd to that the derivative of first function multiply by the second function. Sometimes it helpsto remember the memorize the phrase "First times the derivative of the second plus the secondtimes the derivative of the first."

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Application, proof of the power rule

The product rule can be used to give a proof of the power rule for whole numbers. The proofproceeds bymathematical induction. We begin with the base casen = 1. If f1(x) = x then from

the definition is easy to see that

Next we suppose that for fixed value ofN, we know that forfN(x) = xN, fN'(x) = NxN − 1.

Consider the derivative offN + 1(x) = xN + 1,

We have shown that the statement

is true forn = 1 and that if this statement holds forn = N, then it also holds forn = N + 1.Thus by the principle of mathematical induction, the statement must hold for

.

Quotient rule

There is a similar rule for quotients. To prove it, we go to the definition of the derivative:

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http://en.wikipedia.org/wiki/Mathematical_induction

This leads us to the so-called "quotient rule":

Derivatives of quotients (Quotient Rule)

Which some people remember with the mnemonic "low D-high minus high D-low (over)square the low and away we go!"

Examples

The derivative of (4x − 2) / (x2 + 1) is:

Remember: the derivative of a product/quotientis not the product/quotient of the derivatives.(That is, differentiation does not distribute over multiplication or division.) However one candistribute before taking the derivative. That is

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Derivatives of TrigonometricFunctions

Chain Rule→Calculus← Product and Quotient Rules

Derivatives of TrigonometricFunctions

Sine, Cosine, Tangent, Cosecant, Secant, Cotangent. These are functions that crop up continu-ously in mathematics and engineering and have a lot of practical applications. They also appearin more advanced mathematics, particularly when dealing with things such as line integrals withcomplex numbers and alternate representations of space like spherical and cylindrical coordinatesystems.

We use the definition of the derivative, i.e.,

,

to work these first two out.

Let us find the derivative of sinx, using the above definition.

Definition of derivative

trigonometric identity

factoring

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separation of terms

application of limit

solution

Now for the case ofcos x

Definition of derivative

trigonometric identity

factoring

separation of terms

application of limit

solution

Therefore we have established

Derivative of Sine and Cosine

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To find the derivative of the tangent, we just remember that:

which is a quotient. Applying the quotient rule, we get:

Then, remembering that cos2(x) + sin2(x) = 1, we simplify:

Derivative of the Tangent

For secants, we just need to apply the chain rule to the derivations we have already deter-mined.

So for the secant, we state the equation as:

u(x) = cos(x)

Take the derivative of both equations, we find:

Leaving us with:

Simplifying, we get:

Derivative of the Secant

Using the same procedure on cosecants:

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We get:

Derivative of the Cosecant

Using the same procedure for the cotangent that we used for the tangent, we get:

Derivative of the Cotangent

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Chain Rule

Higher Order Derivatives→Calculus← Derivatives of Trigonomet-ric Functions

Chain Rule

We know how to differentiate regular polynomial functions. For example:

However, there is a useful rule known as thechain method rule. The function above (f(x)

= (x2 + 5)2) can be consolidated into two nested partsf(x) = u2, whereu = m(x) = (x2 + 5).Therefore:

if

and

Then:

Then

Thechain rule states that if we have a function of the formy(u(x)) (i.e. y can be written asa function ofu andu can be written as a function ofx) then:

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Chain RuleIf a function F(x) is composed to two differentiable functions g(x)

and m(x), so that F(x)=g(m(x)), then F(x) is differentiable and,

We can now investigate the original function:

Therefore

This can be performed for more complicated equations. If we consider:

and let

andu=1+x2, so that

and

, then, by applying the chain rule, we find that

So, in just plain words, for the chain rule you take the normal derivative of thewhole thing(make the exponent the coefficient, then multiply by original function but decrease the exponentby 1) then multiply by the derivative of the inside.

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More Differentiation Rules

Higher Order Derivatives→Calculus← Chain Rule


External links

• Online interactive exercises on derivatives• Visual Calculus - Interactive Tutorial on Derivatives, Differentiation, and Integration

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http://wims.unice.fr/wims/wims.cgi?module=home&search_keywords=derivative&search_category=X

http://archives.math.utk.edu/visual.calculus/2/index.html


Implicit differentiation→Calculus← Chain Rule


The second derivative, or second order derivative, is the derivative of the derivative of afunction. The derivative of the functionf(x) may be denoted by

, and its double (or "second") derivative is denoted by

. This is read as "f double prime of x," or "The second derivative of f(x)." Because thederivative of functionf is defined as a function representing the slope of functionf, the doublederivative is the function representing the slope of the first derivative function.

Furthermore, thethird derivative is the derivative of the derivative of the derivative of afunction, which can be represented by

. This is read as "f triple prime ofx", or "The third derivative off(x)". This can continue aslong as the resulting derivative is itself differentiable, with the fourth derivative, the fifthderivative, and so on. Any derivative beyond thefirst derivative can be referred to as ahigherorder derivative.

Notation

Let f(x) be a function in terms ofx. The following are notations for higher order derivatives.

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Notesnth Deriva-tive

4thDerivative3rd Deriva-tive

2nd Deriva-tive

Probably the most common nota-tion.

f(n)(x)f(4)(x)

Leibniz notation.

Another form of Leibniz notation.

Euler's notation.DnfD4fD3fD2f

Warning : You should not writefn(x) to indicate the nth derivative, as this is easily confusedwith the quantityf(x) all raised to the nth power.

The Leibniz notation, which is useful because of its precision, follows from

.

Newton's dot notation extends to the second derivative,

, but typically no further in the applications where this notation is common.

Examples

Example 1:

Find the third derivative of

with respect to x.

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Repeatedly apply thePower Ruleto find the derivatives.

•••

Example 2:

Find the third derivative of

with respect to x.

Use the differentiation rules forexponential expressions, logarithmic expressionsandpolyno-mials.

•

•

•

Applications:

For applications of the second derivative in finding a curve's concavity and points of inflec-tion, see "Extrema and Points of Inflection" and "Extreme Value Theorem". For applications ofhigher order derivatives in physics, see the "Kinematics" section.

Calculus

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http://en.wikibooks.org/wiki/Calculus/Differentiation/Differentiation_Defined#The_Power_Rule

http://en.wikibooks.org/wiki/Calculus/More_Differentation_Rules#Exponents

http://en.wikibooks.org/wiki/Calculus/More_Differentation_Rules#Logarithms



Implicit differentiation

Derivatives of Exponential andLogarithm Functions→

Calculus← Higher Order Derivatives


Implicit differentiation takes a relation and turns it into a rectangular regular equation.

Explicit differentiation

For example, to differentiate a function explicitly,

First we can separate variables to get

Taking the square root of both sides we get a function ofy:

We can rewrite this as a fractional power as

Using the chain rule and simplifying we get,

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Using the same equation

First, differentiate the individual terms of the equation:

Separate the variables:

Divide both sides by

, and simplify to get the same result as above:

Uses

Implicit differentiation is useful when differentiating an equation that cannot be explicitlydifferentiated because it is impossible to isolate variables.

For example, consider the equation,

Differentiate both sides of the equation (remember to use the product rule on the term xy) :

Isolate terms with y':

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Factor out a y' and divide both sides by the other term:


Generally, one will encounter functions expressed in explicit form, that is,y = f(x) form.You might encounter a function that contains a mixture of different variables. Many times it isinconvenient or even impossible to solve fory. A good example is the function

. It is too cumbersome to isolatey in this function. One can utilize implicit differentiation tofind the derivative. To do so, considery to be a nested function that is defined implicitly byx.You need to employ the chain rule whenever you take the derivative of a variable with respectto a different variable: i.e.,

(the derivative with respect tox) of x is 1;

of y is

.

Remember:

Therefore:

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Examples

Example 1

can be solved as:

then differentiated:

However, using implicit differentiation it can also be differentiated like this:

use the product rule:

solve for

:

Note that, if we substitute

into

, we end up with

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again.

Example 2

Find the derivative ofy2 + x2 = 25 with respect tox.

You are seeking

.

Take the derivative of each side of the equation with respect tox.

Inverse Trigonometric Functions

Arcsine, arccosine, arctangent. These are the functions that allow you to determine the anglegiven the sine, cosine, or tangent of that angle.

First, let us start with the arcsine such that:

y = arcsin(x)

To find dy/dxwe first need to break this down into a form we can work with:

x = sin(y)

Then we can take the derivative of that:

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...and solve fordy / dx:

At this point we need to go back to the unit triangle. Sincey is the angle and the oppositeside is sin(y) (which is equal tox), the adjacent side is cos(y) (which is equal to the square root

of 1 minusx2, based on the Pythagorean theorem), and the hypotenuse is 1. Since we have deter-mined the value of cos(y) based on the unit triangle, we can substitute it back in to the aboveequation and get:

Derivative of the Arcsine

We can use an identical procedure for the arccosine and arctangent:

Derivative of the Arccosine

Derivative of the Arctangent

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Derivatives of Exponential andLogarithm Functions

Extrema and Points of Inflec-tion →

Calculus← Implicit differentiation

Derivatives of Exponentialand Logarithm Functions

Exponential Function

First, we determine the derivative ofex using the definition of the derivative:

Then we apply some basic algebra with powers (specifically thatab + c = ab ac):

Sinceex does not depend onh, it is constant ash goes to 0. Thus, we can use the limit rulesto move it to the outside, leaving us with:

Now, the limit can be calculated by techniques we will learn later, for exampleCalculus/Im-proper_Integrals#Definition L'Hopital's rule, and we will see that

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http://en.wikibooks.org/wiki/Calculus/Improper_Integrals#Definition_L.27Hopital.27s_rule

http://en.wikibooks.org/wiki/Calculus/Improper_Integrals#Definition_L.27Hopital.27s_rule

so that we have proved the following rule:

Derivative of the exponential function

Now that we have derived a specific case, let us extend things to the general case. Assumingthata is a positive real constant, we wish to calculate:

One of the oldest tricks in mathematics is to break a problem down into a form that we al-

ready know we can handle. Since we have already determined the derivative ofex, we will at-

tempt to rewriteax in that form.

Using thateln(c) = c and that ln(ab) = b · ln(a), we find that:

Thus, we simply apply the chain rule:

Derivative of the exponential function

Logarithm Function

Closely related to the exponentiation is the logarithm. Just as with exponents, we will derivethe equation for a specific case first (the natural log, where the base ise), and then work to gener-alize it for any logarithm.

First let us create a variabley such that:

It should be noted that what we want to find is the derivative ofy or

.

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Next we will put both sides to the power ofe in an attempt to remove the logarithm fromthe right hand side:

ey = x

Now, applying the chain rule and the property of exponents we derived earlier, we take thederivative of both sides:

This leaves us with the derivative:

Substituting back our original equation ofx = ey, we find that:

Derivative of the Natural Logarithm

If we wanted, we could go through that same process again for a generalized base, but it iseasier just to use properties of logs and realize that:

Since 1 / ln(b) is a constant, we can just take it outside of the derivative:

Which leaves us with the generalized form of:

Derivative of the Logarithm

Logarithmic differentiation

We can use the properties of the logarithm, particularly the natural log, to differentiate moredifficult functions, such a products with many terms, quotients of composed functions, or func-

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tions with variable or function exponents. We do this by taking the natural logarithm of bothsides, re-arranging terms using the logarithm laws below, and then differentiating both sidesimplicitly, before multiplying through by y.

See the examples below.

Example 1

Suppose we wished to differentiate

We take the natural logarithm of both sides

Differentiating implicitly

Multiplying by y

Example 2

Let us differentiate a function

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Taking the natural logarithm of left and right

We then differentiate both sides, recalling the product rule

Multiplying by the original functiony

Example 3

Take a function

Then

We then differentiate

And finally multiply by y

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Extrema and Points of Inflection

Newton's Method→Calculus← Differentiation/Exercises

Extrema and Points of Inflec-tion


The four types of extrema.Maxima andminima are points where a function reaches a highest or lowest value, respec-

tively. There are two kinds ofextrema (a word meaning maximumor minimum): global andlocal, sometimes referred to as "absolute" and "relative", respectively. A global maximum is apoint that takes the largest value on the entire range of the function, while a global minimum isthe point that takes the smallest value on the range of the function. On the other hand, local ex-trema are the largest or smallest values of the function in the immediate vicinity.

All extrema look like the crest of a hill or the bottom of a bowl on a graph of the function.A global extremum is always a local extremum too, because it is the largest or smallest value onthe entire range of the function, and therefore also its vicinity. It is also possible to have a func-tion with no extrema, global or local:y=x is a simple example.

At any extremum, the slope of the graph is necessarily zero, as the graph must stop risingor falling at an extremum, and begin to head in the opposite direction. Because of this, extremaare also commonly calledstationary points or turning points . Therefore, the first derivative ofa function is equal to zero at extrema. If the graph has one or more of thesestationary points,these may be found by setting the first derivative equal to zero and finding the roots of the result-ing equation.

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http://en.wikibooks.org/w/index.php?title=File:Maxima_and_Minima.svg&filetimestamp=20071023190637

http://en.wikipedia.org/wiki/stationary_point


The functionf(x)=x3, which contains a point of inflexion at the point (0,0).However, a slope of zero does not guarantee a maximum or minimum: there is a third class

of stationary point called apoint of inflexion (also spelled point of inflection). Consider thefunction

.

The derivative is

The slope at x=0 is 0. We have a slope of zero, but while this makes it a stationary point,this doesn't mean that it is a maximum or minimum. Looking at the graph of the function youwill see that x=0 is neither, it's just a spot at which the function flattens out. True extrema requirethe a sign change in the first derivative. This makes sense - you have to rise (positive slope) toand fall (negative slope) from a maximum. In between rising and falling, on a smooth curve,there will be a point of zero slope - the maximum. A minimum would exhibit similar properties,just in reverse.


Good (B andC, green) and bad (D andE, blue) points to check in order to classify the extremum(A, black). The bad points lead to an incorrect classification ofA as a minimum.

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http://en.wikibooks.org/w/index.php?title=File:X_cubed_%28narrow%29.svg&filetimestamp=20071023212321

http://en.wikipedia.org/wiki/Point_of_inflexion

http://en.wikibooks.org/w/index.php?title=File:Extremum_Testing.svg&filetimestamp=20071023204543

This leads to a simple method to classify a stationary point - plug x values slightly left andright into the derivative of the function. If the results have opposite signs then it is a true maxi-mum/minimum. You can also use these slopes to figure out if it is a maximum or a minimum:the left side slope will be positive for a maximum and negative for a minimum. However, youmust exercise caution with this method, as, if you pick a point too far from the extremum, youcould take it on the far side ofanotherextremum and incorrectly classify the point.

The Extremum Test

A more rigorous method to classify a stationary point is called theextremum test, or 2ndDerivative Test. As we mentioned before, the sign of the first derivative must change for a sta-tionary point to be a true extremum. Now, thesecondderivative of the function tells us the rateof change of the first derivative. It therefore follows that if the second derivative is positive atthe stationary point, then the gradient is increasing. The fact that it is a stationary point in thefirst place means that this can only be a minimum. Conversely, if the second derivative is nega-tive at that point, then it is a maximum.

Now, if the second derivative is zero, we have a problem. It could be a point of inflexion,or it could still be an extremum. Examples of each of these cases are below - all have a secondderivative equal to zero at the stationary point in question:

• y = x3 has a point of inflexion atx = 0• y = x4 has a minimum atx = 0• y = − x4 has a maximum atx = 0

However, this is not an insoluble problem. What we must do is continue to differentiate untilwe get, at the(n+1)th derivative, a non-zero result at the stationary point:

If n is odd, then the stationary point is a true extremum. If the(n+1)th derivative is positive,it is a minimum; if the(n+1)th derivative is negative, it is a maximum. Ifn is even, then the sta-tionary point is a point of inflexion.

As an example, let us consider the function

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We now differentiate until we get a non-zero result at the stationary point atx=0 (assumewe have already found this point as usual):

Therefore,(n+1) is 4, son is 3. This is odd, and the fourth derivative is negative, so we havea maximum. Note that none of the methods given can tell you if this is a global extremum or justa local one. To do this, you would have to set the function equal to the height of the extremumand look for other roots.

See "Optimization" for a common application of these principles.

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Newton's Method

Related Rates→Calculus← Extrema and Points of In-flection

Newton's Method

Newton's Method (also called the Newton-Raphson method) is a recursive algorithm forapproximating the root of a differentiable function. We know simple formulas for finding theroots of linear and quadratic equations, and there are also more complicated formulae for cubicand quartic equations. At one time it was hoped that there would be formulas found for equationsof quintic and higher-degree, though it was later shown byNeils Henrik Abel that no suchequations exist. The Newton-Raphson method is a method for approximating the roots of polyno-mial equations of any order. In fact the method works for any equation, polynomial or not, aslong as the function is differentiable in a desired interval.

Newton's MethodLet f(x) be a differentiable function. Select a pointx1 based on a firstapproximation to the root, arbitrarily close to the function's root. Toapproximate the root you then recursively calculate using:

As you recursively calculate, thexn's become increasingly better approx-imations of the function's root.For n number of approximations,

In order to explain Newton's method, imagine thatx0 is already very close to a zero off(x).

We know that if we only look at points very close tox0 thenf(x) looks like it's tangent line. Ifx0

was already close to the place wheref(x) was zero, and nearx0 we know thatf(x) looks like its

tangent line, then we hope the zero of the tangent line atx0 is a better approximation thenx0 itself.

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http://en.wikipedia.org/wiki/Niels_Henrik_Abel

The equation for the tangent line tof(x) atx0 is given by

y = f'(x0)(x − x0) + f(x0).

Now we sety = 0 and solve forx.

0 = f'(x0)(x − x0) + f(x0)

− f(x0) = f'(x0)(x − x0)

This value ofx we feel should be a better guess for the value ofx wheref(x) = 0. We chooseto call this value ofx1, and a little algebra we have

If our intuition was correct andx1 is in fact a better approximation for the root off(x), then

our logic should apply equally well atx1. We could look to the place where the tangent line at

x1 is zero. We callx2, following the algebra above we arrive at the formula

And we can continue in this way as long as we wish. At each step, if your current approxima-tion isxn our new approximation will be

Examples

Find the root of the function

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.

As you can seexn is gradually approaching zero (which we know is the root off(x)). One

can approach the function's root with arbitrary accuracy.

Answer:

has a root at

.

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Notes

This method fails whenf'(x) = 0. In that case, one should choose a new starting place. Occa-sionally it may happen thatf(x) = 0 andf'(x) = 0 have a common root. To detect whether this istrue, we should first find the solutions off'(x) = 0, and then check the value off(x) at these places.

Newton's method also may not converge for every function, take as an example:

For this function choosing anyx1 = r − h thenx2 = r + h would cause successive approxima-

tions to alternate back and forth, so no amount of iteration would get us any closer to the rootthan our first guess.

See Also

• Wikipedia:Newton's method• Wikipedia:Abel–Ruffini theorem

Wikimedia Commonshas media related to:category:Newton Method

SeePic-tureLi-censeIn-for-ma-tionHere

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http://en.wikipedia.org/wiki/Newton%27s_method

http://en.wikipedia.org/wiki/Abel�Ruffini_theorem

http://commons.wikimedia.org/wiki/

http://commons.wikimedia.org/wiki/category:Newton_Method

http://commons.wikimedia.org/wiki/Special:Search/Calculus/Newton%27s_Method

http://commons.wikimedia.org/wiki/Special:Search/Calculus/Newton%27s_Method

Related Rates

Kinematics→Calculus← Newton's Method

Related Rates

Introduction

Process for solving related rates problems:

• Write out any relevant formulas and information.• Take the derivative of the primary equation with respect to time.• Solve for the desired variable.• Plug-in known information and simplify.

Related Rates

As stated in the introduction, when doing related rates, you generate a function which com-pares the rate of change of one value with respect to change in time. For example, velocity is therate of change of distance over time. Likewise, acceleration is the rate of change of velocity overtime. Therefore, for the variables for distance, velocity, and acceleration, respectively x, v, anda, and time, t:

Using derivatives, you can find the functions for velocity and acceleration from the distancefunction. This is the basic idea behind related rates: the rate of change of a function is thederivative of that function with respect to time.

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Common Applications

Filling Tank

This is the easiest variant of the most common textbook related rates problem: the fillingwater tank.

• The tank is a cube, with volume 1000L.• You have to fill the tank in ten minutes or you die.• You want to escape with your life and as much money as possible, so you want to find

the smallest pump that can finish the task.

We need a pump that will fill the tank 1000L in ten minutes. So, for pump rate p, volume ofwater pumped v, and minutes t:

Examples

Related rates can get complicated very easily.

Example 1:

• Write out any relevant formulas or pieces of information.

• Take the derivative of the equation above with respect to time. Remember to use theChain Ruleand theProduct Rule.

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http://en.wikibooks.org/wiki/Calculus/More_Differentation_Rules#Chain_Rule

http://en.wikibooks.org/wiki/Calculus/More_Differentation_Rules#Chain_Rule

http://en.wikibooks.org/wiki/Calculus/More_Differentiation_Rules#Product_Rule

Example 2:

• Write out any relevant formulas and pieces of information.

• Take the derivative of both sides of the volume equation with respect to time.

=

=

• Solve for

• Plug-in known information.

Example 3:

Note: Because the vertical distance is downward in nature, the rate of change of y is nega-tive. Similarly, the horizontal distance is decreasing, therefore it is negative (it is getting closerand closer).

The easiest way to describe the horizontal and vertical relationships of the plane's motion isthe Pythagorean Theorem.

• Write out any relevant formulas and pieces of information.

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(wheres is the distance between the plane and the house)

• Take the derivative of both sides of the distance formula with respect to time.

• Solve for

.

• Plug-in known information

=

=

=ft/s

Example 4:

• Write down any relevant formulas and information.

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Substitute

into the volume equation.

=

=

=

• Take the derivative of the volume equation with respect to time.

• Solve for

.

• Plug-in known information and simplify.

=

=ft/min

Example 5:

• Write out any relevant formulas and information.

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Use the Pythagorean Theorem to describe the motion of the ladder.

(wherel is the length of the ladder)

• Take the derivative of the equation with respect to time.

(

so

.)

• Solve for

.

• Plug-in known information and simplify.

=

=ft/sec

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Exercises

Problem Set

Here's a few problems for you to try:

1. A spherical balloon is inflated at a rate of100ft3/min. Assuming the rate of inflationremains constant, how fast is the radius of the balloon increasing at the instant the ra-dius is4 ft?

2. Water is pumped from a cone shaped reservoir (the vertex is pointed down)10 ft in

diameter and10 ft deep at a constant rate of3 ft3/min. How fast is the water levelfalling when the depth of the water is6 ft?

3. A boat is pulled into a dock via a rope with one end attached to the bow of a boat andthe other end held by a man standing6 ft above the bow of the boat. If the man pullsthe rope at a constant rate of2 ft/sec, how fast is the boat moving toward the dockwhen10 ft of rope is out?

Solution Set

Click arrow to show answers1.

2.

3.

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Kinematics

Optimization→Calculus← Related Rates

Kinematics

Introduction

Kinematics or the study of motion is a very relevant topic in calculus.

If x is the position of some moving object, andt is time, this section uses the followingconventions:

•is its position function

•is its velocity function

•is its accelerationfunction

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http://en.wikipedia.org/wiki/velocity

http://en.wikipedia.org/wiki/acceleration

Differentiation

Average Velocity and Acceleration

Average velocity and acceleration problems use the algebraic definitions of velocity andacceleration.

•

•

Examples

Example 1:

A particle's position is defined by the equation

. Find theaverage velocity over the interval [2,7].

• Find the average velocity over the interval[2,7]:

=

=

=

Answer:

.

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Instantaneous Velocity and Acceleration

Instantaneous velocity and acceleration problems use the derivative definitions of velocityand acceleration.

•

•

Examples

Example 2:

A particle moves along a path with a position that can be determined by thefunction

.Determine the acceleration when t = 3.

• Find

• Find

• Find

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=

=

=

Answer:

Integration

•

•



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http://en.wikibooks.org/w/index.php?title=File:Puzzle_stub.png&filetimestamp=20050717004636


http://en.wikibooks.orghttp://en.wikibooks.org/w/index.php?title=Calculus/Kinematics&action=edit

Optimization

Euler's Method→Calculus← Kinematics

Optimization

Introduction

Optimization is one of the uses of Calculus in the real world. Let us assume we are a pizzaparlor and wish to maximize profit. Perhaps we have a flat piece of cardboard and we need tomake a box with the greatest volume. How does one go about this process?

Obviously, this requires the use of maximums and minimums. We know that we find maxi-mums and minimums via derivatives. Therefore, one can conclude that Calculus will be a usefultool for maximizing or minimizing (also known as "Optimizing") a situation.

Examples

Volume Example

A box manufacturer desires to create a box with a surface area of 100 inches squared. Whatis the maximum size volume that can be formed by bending this material into a box? The box isto be closed. The box is to have a square base, square top, and rectangular sides.

• Write out known formulas and information

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• Eliminate the variableh in the volume equation

=

=

• Find the derivative of the volume equation in order to maximize the volume

• Set

and solve for

• Plug-in thex value into the volume equation and simplify

=

=

Answer:

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Sales Example


A small retailer can selln units of a product for a revenue ofr(n) = 8.1n and at a cost ofc(n)

= n3 − 7n2 + 18n, with all amounts in thousands. How many units does it sell to maximize itsprofit?

The retailer's profit is defined by the equationp(n) = r(n) − c(n), which is the revenue generat-ed less the cost. The question asks for the maximum amount of profit which is the maximum ofthe above equation. As previously discussed, the maxima and minima of a graph are found whenthe slope of said graph is equal to zero. To find the slope one finds the derivative ofp(n). By us-ing the subtraction rulep'(n) = r'(n) − c'(n):

=

=

=

Therefore, when

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http://en.wikibooks.org/w/index.php?title=File:Calculus_Graph-Finding_Maximum_Profit.png&filetimestamp=20051024235029

the profit will be maximized or minimized. Use thequadratic formulato find the roots, giving{3.798,0.869}. To find which of these is the maximum and minimum the function can be tested:

p(0.869) = − 3.97321,p(3.798) = 8.58802

Because we only consider the functions for all

(i.e., you can't haven = − 5 units), the only points that can be minima or maxima are thosetwo listed above. To show that 3.798 is in fact a maximum (and that the function doesn't remainconstant past this point) check if the sign ofp'(n) changes at this point. It does, and forn greaterthan 3.798P'(n) the value will remain decreasing. Finally, this shows that for this retailer selling3,798 units would return a profit of $8,588.02.



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http://en.wikipedia.org/wiki/quadratic_formula



http://en.wikibooks.orghttp://en.wikibooks.org/w/index.php?title=Calculus/Optimization&action=edit

Euler's Method

Extreme Value Theorem→Calculus← Optimization

Euler's Method

Euler's Method is a method for estimating the value of a function based upon the values ofthat function's first derivative.

The general algorithm for finding a value of

is:

Examples

The easiest way to keep track of the successive values generated by the algorithm is to drawa table with columns for

.



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http://en.wikibooks.orghttp://en.wikibooks.org/w/index.php?title=Calculus/Euler%27s_Method&action=edit

Extreme Value Theorem

Rolle's Theorem→Calculus← Euler's Method

Extreme Value Theorem

Extreme Value TheoremIf f is a continuous function and closed on the interval [a,b], thenf has both a minimum and amaximum.

This introduces us to the aspect of global extrema and local extrema. (Also known as abso-lute extrema or relative extrema respectively.)

How is this so? Let us use an example.

f(x) = x2 and is closed on the interval [-1,2]. Find all extrema.

A critical point exists at (0,0). Just for practice, let us use the second derivative test to evalu-ate whether or not it is a minimum or maximum. (You should know it is a minimum from lookingat the graph.)

f''(c) > 0, thus it must be a minimum.

As mentioned before, one can find global extrema on a closed interval. How? Evaluate they coordinate at the endpoints of the interval and compare it to they coordinates of the criticalpoint. When you are finding extrema on a closed interval it is called a local extremum and whenit's for the whole graph it's called a global extremum.

1: Critical Point: (0,0) This is the lowest value in the interval. Therefore, it is a local mini-mum which also happens to be the global minimum.

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2: Left Endpoint (-1, 1) This point is not a critical point nor is it the highest/lowest value,therefore it qualifies as nothing.

3: Right Endpoint (2, 4) This is the highest value in the interval, and thus it is a local maxi-mum.

This example was to show you theextreme value theorem. The quintessential point is this:on a closed interval, the function will have both minima and maxima. However, if that intervalwas an open interval of all real numbers, (0,0) would have been a local minimum. On a closedinterval, always remember to evaluate endpoints to obtain global extrema.

First Derivative Test

Recall that the first derivative of a function describes the slope of the graph of the functionat every point along the graph for which the function is defined and differentiable.

Increasing/Decreasing:

• If

, then

is decreasing.

• If

, then

is increasing.

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Local Extrema:

• If

and

changes signs at

, then there exists a local extremum at

.

• If

for

and

for

, then

is a local minimum.

• If

for

and

, then

is a local maximum.

Example 1:

Let

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. Find all local extrema.

• Find

• Set

to find local extrema.

• Determine whether there is a local minimum or maximum at

.

Choose anx value smaller than

:

Choose anx value larger than

:

Therefore, there is a local minimum at

because

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and

changes signs at

.

Answer: local minimum:

.

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Second Derivative Test

Recall that the second derivative of a function describes the concavity of the graph of thatfunction.

• If

and

changes signs at

, then there is a point of inflection (change in concavity) at

.

• If

, then the graph of

is concave down.

• If

, then the graph of

is concave up.

Example 2:

Let

. Find any points of inflection on the graph of

.

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• Find

.

• Set

.

• Determine whether

changes signs at

.

Choose anx value that is smaller than 0:

Choose anx value that is larger than 0:

Therefore, there exists a point of inflection at

because

and

changes signs at

.

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Answer: point of inflection:

.


SeePic-

Extreme value theorem


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http://en.wikipedia.org/wiki/Special:Search/Extreme_value_theorem?go=Go



Rolle's Theorem

Mean Value Theorem forFunctions→

Calculus← Extreme Value Theorem

Rolle's Theorem

Rolle's TheoremIf a function,

, is continuous on the closed interval

, is differentiable on the open interval

, and

, then there exists at least one number c, in the interval

such that

Rolle's Theorem is important in proving theMean Value Theorem.

Examples

Example:

f(x) = x2 − 3x. Show that Rolle's Theorem holds true somewhere within this function. To doso, evaluate thex-intercepts and use those points as your interval.

Solution:

1: The question wishes for us to use thex-intercepts as the endpoints of our interval.

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Factor the expression to obtainx(x − 3) = 0.x = 0 andx = 3 are our two endpoints. We knowthatf(0) andf(3) are the same, thus that satisfies the first part of Rolle's theorem (f(a) = f(b)).

2: Now by Rolle's Theorem, we know that somewhere between these points, the slope willbe zero. Where? Easy: Take the derivative.

= 2x − 3

Thus, atx = 3 / 2, we have a spot with a slope of zero. We know that 3 / 2 (or 1.5) is between0 and 3. Thus, Rolle's Theorem is true for this (as it is for all cases). This was merely a demonstra-tion.

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Mean Value Theorem forFunctions

Integration/Contents→Calculus← Rolle's Theorem

Mean Value Theorem

Mean Value TheoremIf

is continuous on the closed interval

and differentiable on the open interval

, there exists a number,

, in the open interval

such that

.

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Examples


What does this mean? As usual, let us utilize an example to grasp the concept. Visualize (or

graph) the functionf(x) = x3. Choose an interval (anything will work), but for the sake of simplici-ty, [0,2]. Draw a line going from point (0,0) to (2,8). Between the pointsx = 0 andx = 2 exists anumberx = c, where the derivative off at pointc is equal to the slope of the line you drew.

Solution:

1: Using the definition of the mean value theorem

insert values. Our chosen interval is [0,2]. So, we have

2: By the definition of the mean value theorem, we know that somewhere in the intervalexists a point that has the same slope as that point. Thus, let us take the derivative to find thispointx = c.

Now, we know that the slope of the point is 4. So, the derivative at this pointc is 4. Thus, 4

= 3x2. The square root of 4/3 is the point.

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http://en.wikibooks.org/w/index.php?title=File:Lagrange_mean_value_theorem.svg&filetimestamp=20060628113529

Example 2:Find the point that satisifes the mean value theorem on the functionf(x) = sin(x)and the interval [0,π].

Solution:

1: Always start with the definition:

so,

(Remember, sin(π) and sin(0) are both 0.)

2: Now that we have the slope of the line, we must find the pointx = c that has the sameslope. We must now get the derivative!

The cosine function is 0 atπ / 2 + πn (wheren is an integer). Remember, we are bound bythe interval [0,π], soπ / 2 is the pointc that satisfies the Mean Value Theorem.

Differentials

Assume a functiony = f(x) that is differentiable in the open interval (a,b) that contains x.Δy=

The "Differential of x" is theΔx. This is an approximate change in x and can be considered"equivalent" todx. The same holds true for y. What is this saying? One can approximate a changein y by knowing a change in x and a change in x at a point very nearby. Let us view an example.

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Example: A schoolteacher has asked her students to discover what 4.12 is. The students,bereft of their calculators, are too lazy to multiply this out by hand or in their head and desire toutilize calculus. How can they approximate this?

1: Set up a function that mimics the procedure. What are they doing? They are taking anumber (Call it x) and they are squaring it to get a new number (call it y). Thus, y = x^2 Writeyourself a small chart. Make notes of values for x, y,Δx, Δy, and

. We are seeking what y really is, but we need the change in y first.

2: Choose a number close by that is easy to work with. Four is very close to 4.1, so writethat down as x. Yourδx is .1 (This is the "change" in x from the approximation point to the pointyou chose.)

3: Take the derivative of your function.

= 2x. Now, "split" this up (This is not really what is happening, but to keep things simple,assume you are "multiplying"dxover.)

3b. Now you havedy

. We are assumingdyanddxare approximately the same as the change in x, thus we can useΔx and y.

3c. Insert values:dy

. Thus,dy= .8.

4: To findF(4.1), takeF(4) +dy to get an approximation. 16 + .8 = 16.8; This approximationis nearly exact (The real answer is 16.81. This is only one hundredth off!)

Definition of Derivative

The exact value of the derivative at a point is the rate of change over an infinitely smalldistance, approaching zero. Therefore, if h approaches 0 and the function is f(x):

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If h approaches 0, then:

Cauchy's Mean Value Theorem

Cauchy's Mean Value TheoremIf

,

are continuous on the closed interval


,

and

, then there exists a number,


such that

.

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Integration

Indefinite integral→Calculus← Mean Value Theorem

Integration









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• Exercises



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Exercises

Area→Calculus← Improper integrals

Integration/Exercises

Set One: Sums

[Insert Numbered Problems Here]

Solutions to Set One

Set Two: Integration of Polynomials

Given the above rules, practice indefinite integration on the following:

1.

2.

3.

4.

5.

Solutions to Set Two

Indefinite Integration

Antiderivatives

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http://en.wikibooks.org/wiki/Calculus/Integration/Solutions#Solutions_to_Set_Two

1.

2.

3.

4.

5.

6.

7.

8.

Integration by parts

1. Consider the integral

. Find the integral in two different ways. (a) Integrate by parts withu = sin(x) andv' =cos(x). (b) Integrate by parts withu = cos(x) andv' = sin(x).

Compare your answers. Are they the same?

Solutions to Set Three

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http://en.wikibooks.org/wiki/Calculus/Integration/Solutions#Solutions_to_Set_Three

Contents

Indefinite integral→Calculus← Mean Value Theorem

Integration









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• Exercises



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Solutions

Solutions to Set One

Solutions to Set Two

1.

2.3.

4.

5.

Solutions to Set Three

1.

2.

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3.

= tanx + C

4.

5.

6.

7.

8. with the substitutionx = atanθ, we have

, andx2 + a2 = a2sec2θ, so that

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Indefinite integral

Definite integral→Calculus← Integration/Contents

Indefinite integral

Definition

Given functionsF andf such that

for everyx in some intervalI we say thatF is the antiderivative off on I. However,F is notthe only antiderivative. We can add any constant toF without changing the derivative. With this,we define the indefinite integral as follows:

whereF satisfies

andC is any constant.Note that the indefinite integral yields afamilyof functions.

Example

Since the derivative ofx4 is 4x3, the general antiderivative of 4x3 is x4 plus a constant. Thus,

Example: Finding antiderivatives

Let's take a look at 6x2. How would we go about finding the integral of this function? Recallthe rule from differentiation that

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In our circumstance, we have:

This is a start! We now know that the function we seek will have a power of 3 in it. Howwould we get the constant of 6? Well,

Thus, we say that 2x3 is an antiderivative of 6x2.

Indefinite integral identities

Basic Properties of Indefinite Integrals

Constant Rule for indefinite integrals

If c is a constant then

Sum/Difference Rule for indefinite integrals

Indefinite integrals of Polynomials

Say we are given a function of the form,f(x) = xn, and would like to determine the antideriva-tive of f. Considering that

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we have the following rule for indefinite integrals:

Power rule for indefinite integrals

for all

Example

Integral of the Inverse function

To integrate

, we should first remember

Therefore, since

is the derivative of lnx we can conclude that

Note that the polynomial integration rule does not apply when the exponent is -1. Thistechnique of integration must be used instead. Since the argument of the natural logarithm func-tion must be positive (on the real line), the absolute value signs are added around its argumentto ensure that the argument is positive.

Integral of the Exponential function

Since

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we see thatex is its own antiderivative. This allows us to find the integral of an exponentialfunction:

Integral of Sine and Cosine

Recall that

Sosin x is an antiderivative ofcos xand-cos xis an antiderivative ofsin x. Hence we getthe following rules for integratingsin xandcos x

We will find how to integrate more complicated trigonometric functions in the chapter onintegration techniques.

Example

Suppose we want to integrate the functionf(x) = x4 + 1 + 2sinx. An application of the sumrule from above allows us to use the power rule and our rule for integrating sinx as follows,

Example

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The Substitution Rule

The substitution rule is a valuable asset in the toolbox of any integration greasemonkey. Itis essentially the chain rule (a differentiation technique you should be familiar with) in reverse.First, let's take a look at an example:

Preliminary Example

Suppose we want to find

. That is, we want to find a function such that its derivative equals

. Stated yet another way, we want to find an antiderivative of

. Since sin(x) differentiates to cos(x), as a first guess we might try the function sin(x2). Butby the Chain Rule,

Which is almost what we want apart from the fact that there is an extra factor of 2 in front.But this is easily dealt with because we can divide by a constant (in this case 2). So,

Thus, we have discovered a function,

, whose derivative is

. That is,F is an antiderivative of

. This gives us

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Generalization

In fact, this technique will work for more general integrands. Supposeu is a differentiablefunction. Then to evaluate

we just have to notice that by the Chain Rule

As long asu' is continuous we have that

Now the right hand side of this equation is just the integral of cos(u) but with respect tou.If we write u instead ofu(x) this becomes

So, for instance, ifu(x) = x3 we have worked out that

General Substitution Rule

Now there was nothing special about using the cosine function in the discussion above, andit could be replaced by any other function. Doing this gives us the substitution rule for indefiniteintegrals:

Substitution rule for indefinite integralsAssumeu is differentiable with continuous derivative and thatf is continuous on the range

of u. Then

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Notice that it looks like you can "cancel" in the expression

to leave just adu. This does not really make any sense because

is not a fraction. But it's a good way to remember the substitution rule.

Examples

The following example shows how powerful a technique substitution can be. At first glancethe following integral seems intractable, but after a little simplification, it's possible to tackleusing substitution.

Example

We will show that

First, we re-write the integral:

.

Now we preform the following substitution:

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Which yields:

.

Integration by Parts

Integration by parts is another powerful tool for integration. It was mentioned above thatone could consider integration by substitution as an application of the chain rule in reverse. In asimilar manner, one may consider integration by parts as the product rule in reverse.

Preliminary Example

General Integration by Parts

Integration by parts for indefinite integrals

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Supposef andg are differentiable and their derivatives are continuous. Then

If we write u=f(x) andv=g(x), then by using the Leibnitz notationdu=f'(x) dxanddv=g'(x)dx the integration by parts rule becomes

Examples

Example

Find

Here we let:

u = x, so thatdu= dx,

dv= cos(x)dx , so thatv = sin(x).

Then:

Example

Find

In this example we will have to use integration by parts twice.

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Here we let

u = x2, so thatdu= 2xdx,

dv= exdx, so thatv = ex.

Then:

Now to calculate the last integral we use integration by parts again. Let

u = ''x, so thatdu= dx,

dv= exdx, so thatv = ex

and integrating by parts gives

So, finally we obtain

Example

Find

The trick here is to write this integral as

Now let

u = ln(x) sodu= 1 / xdx,

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v = x sodv= 1dx.

Then using integration by parts,

Example

Find

Again the trick here is to write the integrand as

. Then let

u = arctan(x); du = 1/(1+x2) dx

v = x; dv = 1·dx

so using integration by parts,

Example

Find

This example uses integration by parts twice. First let,

u = ex; thus du = exdx

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dv = cos(x)dx; thusv = sin(x)

so

Now, to evaluate the remaining integral, we use integration by parts again, with

u = ex; du = exdx

v = -cos(x); dv = sin(x)dx

Then

Putting these together, we have

Notice that the same integral shows up on both sides of this equation. We can simply addthe integral to both sides to get

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Definite integral

Fundamental Theorem of Cal-culus→

Calculus← Indefinite integral

Definite integral

Suppose we are given a function and would like to determine the area underneath its graphover an interval. We could guess, but how could we figure out the exact area? Below, using afew clever ideas, we actuallydefinesuch an area and show that by using what is called theDefi-nite integral we can indeed determine the exact area underneath a curve.

Definition of the Definite Integral


Figure 1


Figure 2

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http://en.wikibooks.org/w/index.php?title=File:Riemann_Integration_3.png&filetimestamp=20051105162933

http://en.wikibooks.org/w/index.php?title=File:Riemann_Integration_2.png&filetimestamp=20051105190811

The rough idea of defining the area under the graph off is to approximate this area with afinite number of rectangles. Since we can easily work out the area of the rectangles, we get anestimate of the area under the graph. If we use a larger number of rectangles we expect a betterapproximation. Somehow, it seems that we could use our old friend, the limit, and "approach"an infinite number of rectangles to get the exact area. Let's look at such an idea more closely.

Suppose we have a functionf that is positive and two numbersa,bsuch thata<b. Let's pickan integern and divide the interval [a,b] into n subintervals of equal width (see Figure 1). As theinterval [a,b] has widthb-a, each subinterval has width

We denote the endpoints of the subintervals by

. This gives us


Figure 3Now for each

pick asample point

in the interval

and consider the rectangle of height

and widthΔx (see Figure 2). The area of this rectangle is

. By adding up the area of all the rectangles for

we get that the areaS is approximated by

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http://en.wikibooks.org/w/index.php?title=File:Riemann.gif&filetimestamp=20060128192838

A more convenient way to write this is withsummation notation:

For each numbern we get a different approximation. Asn gets larger the width of the rectan-gles gets smaller which yields a better approximation (see Figure 3). In the limit ofAn asn tends

to infinity we get the area ofS.

Definition of the Definite IntegralSupposef is a continuous function on [a,b] and

. Then thedefinite integralof f betweena andb is

where

are any sample points in the interval [xi − 1,xi].It is a fact that iff is continuous on[a,b] then this limit always exists and does not depend

on the choice of the points

. For instance they may be evenly spaced, or distributed ambiguously throughout the interval.The proof of this is technical and is beyond the scope of this section.

Notation

When considering the expression,

, the functionf is called theintegrandand the interval[a,b] is the interval of integration. Alsoais called thelower limit andb theupper limitof integration.

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Figure 6One important feature of this definition is that we also allow functions which take negative

values. Iff(x)<0 for all x then

so

. So the definite integral off will be strictly negative. More generally iff takes on both posi-tive an negative values then

will be the area under the positive part of the graph off minus the area under the graph ofthe negative part of the graph (see Figure 6). For this reason we say that

is thesigned areaunder the graph.


The area spanned byf(x)

A geometrical proof that anti-derivative gives the area

Suppose we have a function F(x) which returns the area between x and some unknown pointu. (Actually, u is the first number before x which satisfies F(u) = 0, but our solution is indepen-

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http://en.wikibooks.org/w/index.php?title=File:Signed-area.png&filetimestamp=20051111151107

http://en.wikibooks.org/w/index.php?title=File:Fundamental-theorem-1.png&filetimestamp=20051119190428

dent from u, so we won't bother ourselves with it.) We don't even know if something like F existsor not, but we're going to investigate what clue do we have if itdoesexist.

We can use F to calculate the area between a and b, for instance, which is obviously F(b)-F(a); F is something general. Now, consider a rather peculiar situation, the area bounded atx andx + Δx, in the limit of

. Of course it can be calculated by using F, but we're looking for another solution this time.As the right border approaches the left one, the shape seems to be an infinitesimal rectangle, withthe height of f(x) and width ofΔx. So, the area reads:

Of course, we could use F to calculate this area as well:

By combining these equations, we have

If we divide both sides byΔx, we get

which is an interesting result, because the left-hand side is the derivative of F with respectto x. This remarkable result doesn't tell us what F itself is, however it tells us what thederivativeof F is, and it isf.

Independence of Variable

It is important to notice that the variablex did not play an important role in the definition ofthe integral. In fact we can replace it with any other letter, so the following are all equal:

Each of these is the signed area under the graph off betweena andb.

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Left and Right Handed Riemann Sums

These methods are sometimes referred to as L-RAM and R-RAM, RAM standing for"Rectangular Approximation Method."

We could have decided to choose all our sample points

to be on the right hand side of the interval [xi − 1,xi] (see Figure 7). Then

for all i and the approximation that we calledAn for the area becomes

This is called theright-handed Riemann sum, and the integral is the limit

Alternatively we could have taken each sample point on the left hand side of the interval. Inthis case

(see Figure 8) and the approximation becomes

Then the integral off is

The key point is that, as long asf is continuous, these two definitions give the same answerfor the integral.

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Figure 7


Figure 8

Example 1

In this example we will calculate the area under the curve given by the graph off(x) = x forx between 0 and 1. First we fix an integern and divide the interval [0,1] inton subintervals ofequal width. So each subinterval has width

To calculate the integral we will use the right-handed Riemann Sum. (We could have usedthe left-handed sum instead, and this would give the same answer in the end). For the right-handed sum the sample points are

Notice that

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http://en.wikibooks.org/w/index.php?title=File:Riemann-Sum-right-hand.png&filetimestamp=20051112132656

http://en.wikibooks.org/w/index.php?title=File:Riemann_Sum_Left_Hand.png&filetimestamp=20051112132620

. Putting this into the formula for the approximation,

Now we use theformula

to get

To calculate the integral off between0 and1 we take the limit asn tends to infinity,

Example 2

Next we show how to find the integral of the functionf(x) = x2 betweenx=a andx=b. Thistime the interval[a,b] has widthb-aso

Once again we will use the right-handed Riemann Sum. So the sample points we choose are

Thus

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We have to calculate each piece on the right hand side of this equation. For the first two,

For the third sum we have to use aformula

to get

Putting this together

Taking the limit asn tend to infinity gives

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Basic Properties of the Integral

From the definition of the integral we can deduce some basic properties. For all the followingrules, suppose thatf andg are continuous on[a,b].

The Constant Rule

Constant Rule

Whenf is positive, the height of the functioncf at a pointx is c times the height of the func-tion f. So the area undercf betweena andb is c times the area underf. We can also give a proofusing the definition of the integral, using the constant rule for limits,

Example

We saw in the previous section that

.

Using the constant rule we can use this to calculate that

Example

We saw in the previous section that

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We can use this and the constant rule to calculate that

There is a special case of this rule used for integrating constants:

Integrating Constants

If c is constant then

Whenc > 0 anda < b this integral is the area of a rectangle of heightc and widthb-awhichequalsc(b-a).

Example

The addition and subtraction rule

Addition and Subtraction Rules of Integration

As with the constant rule, the addition rule follows from the addition rule for limits:

=

=

=

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The subtraction rule can be proved in a similar way.

Example

From above

and

so

Example

The Comparison Rule


Figure 9Comparison Rule

• Suppose

for all x in [a,b]. Then

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http://en.wikibooks.org/w/index.php?title=File:Integral_comparison_1.png&filetimestamp=20051106144944

• Suppose


• Suppose


If

then each of the rectangles in the Riemann sum to calculate the integral off will be abovethey axis, so the area will be non-negative. If

then

and by linearity of the integral we get the second property. Finally if

then the area under the graph off will be greater than the area of rectangle with heightmandless than the area of the rectangle with heightM (see Figure 9). So

Example

Linearity with respect to endpoints

Additivity with respect to endpoints Supposea < c < b. Then

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Again suppose thatf is positive. Then this property should be interpreted as saying that thearea under the graph off betweena andb is the area betweena andc plus the area betweenc andb (see Figure 8)


Figure 8Extension of Additivity with respect to limits of integrationWhena = b we have that

so

Also in defining the integral we assumed thata<b. But the definition makes sense even whenb<a in which case

so has changed sign. This gives

With these definitions,

whatever the order ofa,b,c.Example

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http://en.wikibooks.org/w/index.php?title=File:Integral_linear_endpoints.png&filetimestamp=20051106150447

Even and odd functions

Recall that a functionf is called odd if it satisfiesf( − x) = − f(x) and is called even iff( − x)= f(x).

Supposef is a continuous odd function then for anya,

If f is a continuous even function then for anya,

Caution: For improper integrals (e.g. ifa is infinity, or if the function approaches infinityat 0 ora, etc.), the first equation above is only true if

exists. Otherwise the integral is undefined, and only the Cauchy principal value is 0.

Supposef is an odd function and consider first just the integral from-a to 0. We make thesubstitutionu=-x sodu=-dx. Notice that ifx=-a thenu=a and ifx=0 thenu=0. Hence

Now asf is odd,f( − u) = − f(u) so the integral becomes

Now we can replace the dummy variableu with any other variable. So we can replace it withthe letterx to give

Now we split the integral into two pieces

The proof of the formula for even functions is similar, and is left as an exercise.

Example

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Fundamental Theorem ofCalculus

Integration techniques→Calculus← Definite integral

Fundamental Theorem ofCalculus

Fundamental Theorem of Calculus


SeePic-

Fundamental theorem of calculus


Statement of the Fundamental Theorem

Suppose thatf is continuous on[a,b]. We can define a functionF by

Fundamental Theorem of Calculus Part ISupposef is continuous on[a,b] andF is de-fined by

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http://en.wikipedia.org/wiki/Special:Search/Fundamental_theorem_of_calculus?go=Go



ThenF is differentiable on(a,b)and for all

,

Now recall thatF is said to be an antiderivative off if

.

Fundamental Theorem of Calculus Part II Suppose thatf is continuous on[a,b] and thatF is any antiderivative off. Then


Figure 1Note: a minority of mathematicians refer to part one as two and part two as one. All mathe-

maticians refer to what is stated here as part 2 as The Fundamental Theorem of Calculus.

Proofs

Proof of Fundamental Theorem of Calculus Part I

Supposex is in (a,b). PickΔx so thatx + Δx is also in (a, b). Then

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http://en.wikibooks.org/w/index.php?title=File:Fundamental-theorem-1.png&filetimestamp=20051119190428

and

.

Subtracting the two equations gives

Now

so rearranging this we have

According to themean value theoremfor integration, there exists ac in [x, x + Δx] such that

.

Notice thatc depends onΔx. Anyway what we have shown is that

,

and dividing both sides byΔx gives

.

Take the limit as

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http://en.wikipedia.org/wiki/mean_value_theorem

we get the definition of the derivative ofF atx so we have

.

To find the other limit, we will use the squeeze theorem. The numberc is in the interval [x,x + Δx], sox≤ c ≤ x + Δx. Also,

and

. Therefore, according to the squeeze theorem,

.

As f is continuous we have

which completes the proof.

Proof of Fundamental Theorem of Calculus Part II

Define

Then by the Fundamental Theorem of Calculus part I we know thatP is differentiable on(a,b)and for all

SoP is an antiderivative off. Now we were assuming thatF was also an antiderivative sofor all

,

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A consequence of the Mean Value Theorem is that this implies there is a constantC suchthat for all

,

,

and asP andF are continuous we see this holds whenx=a and whenx=b as well. Since weknow thatP(a)=0 we can putx=a into the equation to get0=F(a) +C soC=-F(a). And puttingx=b gives

Integration of Polynomials

Using the power rule for differentiation we can find a formula for the integral of a power

using the Fundamental Theorem of Calculus. Letf(x) = xn. We want to find an antiderivative forf. Since the differentiation rule for powers lowers the power by 1 we have that

As long as

we can divide byn+1 to get

So the function

is an antiderivative off. If a,b>0 thenF is continuous on[a,b] and we can apply the Funda-mental Theorem of Calculus we can calculate the integral off to get the following rule.

Power Rule of Integration I

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as long as

anda,b > 0.Notice that we allow all values ofn, even negative or fractional. Ifn>0 then this works even

if a or b are negative.

Power Rule of Integration II

as long asn > 0.Examples

• To find

we raise the power by 1 and have to divide by 4. So

• The power rule also works for negative powers. For instance

• We can also use the power rule for fractional powers. For instance

• Using linearity the power rule can also be thought of as applying to constants. For exam-ple,

.

• Using the linearity rule we can now integrate any polynomial. For example

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Infinite Sums

Integration techniques/Recog-nizing Derivatives and the

Substitution Rule→

Calculus← Integration techniques

Integration techniques/Infi-nite Sums

The most basic, and arguably the most difficult, type of evaluation is to use the formal defini-tion of a Riemann integral.

Exact Integrals as Limits of Sums

Using the definition of an integral, we can evaluate the limit as n goes to infinity. Thistechnique requires a fairly high degree of familiarity with summationidentities. This techniqueis often referred to as evaluation "by definition," and can be used to find definite integrals, aslong as the integrands are fairly simple. We start with definition of the integral:

Thenpicking

to be

we get,

In some simple cases, this expression can be reduced to a real number, which can be interpret-ed as the area under the curve if f(x) is positive on [a,b].

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http://en.wikipedia.org/wiki/Sigma_notation#Identities

Example 1

Find

by writing the integral as a limit of Riemann sums.

In other cases, it is even possible to evaluate indefinite integrals using the formal definition.We can define the indefinite integral as follows:

Example 2

Supposef(x) = x2, then we can evaluate the indefinite integral as follows.

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Recognizing Derivatives and theSubstitution Rule

Integration techniques/Integra-tion by Parts→

Calculus← Integration techniques/Infi-nite Sums

Integrationtechniques/Recog-nizing Derivatives and the

Substitution Rule

After learning a simple list of antiderivatives, it is time to move on to more complex inte-grands, which are not at first readily integrable. In these first steps, we notice certain special caseintegrands which can be easily integrated in a few steps.

Recognizing Derivatives and ReversingDerivative Rules

If we recognize a functiong(x) as being the derivative of a functionf(x), then we can easilyexpress the antiderivative ofg(x):

For example, since

we can conclude that

Similarly, since we knowex is its own derivative,

The power rule for derivatives can be reversed to give us a way to handle integrals of powersof x. Since

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,

we can conclude that

or, a little more usefully,

.

Integration by Substitution


Integration by SubstitutionFor many integrals, a substitution can be used to transform the integrand and make possible

the finding of an antiderivative. There are a variety of such substitutions, each depending on theform of the integrand.

Integrating with the derivative present

If a component of the integrand can be viewed as the derivative of another component ofthe integrand, a substitution can be made to simplify the integrand.

For example, in the integral

we see that 3x2 is the derivative ofx3 + 1. Letting

u = x3 + 1

we have

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http://en.wikiversity.org/wiki/Integration_by_Substitution

or, in order to apply it to the integral,

du= 3x2dx.

With this we may write

Note that it was not necessary that we hadexactlythe derivative ofu in our integrand. Itwould have been sufficient to have any constant multiple of the derivative.

For instance, to treat the integral

we may letu = x5. Then

and so

the right-hand side of which is a factor of our integrand. Thus,

In general, the integral of a power of a function times that function's derivative may be inte-grated in this way. Since

,

we have

Calculus

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Therefore,

There is a similar rule for definite integrals, but we have to change the endpoints.

Substitution rule for definite integrals

Assumeu is differentiable with continuous derivative and thatf is continuous on the range ofu.Supposec = u(a) andd = u(b). Then

Examples

Consider the integral

By using the substitutionu = x2 + 1, we obtaindu= 2x dxand

Note how the lower limitx = 0 was transformed intou = 02 + 1 = 1 and the upper limitx =

2 intou = 22 + 1 = 5.

Proof of the substitution rule

We will now prove the substitution rule for definite integrals. LetF be an anti derivative off so

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F'(x) = f(x). By the Fundamental Theorem of Calculus

Next we define a functionG by the rule

Then by the Chain ruleG is differentiable with derivative

Integrating both sides with respect tox and using the Fundamental Theorem of Calculus weget

But by the definition ofF this equals

Hence

which is the substitution rule for definite integrals.

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Integration techniques/Integra-tion by Complexifying→

Calculus← Integration techniques/Rec-ognizing Derivatives and the

Substitution Rule

Integration techniques/Inte-gration by Parts

Continuing on the path of reversing derivative rules in order to make them useful for integra-tion, we reverse the product rule.


If y = uvwhereu andv are functions ofx,

Then

Rearranging,

Therefore,

Therefore,

, or

.

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This is the integration by parts formula. It is very useful in many integrals involving productsof functions, as well as others.

For instance, to treat

we chooseu = x and

. With these choices, we havedu= dxandv = − cosx, and we have

Note that the choice ofu anddvwas critical. Had we chosen the reverse, so thatu = sinx and

, the result would have been

The resulting integral is no easier to work with than the original; we might say that this appli-cation of integration by parts took us in the wrong direction.

So the choice is important. One general guideline to help us make that choice is, if possible,to chooseu to be the factor of the integrand whichbecomes simplerwhen we differentiate it. Inthe last example, we see that sinx does not become simpler when we differentiate it: cosx is nosimpler than sinx.

An important feature of the integration by parts method is that we often need to apply it morethan once. For instance, to integrate

,

we start by choosingu = x2 anddv= ex to get

Note that we still have an integral to take care of, and we do this by applying integration byparts again, withu = x and

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, which gives us

So, two applications of integration by parts were necessary, owing to the power ofx2 in theintegrand.

Note thatany power of xdoes become simpler when we differentiate it, so when we see anintegral of the form

one of our first thoughts ought to be to consider using integration by parts withu = xn. Ofcourse, in order for it to work, we need to be able to write down an antiderivative fordv.

Example

Use integration by parts to evaluate the integral

Solution: If we letu = sin(x) andv' = ex, then we haveu' = cos(x) andv = ex. Using our rulefor integration by parts gives

We do not seem to have made much progress. But if we integrate by parts again withu =

cos(x) andv' = ex and henceu' = − sin(x) andv = ex, we obtain

We may solve this identity to find the anti-derivative ofexsin(x) and obtain

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With definite integral

For definite integrals the rule is essentially the same, as long as we keep the endpoints.

Integration by parts for definite integrals Supposef andg are differen-tiable and their derivatives are continuous. Then

.

This can also be expressed in Leibniz notation.


SeePic-

Integration by parts


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http://en.wikipedia.org/wiki/Special:Search/Integration_by_parts?go=Go



Integration by Complexifying

Integration techniques/PartialFraction Decomposition→

Calculus← Integration techniques/Inte-gration by Parts

Integration techniques/Inte-gration by Complexifying

This technique requires an understanding and recognition of complex numbers. SpecificallyEuler's formula:

Recognize, for example, that the real portion:

Given an integral of the general form:

We can complexify it:

With basic rules of exponents:

It can be proven that the "real portion" operator can be moved outside the integral:

The integral easily evaluates:

Multiplying and dividing by (1-2i):

Which can be rewritten as:

Applying Euler's forumula:

Expanding:

Taking the Real part of this expression:

So:

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Partial Fraction Decomposition

Integrationtechniques/Trigono-metric Substitution→

Calculus← Integration techniques/Inte-gration by Complexifying

Integration techniques/Par-tial Fraction Decomposition

Suppose we want to find

. One way to do this is to simplify the integrand by finding constantsA andB so that

This can be done by cross multiplying the fraction which gives

As both sides have the same denominator we must have 3x + 1 = A(x + 1) + Bx. This is anequation forx so must hold whatever valuex is. If we put inx = 0 we get 1 =A and puttingx =- 1 gives − 2 = −B soB = 2. So we see that

Returning to the original integral

=

=

Rewriting the integrand as a sum of simpler fractions has allowed us to reduce the initialintegral to a sum of simpler integrals. In fact this method works to integrate any rational function.

Calculus

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Method of Partial Fractions:

• Step 1Use long division to ensure that the degree ofP(x) less than the degreeof Q(x).

• Step 2Factor Q(x) as far as possible.• Step 3Write down the correct form for the partial fraction decomposition (see

below) and solve for the constants.

To factor Q(x) we have to write it as a product of linear factors (of the formax+ b) and irre-

ducible quadratic factors (of the formax2 + bx+ c with b2 − 4ac< 0).

Some of the factors could be repeated. For instance ifQ(x) = x3 − 6x2 + 9x we factorQ(x)as

Q(x) = x(x2 − 6x + 9) = x(x − 3)(x − 3) = x(x − 3)2.

It is important that in each quadratic factor we haveb2 − 4ac< 0, otherwise it is possible to

factor that quadratic piece further. For example ifQ(x) = x3 − 3x2 − 2x then we can write

Q(x) = x(x2 − 3x + 2) = x(x − 1)(x + 2)

We will now show how to writeP(x) / Q(x) as a sum of terms of the form

and

Exactly how to do this depends on the factorization ofQ(x) and we now give four cases thatcan occur.

Case (a)Q(x) is a product of linear factors with no repeats.

This means thatQ(x) = (a1x + b1)(a2x + b2)...(anx + bn) where no factor is repeated and no

factor is a multiple of another.

For each linear term we write down something of the form

Calculus

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, so in total we write

Example 1

Find

Here we haveP(x) = 1 + x2,Q(x) = (x + 3)(x + 5)(x + 7) andQ(x) is a product of linear fac-tors. So we write

Multiply both sides by the denominator

1 + x2 = A(x + 5)(x + 7) + B(x + 3)(x + 7) + C(x + 3)(x + 5)

Substitute in three values ofx to get three equations for the unknown constants,

soA = 5 / 4,B = − 13 / 2,C = 25 / 4, and

We can now integrate the left hand side.

Case (b)Q(x) is a product of linear factors some of which are repeated.

If (ax+ b) appears in the factorisation ofQ(x) k-times. Then instead of writing the piece

we use the more complicated expression

Example 2

Find

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HereP(x)=1" and "Q(x)=(x+1)(x+2)2 We write

Multiply both sides by the denominator 1 =A(x + 2)2 + B(x + 1)(x + 2) + C(x + 1)

Substitute in three values ofx to get 3 equations for the unknown constants,

soA=1, B=-1, C=-1, and

We can now integrate the left hand side.

Case (c)Q(x) contains some quadratic pieces which are not repeated.

If (ax2 + bx+ c) appears we use

Case (d)Q(x) contains some repeated quadratic factors.

If (ax2 + bx+ c) appears k-times then use

Calculus

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Trigonometric Substitution

A Wikibookian suggests thatSolving Integrals by Trigonometric substitutionbemergedinto this book or chapter.Discuss whether or not this merger should happen on thediscussion page.

Integration techniques/TangentHalf Angle→

Calculus← Integration techniques/Par-tial Fraction Decomposition

Integration tech-niques/TrigonometricSubsti-

tution

If the integrand contains a single factor of one of the forms

we can try a trigonometric substitution.

• If the integrand contains

let x = asinθ and use theidentity1 − sin2θ = cos2θ.• If the integrand contains

let x = atanθ and use the identity 1 + tan2θ = sec2θ.• If the integrand contains

let x = asecθ and use the identity sec2θ − 1 = tan2θ.


Trigonometric Substitutions

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http://en.wikibooks.org/wiki/Solving_Integrals_by_Trigonometric_substitution

http://en.wikibooks.org/wiki/Help:Merging_pages

http://en.wikibooks.org/wiki/Help:Merging_pages

http://en.wikibooks.org/wiki/Talk:Calculus/Integration_techniques/Trigonometric_Substitution


http://en.wikiversity.org/wiki/Trigonometric_Substitutions





Sine substitution


This substitution is easily derived from a triangle, using thePythagorean Theorem.If the integrand contains a piece of the form

we use the substitution

This will transform the integrand to a trigonometic function. If the new integrand can't beintegrated on sight then the tan-half-angle substitution described below will generally transformit into a more tractable algebraic integrand.

Eg, if the integrand is√(1-x2),

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http://en.wikibooks.org/w/index.php?title=File:Wikibooks-logo.svg&filetimestamp=20090115175128

http://en.wikibooks.org/w/index.php?title=B:Calculus/Integration_techniques/Trigonometric_Substitution&action=edit&redlink=1

http://en.wikibooks.org/w/index.php?title=File:Wikipedia-logo-en.png&filetimestamp=20080610233540

http://en.wikipedia.org/wiki/Trigonometric_substitution

http://en.wikibooks.org/w/index.php?title=File:Trig_Sub_Triangle_1.png&filetimestamp=20100108122114

http://en.wikipedia.org/wiki/Pythagorean_Theorem

If the integrand is√(1+x)/√(1-x), we can rewrite it as

Then we can make the substitution

Tangent substitution


This substitution is easily derived from a triangle, using thePythagorean Theorem.When the integrand contains a piece of the form


E.g, if the integrand is (x2+a2)-3/2 then on making this substitution we find

Calculus

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If the integral is

then on making this substitution we find

After integrating by parts, and using trigonometric identities, we've ended up with an expres-sion involving the original integral. In cases like this we must now rearrange the equation so thatthe original integral is on one side only

As we would expect from the integrand, this is approximatelyz2/2 for largez.

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Secant substitution


This substitution is easily derived from a triangle, using thePythagorean Theorem.If the integrand contains a factor of the form


Example 1

Find

Example 2

Find

We can now integrate by parts

Calculus

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Tangent Half Angle

Integrationtechniques/Trigono-metric Integrals→

Calculus← Integration tech-niques/TrigonometricSubstitu-

tion

Integration techniques/Tan-gent Half Angle

Another useful change of variables is

With this transformation, using the double-angle trig identities,

This transforms a trigonometric integral into a algebraic integral, which may be easier tointegrate.

For example, if the integrand is 1/(1 + sinx ) then

This method can be used to further simplify trigonometric integrals produced by the changesof variables described earlier.

For example, if we are considering the integral

we can first use the substitutionx= sinθ, which gives

Calculus

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then use the tan-half-angle substition to obtain

In effect, we've removed the square root from the original integrand. We could do this witha single change of variables, but doing it in two steps gives us the opportunity of doing thetrigonometric integral another way.

Having done this, we can split the new integrand into partial fractions, and integrate.

This result can be further simplified by use of the identities

ultimately leading to

In principle, this approach will work with any integrand which is the square root of aquadratic multiplied by the ratio of two polynomials. However, it should not be applied automati-cally.

E.g, in this last example, once we deduced

we could have used the double angle formula, since this contains only even powers of cosand sin. Doing that gives

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Using tan-half-angle on this new, simpler, integrand gives

This can be integrated on sight to give

This is the same result as before, but obtained with less algebra, which shows why it is bestto look for the most straightforward methods at every stage.

A more direct way of evaluating the integralI is to substitutet = tanθ right from the start,which will directly bring us to the line

above. More generally, the substitutiont = tanx gives us

so this substitution is the preferable one to use if the integrand is such that all the squareroots would disappear after substitution, as is the case in the above integral.

Example

Using the trigonometric substitutiont = atanx, then

and

(when −π / 2 < x < π / 2). So,

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Alternate Method

In general, to evaluate integrals of the form

,

it is extremely tedious to use the aforementioned "tan half angle" substitution directly, asone easily ends up with a rational function with a 4th degree denominator. Instead, we may firstwrite the numerator as

.

Then the integral can be written as

which can be evaluated much more easily.

Example

Evaluate

.

Let

Calculus

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.

Then

.

Comparing coefficients of cosx, sinx and the constants on both sides, we obtain

yieldingp = q = 1/2, r = 2. Substituting back into the integrand,

.

The last integral can now be evaluated using the "tan half angle" substitution describedabove, and we obtain

.

The original integral is thus

.

Calculus

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Trigonometric Integrals

Integration techniques/Reduc-tion Formula→

Calculus← Integration techniques/Tan-gent Half Angle

Integration tech-niques/Trigonometric Inte-

grals

When the integrand is primarily or exclusively based on trigonometric functions, the follow-ing techniques are useful.

Powers of Sine and Cosine

We will give a general method to solve generally integrands of the form cosm (x)sinn(x).First let us work through an example.

Notice that the integrand contains an odd power of cos. So rewrite it as

We can solve this by making the substitutionu = sin(x) so du = cos(x) dx. Then we can writethe whole integrand in terms of u by using the identity

cos(x)2 = 1 - sin2(x)=1-u2.

So

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This method works whenever there is an odd power of sine or cosine.

To evaluate

wheneither m or n isodd.

• If m is odd substituteu=sinx and use the identity cos2x = 1 - sin2x=1-u2.• If n is odd substituteu=cosx and use the identity sin2x = 1 - cos2x=1-u2.

Example

Find

.

As there is an odd power of sin we letu = cosx so du = - sin(x)dx. Notice that whenx=0 wehaveu=cos(0)=1and whenx = π / 2 we haveu = cos(π / 2) = 0.

When bothm andn are even things get a little more complicated.

To evaluate

when bothm andn areeven.

Use theidentitiessin2x = 1/2 (1- cos 2x) and cos2x = 1/2 (1+ cos 2x).

Example

Find

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As sin2x = 1/2 (1- cos 2x) and cos2x = 1/2 (1+ cos 2x) we have

and expanding, the integrand becomes

Using the multiple angle identities

then we obtain on evaluating

Powers of Tan and Secant

To evaluate

.

1. If n is even and

then substituteu=tanx and use theidentity sec2x = 1 + tan2x.2. If n andm are both odd then substituteu=secx and use theidentity tan2x =

sec2x-1.3. If n is odd andm is even then use theidentity tan2x = sec2x-1 and apply a re-

duction formula to integrate

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Example 1

Find

.

There is an even power of secx. Substitutingu = tanx givesdu= sec2xdxso

Example 2

Find

.

Let u = cosx sodu= − sinxdx. Then

Example 3

Find

.

The trick to do this is to multiply and divide by the same thing like this:

Making the substitutionu = secx + tanx sodu= secxtanx + sec2xdx,

More trigonometric combinations

For the integrals

or

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or

use theidentities

•

•

•

Example 1

Find

We can use the fact that sina cosb=(1/2)(sin(a+b)+sin(a-b)), so

Now use the oddness property of sin(x) to simplify

And now we can integrate

Example 2

Find:

.

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- 304-


Using the identities

Then

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Reduction Formula

Integration techniques/Irra-tional Functions→

Calculus← Integration tech-niques/Trigonometric Integrals

Integration techniques/Re-duction Formula

A reduction formulais one that enables us to solve an integral problem byreducingit to aproblem of solving an easier integral problem, and then reducing that to the problem of solvingan easier problem, and so on.

For example, if we let

Integration by parts allows us to simplify this to

which is our desired reduction formula. Note that we stop at

.

Similarly, if we let

then integration by parts lets us simplify this to

Calculus

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Using the trigonometric identity, tan2=sec2-1, we can now write

Rearranging, we get

Note that we stop atn=1 or 2 if n is odd or even respectively.

As in these two examples, integrating by parts when the integrand contains a power oftenresults in a reduction formula.

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Irrational Functions

Integrationtechniques/Numeri-cal Approximations→

Calculus← Integration techniques/Re-duction Formula

Integration techniques/Irra-tional Functions

Integration of irrational functions is more difficult than rational functions, and many cannotbe done. However, there are some particular types that can be reduced to rational forms by suit-able substitutions.

Type 1

Integrand contains

Use the substitution

.

Example

Find

.

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Type 2

Integral is of the form

Write Px+ Q as

.

Example

Find

.

Type 3

Integrand contains

,

or

This was discussed in "trigonometric substitutions above". Here is a summary:

1. For

, usex = asinθ.2. For

, usex = atanθ.3. For

, usex = asecθ.

Calculus

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Type 4

Integral is of the form

Use the substitution

.

Example

Find

.

Type 5

Other rational expressions with the irrational function

1. If a > 0, we can use

.2. If c > 0, we can use

.3. If ax2 + bx+ c can be factored asa(x − α)(x − β), we can use

.4. If a < 0 andax2 + bx+ c can be factored as −a(α − x)(x − β), we can usex = αcos2θ

+ βsin2θ, / theta+ β.

Calculus

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Numerical Approximations

Improper integrals→Calculus← Integration techniques/Irra-tional Functions

Integration techniques/Nu-merical Approximations

It is often the case, when evaluating definite integrals, that an antiderivative for the integrandcannot be found, or is extremely difficult to find. In some instances, a numerical approximationto the value of the definite value will suffice. The following techniques can be used, and arelisted in rough order of ascending complexity.

Riemann Sum

This comes from the definition of an integral. If we pick n to be finite, then we have:

where

is any point in the i-th sub-interval [xi − 1,xi] on [a,b].

Right Rectangle

A special case of the Riemann sum, where we let

, in other words the point on the far right-side of each sub-interval on, [a,b]. Again if we pickn to be finite, then we have:

Left Rectangle

Another special case of the Riemann sum, this time we let

Calculus

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, which is the the point on the far left side of each sub-interval on [a,b]. As always, this is an ap-proximation when n is finite. Thus, we have:

Trapezoidal Rule

Simpson's Rule

Remember, n must be even,

Further reading

• Numerical Methods/Numerical Integration

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http://en.wikibooks.org/wiki/Numerical_Methods/Numerical_Integration

Improper integrals

Integration/Exercises→Calculus← Integration techniques/Nu-merical Approximations

Improper Integrals

The definition of a definite integral:

requires the interval [a,b] be finite. The Fundamental Theorem of Calculus requires thatf becontinuous on [a,b]. In this section, you will be studying a method of evaluating integrals thatfail these requirements—either because their limits of integration are infinite, or because a finitenumber of discontinuities exist on the interval [a,b]. Integrals that fail either of these requirementsareimproper integrals.

L'Hopital's Rule

L'Hopital's Rule is included in this section because limits involving infinity often appear inimproper integration. L'Hopital's Rule describes how to evaluate limits involving infinity and or0 if the limit evaluates to an indeterminate form.

All of the following expressions are indeterminate forms.

These expressions are calledindeterminatebecause you cannot determine their exact valuein the indeterminate form. Depending on the situation, each indeterminate form could evaluateto a variety of values.

Calculus

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Theorem

If

is indeterminate of type

or

,

then

Note:

can approach a finite valuec,

or

.

Example:

One might think the value

Consider

Plugging the value ofx into the limit yields

Calculus

- 314-


(indeterminate form).

Let

=

=

=

(indeterminateform)

We now apply L'Hopital's Rule by taking the derivative of the top and bottom with respectto x.

Returning to the expression above

=

=

(indeterminateform)

We apply L'Hopital's Rule once again

Therefore

Calculus

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And

Careful: this does not prove that

because

Improper Integrals with Infinite Limits ofIntegration

Consider the integral

Assigning a finite upper boundb in place of infinity gives

This improper integral can be interpreted as the area of the unbounded region betweenf(x)=1/x^2, y=0 (thex-axis), andx=1.

Definition

1. Suppose

exists for all

Calculus

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. Then we define

=

as long as this limit exists and is finite.

If it does exist we say the integral isconvergentand otherwise we say itis divergent.

2. Similarly if

exists for all

we define

=

3. Finally supposec is a fixed real number and that

and

are both convergent. Then we define

=

Example: Convergent Improper Integral

We claim that

Calculus

- 317 -


To do this we calculate

=

=

=

=

Example: Divergent Improper Integral

We claim that the integral

diverges.

This follows as

=

=

=

=

Therefore

diverges.

Example: Improper Integral

Find

Calculus

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To calculate the integral use integration by parts twice to get

Now

and because exponentials overpower polynomials, we see that

and

as well. Hence,

Example: Powers

Show

}}

If

then

=

=

=

=

Notice that we had to assume that

do avoid dividing by zero. However thep = 1 case was done in a previous example.

Calculus

- 319 -


Improper Integrals with a Finite NumberDiscontinuities

First we give a definition for the integral of functions which have a discontinuity at onepoint.

Definition of improper integrals with a singlediscontinuity

If f is continuous on the interval [a,b) and is discontinuous atb, we define :

=

If the limit in question exists we say the integralconvergesand otherwisewe say itdiverges.

Similarly if f is continuous on the interval(a,b] and is discontinuous ata,we define

=

Finally supposef has an discontinuity at a point c in (a,b) and is continu-ous at all other points in [a,b]. If

and

converge we define

Calculus

- 320-


=

Example 1

Show

}}

If

then

=

=

=

=

Notice that we had to assume that

do avoid dividing by zero. So instead we do thep = 1 case separately,

which diverges.

We can also give a definition of the integral of a function with a finite number of discontinu-ities

Calculus

- 321 -


Definition: Improper integrals with finitenumber of discontinuities

Supposef is continuous on [a,b] except at points

in [a,b]. We define

as long as each integral on the right converges.

Notice that by combining this definition with the definition for improper integrals with infi-nite endpoints, we can define the integral of a function with a finite number of discontinuitieswith one or more infinite endpoints.

Example 2

The integral

is improper because the integrand is not continuous at x=2. However had we not notice thatwe might have been tempted to apply the fundamental theorem of calculus and conclude that itequals

which is not correct. In fact the integral diverges.

Comparison Test

There are integrals which cannot easily be evaluated. However it may still be possible toshow they are convergent by comparing them to an integral we already know converges.

Theorem (Comparison Test)Let f andg be continuous functions definedfor all

Calculus

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.

1. Suppose

for all

. Then if

converges so does

2. Suppose

for all

. Then if

diverges so does

A similar theorem holds for improper integrals of the form

and for improper integrals with discontinuities.

Example: Use of comparsion test to show convergence

Show that

converges.

For allx we know that

so

. This implies that

Calculus

- 323 -


.

We have seen that

converges. So putting

and

into the comparison test we get that the integral

converges as well.

Example: Use of Comparsion Test to show divergence

Show that

diverges.

Just as in the previous example we know that

for all x. Thus

We have seen that

diverges. So putting

and

into the comparison test we get that

Calculus

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diverges as well.

An extension of the comparison theorem

To apply the comparison theorem you do not really need

for all

. What we actually need is this inequality holds for sufficiently largex (i.e. there is a numberc such that

for all

). For then

so the first integral converges if and only if third does, and we can apply the comparisontheorem to the

piece.

Example

Show that

converges.

The reason that this integral converges is because for largex thee − x factor in the integrand

is dominant. We could try comparingx7 / 2e − x with e − x, but as

, the inequality

Calculus

- 325 -


is the wrong way around to show convergence.

Instead we rewrite the integrand as

Since the limit

we know that forx sufficiently large we have

. So for largex,

Since the integral

converges the comparison test tells us that

converges as well.


SeePic-

Improper integrals


Calculus

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http://en.wikipedia.org/wiki/Special:Search/Improper_integrals?go=Go



Area

Volume→Calculus← Integration/Exercises

Area

Introduction

Finding the area between two curves, usually given by two explicit functions, is often usefulin calculus.

In general the rule for finding the area between two curves is

or

If f(x) is the upper function and g(x) is the lower function

This is true whether the functions are in the first quadrant or not.

Area between two curves

Suppose we are given two functionsy1=f(x) andy2=g(x) and we want to find the area be-

tween them on the interval[a,b]. Also assume thatf(x)≥ g(x) for all x on the interval[a,b]. Beginby partitioning the interval[a,b] into n equal subintervals each having a length ofΔx=(b-a)/n.Next choose any point in each subinterval,xi* . Now we can 'create' rectangles on each interval.

At the pointxi* , the height of each rectangle isf(xi*)-g(xi*) and the width isΔx. Thus the area

of each rectangle is[f(x i*)-g(xi*)] Δx. An approximationof the area,A, between the two curves

is

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.

Now we take the limit asn approaches infinity and get

which gives the exact area. Recalling the definition of the definite integral we notice that

.

This formula of finding the area between two curves is sometimes known as applying integra-tion with respect to thex-axis since the rectangles used to approximate the area have their baseslying parallel to thex-axis. It will be most useful when the two functions are of the formy1=f(x)

andy2=g(x). Sometimes however, one may find it simpler to integrate with respect to they-axis.

This occurs when integrating with respect to thex-axis would result in more than one integral tobe evaluated. These functions take the formx1=f(y) andx2=g(y) on the interval[c,d]. Note that

[c,d] are values ofy. The derivation of this case is completely identical. Similar to before, wewill assume thatf(y)≥ g(y) for all y on [c,d]. Now, as before we can divide the interval intonsubintervals and create rectangles to approximate the area betweenf(y) andg(y). It may be usefulto picture each rectangle having their 'width',Δy, parallel to they-axis and 'height',f(yi*)-g(yi*)

at the pointyi* , parallel to thex-axis. Following from the work above we may reason that an

approximationof the area,A, between the two curves is

.

As before, we take the limit asn approaches infinity to arrive at

,

which is nothing more than a definite integral, so

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.

Regardless of the form of the functions, we basically use the same formula.

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Volume

Volume of solids of revolution→

Calculus← Area

Volume

When we think about volume from an intuitive point of view, we typically think of it as theamount of "space" an item occupies. Unfortunately assigning a number that measures this amountof space can prove difficult for all but the simplest geometric shapes. Calculus provides a newtool for calculating volume that can greatly extend our ability to calculate volume. In order tounderstand the ideas involved it helps to think about the volume of a cylinder. The volume of a

cylinder is calculated using the formulaV = πr2h. The base of the cylinder is a circle whose area

is given byA = πr2. Notice that the volume of a cylinder is derived by taking the area of its baseand multiplying by the heighth. For more complicated shapes, we could think of approximatingthe volume by taking the area of some cross section at some heightx and multiplying by somesmall change in heightΔx them adding up the heights of all of these approximations from thebottom to the top of the object. This would appear to be a Riemann sum. Keeping this in mind,we can develop a more general formula for the volume of solids in

(3 dimensional space).

Formal Definition

Formally the ideas above suggest that we can calculate the volume of a solid by calculatingthe integral of the cross-sectional area along some dimension. In the above example of a cylinder,the every cross section was given by the same circle, so the cross-sectional area is therefore aconstant function, and the dimension of integration was vertical (although it could have been anyone we desired). Generally, ifS is a solid that lies in

betweenx = a andx = b, letA(x) denote the area of a cross section taken in the plane perpen-dicular to thex direction, and passing through the pointx. If the functionA(x) is continuous on[a,b], then the volumeVS of the solidS is given by:

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Examples

Example 1: A right cylinder


Now we will calculate the volume of a right cylinder using our new ideas about how to calcu-late volume. Since we already know the formula for the volume of a cylinder this will give us a"sanity check" that our formulas make sense. First, we choose a dimension along which to inte-grate. In this case, it will greatly simplify the calculations to integrate along the height of thecylinder, so this is the direction we will choose. Thus we will call the vertical direction (see dia-gram)x. Now we find the function,A(x), which will describe the cross-sectional area of ourcylinder at a height ofx. The cross-sectional area of a cylinder is simply a circle. Now simply

recall that the area of a circle isπr2, and soA(x) = πr2. Before performing the computation, wemust choose our bounds of integration. In this case, we simply definex = 0 to be the base of thecylinder, and so we will integrate fromx = 0 tox = h, whereh is the height of the cylinder. Final-ly, we integrate:

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http://en.wikibooks.org/w/index.php?title=File:Cylinder_with_cross_section_at_height_x.svg&filetimestamp=20090222142953

Vcylin-

der

=

πr2(h− 0)

=

πr2h.

This is exactly the familiar formula for the volume of a cylinder.

Example 2: A right circular cone


For our next example we will look at an example where the cross sectional area is not con-stant. Consider a right circular cone. Once again the cross sections are simply circles. But nowthe radius varies from the base of the cone to the tip. Once again we choosex to be the verticaldirection, with the base atx = 0 and the tip atx = h, and we will letR denote the radius of thebase. While we know the cross sections are just circles we cannot calculate the area of the crosssections unless we find some way to determine the radius of the circle at heightx.

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http://en.wikibooks.org/w/index.php?title=File:Plane_intersecting_cone_2.png&filetimestamp=20090222101722


Luckily in this case it is possible to use some of what we know from geometry. We canimagine cutting the cone perpendicular to the base through some diameter of the circle all theway to the tip of the cone. If we then look at the flat side we just created, we will see simply atriangle, whose geometry we understand well. The right triangle from the tip the base at heightx is similar to the right triangle with from the tip with heighth triangle. This tells us that

. So that we see that the radius of the circle at heightx is

. Now using the familiar formula for the area of a circle we see that

.

Now we are ready to integrate.

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http://en.wikibooks.org/w/index.php?title=File:Similar_triangles_for_cone.svg&filetimestamp=20090222150543

Vcone

By u-substitution we may letu = h − x, thendu= − dxand our integralbecomes

Example 3: A sphere


In a similar fashion, we can use our definition to prove the well known formula for the vol-ume of a sphere. First, we must find our cross-sectional area function,A(x). Consider a sphereof radiusRwhich is centered at the origin in

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http://en.wikibooks.org/w/index.php?title=File:Sphere_with_cross_section.svg&filetimestamp=20090220141329

. If we again integrate vertically thenx will vary from − R to R. In order to find the area of aparticular cross section it helps to draw a right triangle whose between the center of the sphere,the center of the circular cross section, and a point along the circumference of the cross section.As shown in the diagram the side lengths of this triangle will beR, | x | , andr. Wherer is theradius of the circular cross section. Then by the Pythagorean theorem

and find thatA(x) = π(R2 − | x | 2). It is slightly helpful to notice that |x | 2 = x2 so we do not needto keep the absolute value.

So we have that

Vsphere

Extension to Non-trivial Solids

Now that we have shown our definition agrees with our prior knowledge, we will see howit can help us extend our horizons to solids whose volumes are not possible to calculate usingelementary geometry.

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Volume of solids of revolution

Arc length→Calculus← Volume

Volume of solids of revolu-tion

Revolution solids

A solid is said to be of revolution when it is formed by rotating a given curve against an axis.For example, rotating a circle positioned at(0,0) against they-axiswould create a revolutionsolid, namely, a sphere.

Calculating the volume

Calculating the volume of this kind of solid is very similar to calculating anyvolume: wecalculate the basal area, and then we integrate through the height of the volume.

Say we want to calculate the volume of the shape formed rotating over thex-axisthe areacontained between the curvesf(x) andg(x) in the range [a,b]. First calculate the basal area:

| πf(x)2 − πg(x)2 |

And then integrate in the range [a,b]:

Alternatively, if we want to rotate in they-axisinstead,f andg must be invertible in the range[a,b], and, following the same logic as before:

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SeePic-

Solid of revolution


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http://en.wikibooks.orghttp://en.wikibooks.org/w/index.php?title=Calculus/Volume_of_solids_of_revolution&action=edit


http://en.wikipedia.org/wiki/Special:Search/Solid_of_revolution?go=Go



Arc length

Surface area→Calculus← Volume of solids of revolu-tion

Arc length

Suppose that we are given a functionf and we want to calculate the length of the curve drawnout by the graph off. If the graph were a straight line this would be easy — the formula for thelength of the line is given by Pythagoras' theorem. And if the graph were a polygon we can calcu-late the length by adding up the length of each piece.

The problem is that most graphs are not polygons. Nevertheless we can estimate the lengthof the curve by approximating it with straight lines. Suppose the curveC is given by the formulay = f(x) for

. We divide the interval [a,b] into n subintervals with equal widthΔx and endpoints

. Now letyi = f(xi) soPi = (xi,yi) is the point on the curve abovexi. The length of the straight

line betweenPi andPi + 1 is

So an estimate of the length of the curveC is the sum

As we divide the interval [a,b] into more pieces this gives a better estimate for the length ofC. In fact we make that a definition.

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Definition (Length of a Curve)

The length of the curvey = f(x) for

is defined to be

The Arclength Formula

Suppose thatf' is continuous on [a,b]. Then the length of the curve given byy = f(x) betweena andb is given by

And in Leibniz notation

Proof: Consideryi + 1 − yi = f(xi + 1) − f(xi). By theMean Value Theoremthere is a pointzi

in (xi + 1,xi) such that

.

So

=

=

=

=

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Putting this into the definiton of the length ofC gives

Now this is the definition of the integral of the function

betweena andb (notice thatg is continuous because we are assuming thatf' is continuous).Hence

as claimed.

Arclength of a parametric curve

For a parametric curve, that is, a curve defined byx = f(t) andy = g(t), the formula is slightlydifferent:

Proof: The proof is analogous to the previous one: Consideryi + 1 − yi = g(ti + 1) − g(ti) and

xi + 1 − xi = f(ti + 1) − f(ti). By the Mean Value Theorem there are pointsci anddi in (ti + 1,ti) such

that

and

.

So

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=

=

=

=

Putting this into the definiton of the length of the curve gives

This is equivalent to:


SeePic-

Arc length


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http://en.wikipedia.org/wiki/Special:Search/Arc_length?go=Go



Surface area

Work →Calculus← Arc length

Surface area




SeePic-

Surface_of_revolution


Suppose we are given a functionf and we want to calculate the surface area of the functionf rotated around a given line. The calculation of surface area of revolution is related to the arclength calculation.

If the functionf is a straight line, other methods such as surface area formulas for cylindersand conical frustra can be used. However, iff is not linear, an integration technique must be used.

Recall the formula for the lateral surface area of a conical frustrum:

wherer is the average radius andl is the slant height of the frustrum.

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http://en.wikipedia.org/wiki/Special:Search/Surface_of_revolution?go=Go



For y=f(x) and

, we divide[a,b] into subintervals with equal widthΔx and endpoints

. We map each point

to a conical frustrum of widthΔx and lateral surface area

.

We can estimate the surface area of revolution with the sum

As we divide[a,b] into smaller and smaller pieces, the estimate gives a better value for thesurface area.

Definition (Surface of Revolution)

The surface area of revolution of the curvey=f(x) about a line for

is defined to be

The Surface Area Formula

Supposef is a continuous function on the interval[a,b] andr(x) represents the distance fromf(x) to the axis of rotation. Then the lateral surface area of revolution about a line is given by

And in Leibniz notation

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Proof:

=

=

=

As

and

, we know two things:

1. the average radius of each conical frustrumr i approaches a single value

2. the slant height of each conical frustruml i equals an infitesmal segment of arc length

From the arc length formula discussed in the previous section, we know that

Therefore

=

=

Because of the definition of an integral

, we can simplify the sigma operation to an integral.

Or if f is in terms ofy on the interval[c,d]

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Work

Centre of mass→Calculus← Surface area

Work



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Centre of mass

Parametric Introduction→Calculus← Work

Centre of mass



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Parametric and Polar Equations

Parametric Introduction→Calculus← Probability

Parametric and Polar Equa-tions



• Introduction to Parametric Equations• Differentiation and Parametric Equations• Integration and Parametric Equations

Polar Equations

• Introduction to Polar Equations• Differentiation and Polar Equations• Integration and Polar Equations

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http://en.wikibooks.org/wiki/File:HuggingRoseStarconstructionJohnManuel.png

Parametric Introduction

Parametric Differentiation→Calculus← Parametric and Polar Equa-tions

Parametric Introduction

Introduction

Parametric equations are typically definied by two equations that specify both the x and ycoordinates of a graph using a parameter. They are graphed using the parameter (usually t) tofigure out both the x and y coordinates.

Example 1:

Note: This parametric equation is equivalent to the rectangular equation

.

Example 2:

Note: This parametric equation is equivalent to the rectangular equation

and the polar equation

.

Parametric equations can be plotted by using a t-table to show values of x and y for eachvalue of t. They can also be plotted by eliminating the parameter though this method removesthe parameter's importance.

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Forms of Parametric Equations

Parametric equations can be described in three ways:

• Parametric form• Vector form• An equality

The first two forms are used far more often, as they allow us to find the value of the compo-nent at the given value of the parameter. The final form is used less often; it allows us to verifya solution to the equation, or find the parameter (or some constant multiple thereof).

Parametric Form

A parametric equation can be shown inparametric formby describing it with a system ofequations. For instance:

Vector Form

Vector formcan be used to describe a parametric equation in a similar manner toparametricform. In this case, a position vector is given:

Equalities

A parametric equation can also be described with a set of equalities. This is done by solvingfor the parameter, and equating the components. For example:

From here, we can solve for t:

And hence equate the two right-hand sides:

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Converting Parametric Equations

There are a few common place methods used to change a parametric equation to rectangularform. The first involves solving for t in one of the two equations and then replacing the new ex-pression for t with the variable found in the second equation.

Example 1:

becomes

Example 2:

Given

Isolate the trigonometric functions

Use the "Beloved Identity"

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Parametric Differentiation

Taking Derivatives of Parametric Systems

Just as we are able to differentiate functions of x, we are able to differentiate x and y, whichare functions of t. Consider:

We would find the derivative of x with respect to t, and the derivative of y with respect tot:

In general, we say that if

and

then:

and

It's that simple.

This process works for any amount of variables.

Slope of Parametric Equations

In the above process, x' has told us only the rate at which x is changing, not the rate for y,and vice versa. Neither is the slope.

In order to find the slope, we need something of the form

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.

We can discover a way to do this by simple algebraic manipulation:

So, for the example in section 1, the slope at any time t:

In order to find a vertical tangent line, set the horizontal change, or x', equal to 0 and solve.

In order to find a horizontal tangent line, set the vertical change, or y', equal to 0 and solve.

If there is a time when both x' and y' are 0, that point is called a singular point.

Concavity of Parametric Equations

Solving for the second derivative of a parametric equation can be more complex that it mayseem at first glance. When you have take the derivative of

in terms of t, you are left with

:

.

By multiplying this expression by

, we are able to solve for the second derivative of the parametric equation:

.

Thus, the concavity of a parametric equation can be described as:

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So for the example in sections 1 and 2, the concavity at any time t:

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Parametric Integration

Introduction

Because most parametric equations are given in explicit form, they can be integrated likemany other equations. Integration has a variety of applications with respect to parametric equa-tions, especially in kinematics and vector calculus.



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Polar Introduction


A polar grid with several angles labeled in degreesThepolar coordinate systemis a two-dimensional coordinate system in which each point

on a plane is determined by an angle and a distance. The polar coordinate system is especiallyuseful in situations where the relationship between two points is most easily expressed in termsof angles and distance; in the more familiar Cartesian coordinate system or rectangular coordinatesystem, such a relationship can only be found through trigonometric formulae.

As the coordinate system is two-dimensional, each point is determined by two polar coordi-nates: the radial coordinate and the angular coordinate. The radial coordinate (usually denotedasr) denotes the point's distance from a central point known as thepole(equivalent to theoriginin the Cartesian system). The angular coordinate (also known as the polar angle or the azimuthangle, and usually denoted byθ or t) denotes the positive or anticlockwise (counterclockwise)angle required to reach the point from the 0° ray orpolar axis(which is equivalent to the positivex-axis in the Cartesian coordinate plane).

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http://en.wikibooks.org/wiki/File:Polar_graph_paper.svg

Plotting points with polar coordinates


The points (3,60°) and (4,210°) on a polar coordinate systemEach point in the polar coordinate system can be described with the two polar coordinates,

which are usually calledr (the radial coordinate) andθ (the angular coordinate, polar angle, orazimuth angle, sometimes represented asφ or t). Ther coordinate represents the radial distancefrom the pole, and theθ coordinate represents the anticlockwise (counterclockwise) angle fromthe 0° ray (sometimes called the polar axis), known as the positive x-axis on the Cartesian coordi-nate plane.

For example, the polar coordinates (3, 60°) would be plotted as a point 3 units from the poleon the 60° ray. The coordinates (−3, 240°) would also be plotted at this point because a negativeradial distance is measured as a positive distance on the opposite ray (the ray reflected about theorigin, which differs from the original ray by 180°).

One important aspect of the polar coordinate system, not present in the Cartesian coordinatesystem, is that a single point can be expressed with an infinite number of different coordinates.This is because any number of multiple revolutions can be made around the central pole withoutaffecting the actual location of the point plotted. In general, the point (r, θ) can be representedas (r, θ ± n×360°) or (−r, θ ± (2n + 1)180°), wheren is any integer.

The arbitrary coordinates (0,θ) are conventionally used to represent the pole, as regardlessof theθ coordinate, a point with radius 0 will always be on the pole. To get a unique representa-tion of a point, it is usual to limitr to negative and non-negative numbersr ≥ 0 andθ to the inter-val [0, 360°) or (−180°, 180°] (or, in radian measure, [0, 2π) or (−π, π]).

Angles in polar notation are generally expressed in either degrees or radians, using the con-version 2π rad = 360°. The choice depends largely on the context. Navigation applications use

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http://en.wikibooks.org/wiki/File:CircularCoordinates.png

degree measure, while some physics applications (specifically rotational mechanics) and almostall mathematical literature on calculus use radian measure.

Converting between polar and Cartesian coordinates


A diagram illustrating the conversion formulaeThe two polar coordinatesr andθ can be converted to the Cartesian coordinatesx andy by

using the trigonometric functions sine and cosine:

while the two Cartesian coordinatesx andy can be converted to polar coordinater by

(by a simple application of the Pythagorean theorem).

To determine the angular coordinateθ, the following two ideas must be considered:

• For r = 0,θ can be set to any real value.• For r ≠ 0, to get a unique representation forθ, it must be limited to an interval of size

2π. Conventional choices for such an interval are [0, 2π) and (−π, π].

To obtainθ in the interval [0, 2π), the following may be used (arctan denotes the inverse ofthe tangent function):

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http://en.wikibooks.org/wiki/File:Polar_to_cartesian.svg

To obtainθ in the interval (−π, π], the following may be used:

One may avoid having to keep track of the numerator and denominator signs by use of theatan2 function, which has separate arguments for the numerator and the denominator.

Polar equations

The equation defining an algebraic curve expressed in polar coordinates is known as apolarequation. In many cases, such an equation can simply be specified by definingr as a function ofθ. The resulting curve then consists of points of the form (r(θ), θ) and can be regarded as thegraph of the polar functionr.

Different forms of symmetry can be deduced from the equation of a polar functionr. Ifr(−θ) = r(θ) the curve will be symmetrical about the horizontal (0°/180°) ray, ifr(π−θ) = r(θ) itwill be symmetric about the vertical (90°/270°) ray, and ifr(θ−α°) = r(θ) it will be rotationallysymmetricα° counterclockwise about the pole.

Because of the circular nature of the polar coordinate system, many curves can be describedby a rather simple polar equation, whereas their Cartesian form is much more intricate. Amongthe best known of these curves are the polar rose, Archimedean spiral, lemniscate, limaçon, andcardioid.

For the circle, line, and polar rose below, it is understood that there are no restrictions onthe domain and range of the curve.

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Circle


A circle with equationr(θ) = 1The general equation for a circle with a center at (r0, φ) and radiusa is

This can be simplified in various ways, to conform to more specific cases, such as theequation

for a circle with a center at the pole and radiusa.

Line

Radial lines (those running through the pole) are represented by the equation

,

whereφ is the angle of elevation of the line; that is,φ = arctanm wherem is the slope ofthe line in the Cartesian coordinate system. The non-radial line that crosses the radial lineθ = φperpendicularly at the point (r0, φ) has the equation

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http://en.wikibooks.org/wiki/File:Circle_r=1.PNG

Polar rose


A polar rose with equationr(θ) = 2 sin 4θA polar rose is a famous mathematical curve that looks like a petaled flower, and that can

be expressed as a simple polar equation,

for any constantφ0 (including 0). Ifk is an integer, these equations will produce ak-petaled

rose if k is odd, or a 2k-petaled rose ifk is even. Ifk is rational but not an integer, a rose-likeshape may form but with overlapping petals. Note that these equations never define a rose with2, 6, 10, 14, etc. petals. The variablea represents the length of the petals of the rose.

Archimedean spiral


One arm of an Archimedean spiral with equationr(θ) = θ for 0 < θ < 6πThe Archimedean spiral is a famous spiral that was discovered by Archimedes, which also

can be expressed as a simple polar equation. It is represented by the equation

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http://en.wikibooks.org/wiki/File:Rose_r=2sin%284theta%29.PNG

http://en.wikibooks.org/wiki/File:Archimedian_spiral.PNG

Changing the parameterawill turn the spiral, whilebcontrols the distance between the arms,which for a given spiral is always constant. The Archimedean spiral has two arms, one forθ > 0and one forθ < 0. The two arms are smoothly connected at the pole. Taking the mirror image ofone arm across the 90°/270° line will yield the other arm. This curve is notable as one of the firstcurves, after theConic Sections, to be described in a mathematical treatise, and as being a primeexample of a curve that is best defined by a polar equation.

Conic sections


Ellipse, showing semi-latus rectumA conic section with one focus on the pole and the other somewhere on the 0° ray (so that

the conic's semi-major axis lies along the polar axis) is given by:

wheree is the eccentricity and

is the semi-latus rectum (the perpendicular distance at a focus from the major axis to thecurve). Ife > 1, this equation defines a hyperbola; ife = 1, it defines a parabola; and ife < 1, itdefines an ellipse. The special casee= 0 of the latter results in a circle of radius

.

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http://en.wikibooks.org/wiki/Conic_Sections

http://en.wikibooks.org/wiki/File:Elps-slr.svg

Polar Differentiation

Differential calculus

We have the following formulas:

To find the Cartesian slope of the tangent line to a polar curver(θ) at any given point, thecurve is first expressed as a system of parametric equations.

Differentiating both equations with respect toθ yields

Dividing the second equation by the first yields the Cartesian slope of the tangent line to thecurve at the point (r, r(θ)):

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Polar Integration

Introduction

Integrating a polar equation requires a different approach than integration under the Cartesiansystem, hence yielding a different formula, which is not as straightforward as integrating thefunctionf(x).

Proof

In creating the concept of integration, we used Riemann sums on rectangles to approximatethe area under the curve. However, with polar graphs, one must use triangles that start from theorigin, and have a radius ending on the curve. If you don't mind skipping the proof, this is theform to use to integrate a polar expression of the formr = f(θ):

,

where (a,f(a)) and (b,f(b)) are the ends of the curve that you wish to integrate.

Integral calculus


The integration regionR is bounded by the curver = f(θ) and the raysθ = a andθ = b.

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http://en.wikibooks.org/wiki/File:Polar_coordinates_integration_region.svg

Let R denote the region enclosed by a curver = f(θ) and the raysθ = a andθ = b, where 0< b − a < 2π. Then, the area ofR is


The regionR is approximated byn sectors (here,n = 5).This result can be found as follows. First, the interval [a,b] is divided inton subintervals,

wheren is an arbitrary positive integer. Thusθ, the length of each subinterval, is equal tob − a(the total length of the interval), divided byn, the number of subintervals. For each subinterval

, let θi be the midpoint of the subinterval, and construct a circular sector with the center at

the origin, radiusr i = f(θi), central angleδθ, and arc lengthr iδθ. The area of each constructed

sector is therefore equal to

. Hence, the total area of all of the sectors is

As the number of subintervalsn is increased, the approximation of the area continues toimprove. In the limit as

, the sum becomes the Riemann integral.

Generalization

Using Cartesian coordinates, an infinitesimal area element can be calculated asdA=

. The substitution rule for multiple integrals states that, when using other coordinates, theJacobian determinant of the coordinate conversion formula has to be considered:

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http://en.wikibooks.org/wiki/File:Polar_coordinates_integration_Riemann_sum.svg

Hence, an area element in polar coordinates can be written as

Now, a function that is given in polar coordinates can be integrated as follows:

Here,R is the same region as above, namely, the region enclosed by a curver = f(θ) and theraysθ = a andθ = b.

The formula for the area ofR mentioned above is retrieved by takingg identically equal to1.

Applications

Polar integration is often useful when the corresponding integral is either difficult or impossi-ble to do with the Cartesian coordinates. For example, let's try to find the area of the closed unit

circle. That is, the area of the region enclosed byx2 + y2 = 1.

In Cartesian

In order to evaluate this, one usually uses trigonometric substitution. By setting sinθ = x, weget both

and

.

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Putting this back into the equation, we get

In Polar

To integrate in polar coordinates, we first realize

and in order to include the whole circle,a = 0 andb = 2π.

An interesting example

A less intuitive application of polar integration yields the Gaussian integral

Try it! (Hint: multiply

and

.)

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Sequences→Calculus← Polar Integration


Basics


Series and calculus


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Exercises

1. Assume that thenth partial sum of aseriesis given by

.a) Does the series converge? If so, to what value?

b) What is the formula for thenth term of the series?

2. Find the value to which each the following series converges:a)

b)

c)

d)

3. Determine whether each the following series converges or diverges:a)

b)

c)

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d)

e)

f)

g)

4. Determine whether each the following series converges conditionally, converges abso-lutely, or diverges:

a)

b)

c)

d)

e)

f)

g)

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Hints

1.a) take a limit

b) sn = sn − 1 + an

2.a) sum of an infinite geometric series

b) sum of an infinite geometric series

c) telescoping series

d) rewrite so that all exponents aren

3.a)p-series

b) geometric series

c) limit comparison test

d) direct comparison test

e) divergence test

f) alternating series test

g) alternating series test

4.a) direct comparison test

b) alternating series test; integral test or direct comparison test

c) divergence test

d) alternating series test; limit comparison test

e) divergence test

f) ratio test

g) divergence test

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Answers only

1.a) The series converges to 2.

b)

2.a) 4

b)

c) 1

d) −1/5

3.a) converges

b) converges

c) diverges

d) diverges

e) diverges

f) converges

g) diverges

4.a) converges conditionally

b) converges conditionally

c) diverges

d) converges absolutely

e) diverges

f) converges absolutely

g) diverges

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Full solutions

1.a) The series converges to 2 since:

b)

2.a) The series is

and so is geometric with first terma = 3 and common ratior = 1/4. So

b)

c) Note that

by partial fractions. So

All but the first and last terms cancel out, so

d) The series simplifies to

and so is geometric. Thus

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http://en.wikibooks.org/wiki/HSE_Partial_Fractions

3.a) This is ap-series withp = 2. Sincep > 1, the series converges.

b) This is a geometric series with common ratior = 1/2, and so converges since |r | < 1.

c) This series can be compared to ap-series:

The

symbol means the two series are "asymptotically equivalent"—that is, they either bothconverge or both diverge because their terms behave so similarly when summed asn getsvery large. This can be shown by thelimit comparison test:

Since the limit is positive and finite, the two series either both converge or both diverge.The simpler series diverges because it is ap-series withp = 1 (harmonic series), and so theoriginal series diverges by the limit comparison test.

d) This series can be compared to a smallerp-series:

Thep-series diverges sincep = 1 (harmonic series), so the larger series diverges by theappropriatedirect comparison test.

e)solution to come

f) solution to come

g) solution to come

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http://en.wikibooks.org/wiki/Calculus/Series#Limit_comparison_test

http://en.wikibooks.org/wiki/Calculus/Series#Direct_comparison_tests

4.a)solution to come

b) solution to come

c) solution to come

d) solution to come

e)solution to come

f) solution to come

g) solution to come

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Sequences

A sequenceis an ordered list of objects (or events). Like a set, it contains members (alsocalledelementsor terms), and the number of terms (possibly infinite) is called thelengthof thesequence. Unlike a set, order matters, and the exact same elements can appear multiple times atdifferent positions in the sequence.

For example, (C, R, Y) is a sequence of letters that differs from (Y, C, R), as the orderingmatters. Sequences can befinite, as in this example, orinfinite, such as the sequence of all evenpositive integers (2, 4, 6,...).


An infinite sequence of real numbers (in blue). This sequence is neither increasing, nor decreas-ing, nor convergent. It is however bounded.

Examples and notation

There are various and quite different notions of sequences in mathematics, some of which(e.g., exact sequence) are not covered by the notations introduced below.

A sequence may be denoted (a1, a2, ...). For shortness, the notation (an) is also used.

A more formal definition of afinite sequencewith terms in a setS is a function from {1, 2,...,n} to Sfor somen ≥ 0. An infinite sequencein Sis a function from {1, 2, ...} (the set of natu-ral numbers without 0) toS.

Sequences may also start from 0, so the first term in the sequence is thena0.

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http://en.wikibooks.org/wiki/File:Cauchy_sequence_illustration2.png

A finite sequence is also called an n-tuple. Finite sequences include theempty sequence( )that has no elements.

A function from all integers into a set is sometimes called abi-infinite sequence, since itmay be thought of as a sequence indexed by negative integers grafted onto a sequence indexedby positive integers.

Types and properties of sequences

A subsequence of a given sequence is a sequence formed from the given sequence by delet-ing some of the elements without disturbing the relative positions of the remaining elements.

If the terms of the sequence are a subset of an ordered set, then amonotonically increasingsequence is one for which each term is greater than or equal to the term before it; if each term isstrictly greater than the one preceding it, the sequence is calledstrictly monotonically increasing.A monotonically decreasing sequence is defined similarly. Any sequence fulfilling the monotonic-ity property is called monotonic ormonotone. This is a special case of the more general notionof monotonic function.

The termsnon-decreasingandnon-increasingare used in order to avoid any possible confu-sion with strictly increasing and strictly decreasing, respectively. If the terms of a sequence areintegers, then the sequence is an integer sequence. If the terms of a sequence are polynomials,then the sequence is a polynomial sequence.

If Sis endowed with a topology, then it becomes possible to considerconvergenceof an infi-nite sequence inS. Such considerations involve the concept of the limit of a sequence.

Sequences in analysis

In analysis, when talking about sequences, one will generally consider sequences of the form

or

which is to say, infinite sequences of elements indexed by natural numbers. (It may be conve-nient to have the sequence start with an index different from 1 or 0. For example, the sequence

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defined byxn = 1/log(n) would be defined only forn ≥ 2. When talking about such infinite se-

quences, it is usually sufficient (and does not change much for most considerations) to assumethat the members of the sequence are defined at least for all indices large enough, that is, greaterthan some givenN.)

The most elementary type of sequences are numerical ones, that is, sequences of real orcomplex numbers.

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Series

Introduction

An arithmetic series is the sum of a sequence of terms with a common difference. A geomet-ric series is the sum of terms with a common ratio. For example, an interesting series which ap-pears in many practical problems in science, engineering, and mathematics is the geometric series

r + r2 + r3 + r4 + ... where the ... indicates that the series continues indefinitely. A common wayto study a particular series (following Cauchy) is to define a sequence consisting of the sum ofthe firstn terms. For example, to study the geometric series we can consider the sequence whichadds together the first n terms:

Generally by studying the sequence of partial sums we can understand the behavior of theentire infinite series.

Two of the most important questions about a series are

• Does it converge?• If so, what does it converge to?

For example, it is fairly easy to see that forr > 1, the geometric seriesSn(r) will not converge

to a finite number (i.e., it will diverge to infinity). To see this, note that each time we increase

the number of terms in the series,Sn(r) increases byrn + 1, sincern + 1 > 1 for all r > 1 (as we de-

fined), Sn(r) must increase by a number greater than one every term. When increasing the sum

by more than one for every term, it will diverge.

Perhaps a more surprising and interesting fact is that for |r | < 1,Sn(r) will converge to a fi-

nite value. Specifically, it is possible to show that

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Indeed, consider the quantity

Since

as

for | r | < 1, this shows that

as

. The quantity 1 -r is non-zero and doesn't depend onn so we can divide by it and arrive atthe formula we want.

We'd like to be able to draw similar conclusions about any series.

Unfortunately, there is no simple way to sum a series. The most we will be able to do inmost cases is determine if it converges. The geometric and the telescoping series are the onlytypes of series in which we can easily find the sum of.

Convergence

It is obvious that for a series to converge, thean must tend to zero (because sum of any infi-

nite terms is infinity, except when the sequence approaches 0), but even if the limit of the se-quence is 0, is not sufficient to say it converges.

Consider the harmonic series, the sum of 1/n, and group terms

As m tends to infinity, so does this final sum, hence the series diverges.

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We can also deduce something about how quickly it diverges. Using the same grouping ofterms, we can get an upper limit on the sum of the first so many terms, thepartial sums.

or

and the partial sums increase like logm, very slowly.

Notice that to discover this, we compared the terms of the harmonic series with a series weknew diverged.

This is aconvergence test(also known as the direct comparison test) we can apply to anypair of series.

• If bn converges and |an|≤|bn| thenan converges.• If bn diverges and |an|≥|bn| thenan diverges.

There are many such tests, the most important of which we'll describe in this chapter.

Absolute convergence

Theorem: If the series ofabsolutevalues,

, converges, then so does the series

We say such a seriesconverges absolutely.

Proof:

Let ε > 0

According to the Cauchy criterion for series convergence, existsN so that for allN < m,n:

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We know that:

And then we get:

Now we get:

Which is exactly the Cauchy criterion for series convergence.

Q.E.D

The converse does not hold.The series 1-1/2+1/3-1/4 ... converges, even though the seriesof its absolute values diverges.

A series like this that converges, but not absolutely, is said toconverge conditionally.

If a series converges absolutely, we can add terms in any order we like. The limit will stillbe the same.

If a series converges conditionally, rearranging the terms changes the limit. In fact, we canmake the series converge to any limit we like by choosing a suitable rearrangement.

E.g, in the series 1-1/2+1/3-1/4 ..., we can add only positive terms until the partial sum ex-ceeds 100, subtract 1/2, add only positive terms until the partial sum exceeds 100, subtract 1/4,and so on, getting a sequence with the same terms that converges to 100.

This makes absolutely convergent series easier to work with. Thus, all but one of conver-gence tests in this chapter will be for series all of whose terms are positive, which must be abso-lutely convergent or divergent series. Other series will be studied by considering the correspond-ing series of absolute values.

Ratio test

For a series with termsan, if

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then

• the series converges (absolutely) ifr<1• the series diverges ifr>1• the series could do either ifr=1, so the test is not conclusive in this case.

E.g, suppose

then

so this series converges.

Integral test

If f(x) is a monotonically decreasing, always positive function, then the series

converges ifand only ifthe integral

converges.

E.g, considerf(x)=1/xp, for a fixedp.

• If p=1 this is the harmonic series, which diverges.• If p<1 each term is larger than the harmonic series, so it diverges.• If p>1 then

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The integral converges, forp>1, so the series converges.

We can prove this test works by writing the integral as

and comparing each of the integrals with rectangles, giving the inequalities

Applying these to the sum then shows convergence.

Limit comparison test

Given an infinite series

with positive terms only, if one can find another infinite series

with positive terms for which

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for a positive and finiteL (i.e., the limit exists and is not zero), then the two series eitherboth converge or both diverge. That is,

•converges if

converges, and•

diverges if

diverges.

Example:

For largen, the terms of this series are similar to, but smaller than, those of the harmonicseries. We compare the limits.

so this series diverges.

Alternating series

If the signs of thean alternate,

then we call this analternatingseries.The series sum converges provided that

and

.

The error in a partial sum of an alternating series is smaller than the first omitted term.

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Geometric series

The geometric series can take either of the following forms

or

As you have seen at the start, the sum of the geometric series is

.

Telescoping series

Expanding (or "telescoping") this type of series is informative. If we expand this series, weget:

Additive cancellation leaves:

Thus,

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and all that remains is to evaluate the limit.

There are other tests that can be used, but these tests are sufficient for all commonly encoun-tered series.

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Taylor series

Taylor Series


sin(x)and Taylor approximations, polynomials of degree1, 3, 5, 7, 9, 11and13.

TheTaylor seriesof an infinitely oftendifferentiablereal (or complex)function f definedon anopen interval(a-r, a+r) is thepower series

Here,n! is thefactorialof n andf (n)(a) denotes thenth derivativeof f at the pointa. If thisseries converges for everyx in the interval (a-r, a+r) and the sum is equal tof(x), then the func-tion f(x) is calledanalytic. To check whether the series converges towardsf(x), one normallyuses estimates for the remainder term ofTaylor's theorem. A function is analytic if and only if apower seriesconverges to the function; the coefficients in that power series are then necessarilythe ones given in the above Taylor series formula.

If a = 0, the series is also called aMaclaurin series.

The importance of such a power series representation is threefold. First, differentiation andintegration of power series can be performed term by term and is hence particularly easy. Second,an analytic function can be uniquely extended to aholomorphic functiondefined on an open disk

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http://en.wikibooks.org/wiki/File:Sintay.png

http://en.wikipedia.org/wiki/differentiable

http://en.wikipedia.org/wiki/function_%28mathematics%29

http://en.wikipedia.org/wiki/interval_%28mathematics%29

http://en.wikipedia.org/wiki/power_series

http://en.wikipedia.org/wiki/Factorial

http://en.wikipedia.org/wiki/Derivative

http://en.wikipedia.org/wiki/Taylor%27s_theorem

http://en.wikipedia.org/wiki/Power_series

http://en.wikipedia.org/wiki/Power_series

http://en.wikipedia.org/wiki/Holomorphic_function

in thecomplex plane, which makes the whole machinery ofcomplex analysisavailable. Third,the (truncated) series can be used to approximate values of the function near the point of expan-sion.


The functione-1/x²is not analytic: the Taylor series is 0, although the function is not.

Note that there are examples ofinfinitely often differentiable functionsf(x) whose Taylorseries converge, but arenot equal tof(x). For instance, for the function defined piecewise bysaying thatf(x) = exp(−1/x²) if x ≠ 0 andf(0) = 0, all the derivatives are zero atx = 0, so theTaylor series off(x) is zero, and itsradius of convergenceis infinite, even though the functionmost definitely is not zero. This particular pathology does not afflictcomplex-valued functionsof a complex variable. Notice that exp(−1/z²) does not approach 0 asz approaches 0 along theimaginary axis.

Some functions cannot be written as Taylor series because they have asingularity; in thesecases, one can often still achieve a series expansion if one allows also negative powers of thevariablex; seeLaurent series. For example,f(x) = exp(−1/x²) can be written as a Laurent series.

TheParker-Sockacki theoremis a recent advance in finding Taylor series which are solutionsto differential equations. This theorem is an expansion on thePicard iteration.

Derivation/why this works

If a function f(x) is written as a infinite power series, it will look like this:

f(x)=c0(x-a)0+c1(x-a)1+c2(x-a)2+c3(x-a)3+c4(x-a)4+c5(x-a)5+c6(x-a)6+c7(x-

a)7+...

where a is half the radius of convergence and c0,c1,c2,c3,c4... are coefficients. If we substitute

a for x:

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http://en.wikipedia.org/wiki/Complex_number

http://en.wikipedia.org/wiki/Complex_analysis

http://en.wikibooks.org/wiki/File:Expinvsq5.svg

http://en.wikipedia.org/wiki/Infinitely_often_differentiable_function

http://en.wikipedia.org/wiki/Radius_of_convergence

http://en.wikipedia.org/wiki/Complex_number

http://en.wikipedia.org/wiki/Singularity_%28mathematics%29

http://en.wikipedia.org/wiki/Laurent_series

http://www.math.jmu.edu/jim/picard.html

http://en.wikipedia.org/wiki/Differential_equations

http://en.wikipedia.org/wiki/Picard_iteration

f(a)=c0

If we differentiate:

f´(x)=1c1(x-a)0+2c2(x-a)1+3c3(x-a)2+4c4(x-a)3+5c5(x-a)4+6c6(x-a)5+7c7(x-

a)6+...

If we substitute a for x:

f´(a)=1c1

If we differentiate:

f´´(x)=2c2+3*2*c3(x-a)1+4*3*c4(x-a)2+5c5*4*(x-a)3+6*5*c6(x-a)4+7c7*6*(x-

a)5+...

If we substitute a for x:

f´´(a)=2c2

Extrapolating:

n!cn=fn(a)

where f0(x)=f(x) and f1(x)=f´(x) and so on.We can actually go ahead and say that the powerapproximation of f(x) is:

f(x)=Sn=0¥((fn(a)/n!)*(x-a)n)

<this needs to be improved>

List of Taylor series

Several important Taylor series expansions follow. All these expansions are also valid forcomplex argumentsx.

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Exponential functionandnatural logarithm:

Geometric series:

Binomial series:

Trigonometric functions:

Hyperbolic functions:

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http://en.wikipedia.org/wiki/Exponential_function

http://en.wikipedia.org/wiki/Natural_logarithm

http://en.wikipedia.org/wiki/Geometric_series

http://en.wikipedia.org/wiki/Binomial_theorem

http://en.wikipedia.org/wiki/Trigonometric_function

http://en.wikipedia.org/wiki/Hyperbolic_function

Lambert's W function:

The numbersBk appearing in the expansions of tan(x) and tanh(x) are theBernoulli numbers.

The C(α,n) in the binomial expansion are thebinomial coefficients. TheEk in the expansion of

sec(x) areEuler numbers.

Multiple dimensions

The Taylor series may be generalized to functions of more than one variable with

History

The Taylor series is named for mathematicianBrook Taylor, who first published the powerseries formula in 1715.

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http://en.wikipedia.org/wiki/Lambert%27s_W_function

http://en.wikipedia.org/wiki/Bernoulli_numbers

http://en.wikipedia.org/wiki/Binomial_coefficient

http://en.wikipedia.org/wiki/Euler_numbers

http://en.wikipedia.org/wiki/Brook_Taylor

Constructing a Taylor Series

Several methods exist for the calculation of Taylor series of a large number of functions.One can attempt to use the Taylor series as-is and generalize the form of the coefficients, or onecan use manipulations such as substitution, multiplication or division, addition or subtraction ofstandard Taylor series (such as those above) to construct the Taylor series of a function, by virtueof Taylor series being power series. In some cases, one can also derive the Taylor series by repeat-edly applyingintegration by parts. The use ofcomputer algebra systemsto calculate Taylor seriesis common, since it eliminates tedious substitution and manipulation.

Example 1

Consider the function

for which we want a Taylor series at 0.

We have for the natural logarithm

and for the cosine function

We can simply substitute the second series into the first. Doing so gives

Expanding by usingmultinomial coefficientsgives the required Taylor series. Note that co-sine and thereforef are even functions, meaning thatf(x) = f( − x), hence the coefficients of the

odd powersx, x3, x5, x7 and so on have to be zero and don't need to be calculated. The first fewterms of the series are

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http://en.wikipedia.org/wiki/Integration_by_parts

http://en.wikipedia.org/wiki/Computer_algebra_system

http://en.wikipedia.org/wiki/Multinomial_coefficient

The general coefficient can be represented usingFaà di Bruno's formula. However, thisrepresentation does not seem to be particularly illuminating and is therefore omitted here.

Example 2

Suppose we want the Taylor series at 0 of the function

We have for the exponential function

and, as in the first example,

Assume the power series is

Then multiplication with the denominator and substitution of the series of the cosine yields

Collecting the terms up to fourth order yields

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http://en.wikipedia.org/wiki/Fa�_di_Bruno%27s_formula

Comparing coefficients with the above series of the exponential function yields the desiredTaylor series

Convergence

Generalized Mean Value Theorem



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http://en.wikibooks.orghttp://en.wikibooks.org/w/index.php?title=Calculus/Taylor_series&action=edit

Power series

The study ofpower seriesis aimed at investigating series which can approximate somefunction over a certain interval.

Motivations

Elementary calculus (differentiation) is used to obtain information on a line which touchesa curve at one point (i.e. a tangent). This is done by calculating the gradient, or slope of the curve,at a single point. However, this does not provide us with reliable information on the curve's actualvalueat given points in a wider interval. This is where the concept of power series becomesuseful.

An example

Consider the curve ofy = cos(x), about the pointx = 0. A naïve approximation would be theline y = 1. However, for a more accurate approximation, observe that cos(x) looks like an invertedparabola aroundx = 0 - therefore, we might think about which parabola could approximate theshape of cos(x) near this point. This curve might well come to mind:

In fact, this is the best estimate for cos(x) which uses polynomials of degree 2 (i.e. a highest

term ofx2) - but how do we know this is true? This is the study of power series: finding optimalapproximations to functions using polynomials.

Definition

A power seriesis aseriesof the form

a0x0 + a1x

1 + ... +anxn

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or, equivalently,

Radius of convergence

When using a power series as an alternative method of calculating a function's value, theequation

can only be used to studyf(x) where the power series converges - this may happen for a finiterange, or for allreal numbers.

The size of the interval (around its center) in which the power series converges to the func-tion is known as theradius of convergence.

An example

(a geometric series)

this converges when |x | < 1, the range -1 <x < +1, so the radius of convergence - centeredat 0 - is1. It should also be observed that at theextremitiesof the radius, that is wherex = 1 andx = -1, the power series does not converge.

Another example

Using theratio test, this series converges when the ratio of successive terms is less than one:

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http://en.wikipedia.org/wiki/Real_number

http://en.wikibooks.org/w/index.php?title=Calculus/Ratio_test&action=edit&redlink=1

or

which is always true - therefore, this power series has an infinite radius of convergence. Ineffect, this means that the power series canalwaysbe used as a valid alternative to the original

function, ex.

Abstraction

If we use the ratio test on an arbitrary power series, we find it converges when

and diverges when

The radius of convergence is therefore

If this limit diverges to infinity, the series has an infinite radius of convergence.

Differentiation and Integration

Within its radius of convergence, a power series can be differentiated and integrated termby term.

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Both the differential and the integral have the same radius of convergence as the originalseries.

This allows us to sum exactly suitable power series. For example,

This is a geometric series, which converges for |x | < 1. Integrating both sides, we get

which will also converge for |x | < 1. Whenx = -1 this is the harmonic series, whichdi-verges'; whenx = 1 this is an alternating series with diminishing terms, whichconvergesto ln2 - this is testing the extremities.

It also lets us write power series for integrals we cannot do exactly such as the error function:

The left hand side can not be integrated exactly, but the right hand side can be.

This gives us a power series for the sum, which has an infinite radius of convergence, lettingus approximate the integral as closely as we like.

Further reading

• "Decoding the Rosetta Stone"article by Jack W. Crenshaw 2005-10-12

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http://embedded.com/showArticle.jhtml?articleID=172300631

Multivariable and differentialcalculus

Vectors→Calculus← Sequences and Series/Exer-cises

Multivariable and differen-tial calculus


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http://en.wikibooks.org/wiki/File:Triple_Integral_Example_2.svg

• Vectors


• Lines and Planes in Space


• Multivariable Calculus


• Ordinary Differential Equations


• Partial Differential Equations


• Exercises

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Vectors

Lines and Planes in Space→Calculus← Multivariable and differen-tial calculus

Vectors

Two-Dimensional Vectors

Introduction

In most mathematics courses up until this point, we deal withscalars. These are quantitieswhich only need one number to express. For instance, the amount of gasoline used to drive tothe grocery store is a scalar quantity because it only needs one number: 2 gallons.

In this unit, we deal withvectors. A vector is adirected line segment-- that is, a line seg-ment that points one direction or the other. As such, it has aninitial point and aterminal point .The vector starts at the initial point and ends at the terminal point, and the vector points towardsthe terminal point. A vector is drawn as a line segment with an arrow at the terminal point:


The same vector can be placed anywhere on the coordinate plane and still be the same vector-- the only two bits of information a vector represents are themagnitudeand thedirection. Themagnitude is simply the length of the vector, and the direction is the angle at which it points.Since neither of these specify a starting or endinglocation, the same vector can be placed any-where. To illustrate, all of the line segments below can be defined as the vector with magnitude

and angle 45 degrees:

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http://en.wikibooks.org/wiki/File:VectorCalculations_1.png


It is customary, however, to place the vector with the initial point at the origin as indicatedby the black vector. This is called thestandard position.

Component Form

In standard practice, we don't express vectors by listing the length and the direction. We in-stead usecomponent form, which lists the height (rise) and width (run) of the vectors. It iswritten as follows:

Other ways of denoting a vector in component form include:

and

From the diagram we can now see the benefits of the standard position: the two numbers forthe terminal point's coordinates are the same numbers for the vector's rise and run. Note that wenamed this vectoru. Just as you can assign numbers to variables in algebra (usually x, y, and z),you can assign vectors to variables in calculus. The letters u, v, and w are usually used, and eitherboldface or an arrow over the letter is used to identify it as a vector.

When expressing a vector in component form, it is no longer obvious what the magnitudeand direction are. Therefore, we have to perform some calculations to find the magnitude anddirection.

Magnitude

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whereux is the width, or run, of the vector;uy is the height, or rise, of the vector. You should

recognize this formula as the Pythagorean theorem. It is -- the magnitude is the distance betweenthe initial point and the terminal point.

The magnitude of a vector can also be called the norm.

Direction

whereθ is the direction of the vector. This formula is simply the tangent formula for righttriangles.

Vector Operations

For these definitions, assume:

Vector Addition

Vector Addition is often calledtip-to-tail addition, because this makes it easier to remember.

The sum of the vectors you are adding is called the resultant vector, and is the vector drawnfrom the initial point (tip) of the first vector to the terminal point (tail) of the second vector. Al-though they look like the arrows, the pointy bit is the tail, not the tip. (Imagine you were walkingthe direction the vector was pointing... you would start at the flat end (tip) and walk toward thepointy end.)

It looks like this:

(Notice, the black lined vector is the sum of the two dotted line vectors!)

Numerically:

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Or more generally:

Scalar Multiplication

Graphically, multiplying a vector by a scalar changes only the magnitude of the vector bythat same scalar. That is, multiplying a vector by 2 will "stretch" the vector to twice its originalmagnitude, keeping the direction the same.


Numerically, you calculate the resultant vector with this formula:

, where c is a constant scalar.As previously stated, the magnitude is changed by the same constant:

Since multiplying a vector by a constant results in a vector in the same direction, we canreason that two vectors are parallel if one is a constant multiple of the other -- that is, that

if

for some constant c.

We can also divide by a non-zero scalar by instead multiplying by the reciprocal, as withdividing regular numbers:

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Dot Product

The dot product, sometimes confusingly called the scalarproduct, of two vectors is givenby:

or which is equivalent to:

whereθ is the angle difference between the two vectors. This provides a convenient way offinding the angle between two vectors:

Applications of Scalar Multiplication and Dot Product

Unit Vectors

A unit vector is a vector with a magnitude of 1. Theunit vector of u is a vector in the samedirection as

, but with a magnitude of 1:


The process of finding the unit vector of u is callednormalization. As mentioned inscalarmultiplication , multiplying a vector by constant C will result in the magnitude being multipliedby C. We know how to calculate the magnitude of

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. We know that dividing a vector by a constant will divide the magnitude by that constant.Therefore, if that constant is the magnitude, dividing the vector by the magnitude will result ina unit vector in the same direction as

:

, where

is the unit vector of

Standard Unit Vectors

A special case ofUnit Vectorsare theStandard Unit Vectorsi andj : i points one unit directlyright in the x direction, andj points one unit directly up in the y direction:

Using the scalar multiplication and vector addition rules, we can then express vectors in adifferent way:

If we work that equation out, it makes sense. Multiplying x byi will result in the vector

. Multiplying y by j will result in the vector

. Adding these two together will give us our original vector,

. Expressing vectors using i and j is calledstandard form.

Projection and Decomposition of Vectors

Sometimes it is necessary to decompose a vector

into two components: one component parallel to a vector

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, which we will call

; and one component perpendicular to it,

.


Since the length of

is (

), it is straightforward to write down the formulas for

and

:

and

Length of a vector

The length of a vector is given by the dot product of a vector with itself, andθ = 0deg:

Perpendicular vectors

If the angleθ between two vectors is 90 degrees or

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(if the two vectors are orthogonal to each other), that is the vectors are perpendicular, then thedot product is 0. This provides us with an easy way to find a perpendicular vector: if you have avector

, a perpendicular vector can easily be found by either

Polar coordinates

Polar coordinates are an alternative two-dimensional coordinate system, which is often usefulwhen rotations are important. Instead of specifying the position along thexandyaxes, we specifythe distance from the origin,r, and the direction, an angleθ.

Looking at this diagram, we can see that the values ofx andy are related to those ofr andθ by the equations

Because tan-1 is multivalued, care must be taken to select the right value.

Just as for Cartesian coordinates the unit vectors that point in thex andy directions are spe-cial, so in polar coordinates the unit vectors that point in ther andθ directions are also special.

We will call these vectors

and

, pronounced r-hat and theta-hat. Putting a circumflex over a vector this way is often usedto mean the unit vector in that direction.

Again, on looking at the diagram we see,

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Three-Dimensional Coordinates and Vectors

Basic definition

Two-dimensional Cartesian coordinates as we've discussed so far can be easily extended tothree-dimensions by adding one more value: 'z'. If the standard (x,y) coordinate axes are drawnon a sheet of paper, the 'z' axis would extend upwards off of the paper.

Similar to the two coordinate axes in two-dimensional coordinates, there are threecoordi-nate planesin space. These are thexy-plane, theyz-plane, and thexz-plane. Each plane is the"sheet of paper" that contains both axes the name mentions. For instance, the yz-plane containsboth the y and z axes and is perpendicular to the x axis.

Therefore, vectors can be extended to three dimensions by simply adding the 'z' value.

To facilitate standard form notation, we add another standard unit vector:

Again, both forms (component and standard) are equivalent.

Magnitude: Magnitude in three dimensions is the same as in two dimensions, with the addi-tion of a 'z' term in the radicand.

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Three dimensions

The polar coordinate system is extended into three dimensions with two different coordinatesystems, the cylindrical and spherical coordinate systems, both of which include two-dimensionalor planar polar coordinates as a subset. In essence, the cylindrical coordinate system extendspolar coordinates by adding an additional distance coordinate, while the spherical system insteadadds an additional angular coordinate.

Cylindrical coordinates


a point plotted with cylindrical coordinatesThecylindrical coordinate systemis a coordinate system that essentially extends the two-

dimensional polar coordinate system by adding a third coordinate measuring the height of a pointabove the plane, similar to the way in which the Cartesian coordinate system is extended intothree dimensions. The third coordinate is usually denotedh, making the three cylindrical coordi-nates (r, θ, h).

The three cylindrical coordinates can be converted to Cartesian coordinates by

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http://en.wikibooks.org/wiki/File:Cylindrical_coordinates.svg

Spherical coordinates


A point plotted using spherical coordinatesPolar coordinates can also be extended into three dimensions using the coordinates (ρ, φ,

θ), whereρ is the distance from the origin,φ is the angle from the z-axis (called the colatitudeor zenith and measured from 0 to 180°) andθ is the angle from the x-axis (as in the polar coordi-nates). This coordinate system, called thespherical coordinate system, is similar to the latitudeand longitude system used for Earth, with the origin in the centre of Earth, the latitudeδ beingthe complement ofφ, determined byδ = 90° −φ, and the longitudel being measured byl = θ −180°.

The three spherical coordinates are converted to Cartesian coordinates by

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http://en.wikibooks.org/wiki/File:Spherical_Coordinates_%28Colatitude,_Longitude%29.svg

Cross Product

The cross product of two vectors is adeterminant:

and is also apseudovector.

The cross product of two vectors is orthogonal to both vectors. The magnitude of the crossproduct is the product of the magnitude of the vectors and the sin of the angle between them.

This magnitude is the area of the parallelogram defined by the two vectors.

The cross product islinear andanticommutative. For any numbersa andb,

If both vectors point in the same direction, their cross product is zero.

Triple Products

If we have three vectors we can combine them in two ways, a triple scalar product,

and a triple vector product

The triple scalar product is a determinant

If the three vectors are listed clockwise, looking from the origin, the sign of this product ispositive. If they are listed anticlockwise the sign is negative.

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http://en.wikipedia.org/wiki/determinant

http://en.wikipedia.org/wiki/pseudovector

The order of the cross and dot products doesn't matter.

Either way, the absolute value of this product is the volume of the parallelepiped defined bythe three vectors,u, v, andw

The triple vector product can be simplified

This form is easier to do calculations with.

The triple vector product isnot associative.

There are special cases where the two sides are equal, but in general the brackets matter.They must not be omitted.

Three-Dimensional Lines and Planes

We will user to denote the position of a point.

The multiples of a vector,a all lie on a line through the origin. Adding a constant vectorbwill shift the line, but leave it straight, so the equation of a line is,

This is aparametric equation. The position is specified in terms of the parameters.

Any linear combination of two vectors,a andb lies on a single plane through the origin,provided the two vectors are not colinear. We can shift this plane by a constant vector again andwrite

If we choosea andb to beorthonormalvectors in the plane (i.e unit vectors at right angles)thens andt are Cartesian coordinates for points in the plane.

These parametric equations can be extended to higher dimensions.

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Instead of giving parametric equations for the line and plane, we could use constraints. E.g,for any point in thexyplanez=0

For a plane through the origin, the single vector normal to the plane,n, is at right angle withevery vector in the plane, by definition, so

is a plane through the origin, normal ton.

For planes not through the origin we get

A line lies on the intersection of two planes, so it must obey the constraint for both planes,i.e

These constraint equations con also be extended to higher dimensions.

Vector-Valued Functions

Vector-Valued Functions are functions that instead of giving a resultant scalar value, givea resultant vector value. These aid in the create of direction and vector fields, and are thereforeused in physics to aid with visualizations of electric, magnetic, and many other fields. They areof the following form:

Introduction



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http://en.wikibooks.orghttp://en.wikibooks.org/w/index.php?title=Calculus/Vectors&action=edit

Limits, Derivatives, and Integrals

Put simply, the limit of a vector-valued function is the limit of its parts.

Proof:

Suppose

Therefore for anyε > 0 there is aφ > 0 such that

But by the triangle inequality

So

Therefore

A similar argument can be used through parts a_n(t)

Now let

again, and that for anyε>0 there is a correspondingφ>0 such 0<|t-c|<φ implies

Then

therefore!:

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From this we can then create an accurate definition of a derivative of a vector-valued func-tion:

The final step was accomplished by taking what we just did with limits.

By the Fundamental Theorem of Calculus integrals can be applied to the vector's compo-nents.

In other words: the limit of a vector function is the limit of its parts, the derivative of a vectorfunction is the derivative of its parts, and the integration of a vector function is the integrationof it parts.

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Velocity, Acceleration, Curvature, and a brief mentionof the Binormal

Assume we have a vector-valued function which starts at the origin and as its independentvariables changes the points that the vectors point at trace a path.

We will call this vector

, which is commonly known as theposition vector.

If

then represents a position and t represents time, then in model with Physics we know thefollowing:

is displacement.

where

is the velocity vector.

is the speed.

where

is the acceleration vector.

The only other vector that comes in use at times is known as the curvature vector.

The vector

used to find it is known as the unit tangent vector, which is defined as

or shorthand

.

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The vector normal

to this then is

.

We can verify this by taking the dot product

Also note that

and

and

Then we can actually verify:

Therefore

is perpendicular to

What this gives rise to is theUnit Normal Vector

of which the top-most vector is the Normal vector, but the bottom half

is known as the curvature. Since the Normal vector points toward the inside of a curve, thesharper a turn, the Normal vector has a large magnitude, therefore the curvature has a small value,and is used as an index in civil engineering to reflect the sharpness of a curve (clover-leaf high-ways, for instance).

The only other thing not mentioned is the Binormal that occurs in 3-d curves

, which is useful in creating planes parallel to the curve.

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Lines and Planes in Space

Multivariable calculus→Calculus← Vectors

Lines and Planes in Space

Introduction

For many practical applications, for example for describing forces in physics and mechanics,you have to work with the mathematical descriptions oflinesandplanesin 3-dimensional space.


Line in Space

A line in space is defined by two points in space, which I will callP1 andP2. Let

be the vector from the origin toP1, and

the vector from the origin toP2. Given these two points, every other pointP on the line can

be reached by

where

is the vector fromP1 andP2:

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Plane in Space

The same idea can be used to describe a plane in 3-dimensional space, which is uniquelydefined by three points (which do not lie on a line) in space (P1,P2,P3). Let

be the vectors from the origin toPi. Then

with:

Note that the starting point does not have to be

, but can be any point in the plane. Similarly, the only requirement on the vectors

and

is that they have to be two non-collinear vectors in our plane.

Vector Equation (of aPlane in Space, or of aLine in a Plane)

An alternative representation of a Plane in Space is obtained by observing that a plane isdefined by a pointP1 in that plane and a direction perpendicular to the plane, which we denote

with the vector

. As above, let

describe the vector from the origin toP1, and

the vector from the origin to another pointP in the plane. Since any vector that lies in theplane is perpendicular to

, thevector equationof the plane is given by

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In 2 dimensions, the same equation uniquely describes a Line.

Scalar Equation (of aPlane in Space, or of aLine in a Plane)

If we express

and

through their components

writing out the scalar product for

provides us with thescalar equationfor a plane in space:

ax+ by+ cz= dwhere

.

In 2d space, the equivalent steps lead to the scalar equation for a line in a plane:

ax+ by= c

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Multivariable calculus

Ordinary differential equations→

Calculus← Lines and Planes in Space

Multivariable calculus

A Wikibookian believes this page should be split into smaller pages with a narrowersubtopic.You can help by splitting this big page into smaller ones. Please make sure to follow thenaming policy. Dividing books into smaller modules can provide more focus and alloweach one to do one thing well, which benefits everyone.You can ask for help in dividing this book in theassistance reading room.

In your previous study of calculus, we have looked at functions and their behavior. Most ofthese functions we have examined have been all in the form

f(x) : R → R,

and only occasional examination of functions of two variables. However, the study of func-tions ofseveralvariables is quite rich in itself, and has applications in several fields.

We write functions of vectors - many variables - as follows:

f : Rm → Rn

andf(x) for the function that maps a vector inRm to a vector inRn.

Before we can do calculus inRn, we must familiarize ourselves with the structure ofRn. We

need to know which properties ofR can be extended toRn

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http://en.wikibooks.org/wiki/Wikibooks:NP


http://en.wikibooks.org/wiki/Wikibooks:Reading_room/Assistance

Topology inRn

We are already familiar with the nature of the regular real number line, which is the setR,

and the two-dimensional plane,R2. This examination oftopologyin Rn attempts to look at a

generalization of the nature ofn-dimensional spaces -R, or R23, or Rn.

Lengths and distances

If we have a vector inR2, we can calculate its length using the Pythagorean theorem. Forinstance, the length of the vector (2, 3) is

We can generalize this toRn. We define a vector's length, written |x|, as the square root ofthe sum of the squares of each of its components. That is, if we have a vectorx=(x1,...,xn),

Now that we have established some concept of length, we can establish the distance betweentwo vectors. We define this distance to be the length of the two vectors' difference. We write thisdistanced(x, y), and it is

This distance function is sometimes referred to as ametric. Other metrics arise in differentcircumstances. The metric we have just defined is known as theEuclideanmetric.

Open and closed balls

In R, we have the concept of aninterval, in that we choose a certain number of other pointsabout some central point. For example, the interval [-1, 1] is centered about the point 0, and in-cludes points to the left and right of zero.

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In R2 and up, the idea is a little more difficult to carry on. ForR2, we need to consider points

to the left, right, above, and below a certain point. This may be fine, but forR3 we need to includepoints in more directions.

We generalize the idea of the interval by considering all the points that are a given, fixed

distance from a certain point - now we know how to calculate distances inRn, we can make ourgeneralization as follows, by introducing the concept of anopen balland aclosed ballrespective-ly, which are analogous to the open and closed interval respectively.

anopen ball

is a set in the form {x ∈ Rn|d(x, a) < r}

aclosed ball

is a set in the form {x ∈ Rn|d(x, a) ≤ r}

In R, we have seen that the open ball is simply an open interval centered about the point

x=a. In R2 this is a circle with no boundary, and inR3 it is a sphere with no outer surface. (Whatwould the closed ball be?)

Boundary points

If we have some area, say a field, then the common sense notion of theboundaryis the points'next to' both the inside and outside of the field. For a set, S, we can define this rigorously bysaying the boundary of the set contains all those points such that we can find points both insideand outside the set. We call the set of such points∂S

Typically, when it exists the dimension of∂S is one lower than the dimension of S. e.g theboundary of a volume is a surface and the boundary of a surface is a curve.

This isn't always true; but it is true of all the sets we will be using.

A setS is boundedif there is some positive number such that we can encompass this set bya closed ball about0. --> if every point in it is within a finite distance of the origin, i.e there existssomer>0 such thatx is in S implies |x|<r.

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Curves and parameterizations

If we have a functionf : R → Rn, we say thatf's image (the set {f(t) | t ∈ R} - or some subset

of R) is acurvein Rn andf is its parametrization.

Parameterizations are not necessarily unique - for example,f(t) = (cost, sin t) such thatt ∈[0, 2π) is one parametrization of the unit circle, andg(t) = (cosat, sinat) such thatt ∈ [0, 2π/a)is a whole family of parameterizations of that circle.

Collision and intersection points

Say we have two different curves. It may be important to consider

• when the two curves cross each other - where theyintersect• when the two curves hit each other at the same time - where theycollide.

Intersection points

Firstly, we have two parameterizationsf(t) andg(t), and we want to find out when they inter-sect, this means that we want to know when the function values of each parametrization are thesame. This means that we need to solve

f(t) = g(s)

because we're seeking the function values independent of the times they intersect.

For example, if we havef(t) = (t, 3t) andg(t) = (t, t2), and we want to find intersection points:

f(t) = g(s)

(t, 3t) = (s, s2)

t = s and 3t = s2

with solutions (t, s) = (0, 0) and (3, 3)

So, the two curves intersect at the points (0, 0) and (3, 3).

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Collision points

However, if we want to know when the points "collide", withf(t) andg(t), we need to knowwhen both the function valuesandthe times are the same, so we need to solve instead

f(t) = g(t)

For example, using the same functions as before,f(t) = (t, 3t) andg(t) = (t, t2), and we wantto find collision points:

f(t) = g(t)

(t, 3t) = (t, t2)

t = t and 3t = t2

which gives solutionst = 0, 3 So the collision points are (0, 0) and (3, 3).

We may want to do this to actually model physical problems, such as in ballistics.

Continuity and differentiability

If we have a parametrizationf : R → Rn, which is built up out ofcomponent functionsin theform f(t) = (f1(t),...,fn(t)), f is continuous if and only if each component function is also.

In this case the derivative off(t) is

ai = (f1′(t),...,fn′(t)). This is actually a specific consequence of a more general fact we willsee later.

Tangent vectors

Recall in single-variable calculus that on a curve, at a certain point, we can draw a line thatis tangent to that curve at exactly at that point. This line is called atangent. In the several variablecase, we can do something similar.

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We can expect thetangent vectorto depend onf′(t) and we know that a line is its own tan-gent, so looking at a parametrised line will show us precisely how to define the tangent vectorfor a curve.

An arbitrary line isf(t)=at+b, with :fi(t)=ait+bi, so

fi′(t)=ai and

f′(t)=a, which is the direction of the line, its tangent vector.

Similarly, for any curve, the tangent vector isf′(t).

Angle between curves


We can then formulate the concept of theanglebetween two curves by considering the anglebetween the two tangent vectors. If two curves, parametrized byf1 andf2 intersect at some point,

which means that

f1(s)=f2(t)=c,

the angle between these two curves atc is the angle between the tangent vectorsf1′(s) and

f2′(t) is given by

Tangent lines

With the concept of the tangent vector as being analogous to being the gradient of the linein the one variable case, we can form the idea of thetangent line. Recall that we need a point onthe line and its direction.

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http://en.wikibooks.org/wiki/File:Inner-product-angle.png

If we want to form the tangent line to a point on the curve, sayp, we have the direction ofthe linef′(p), so we can form the tangent line

x(t)=p+t f′(p)

Different parameterizations

One such parametrization of a curve is not necessarily unique. Curves can have several differ-ent parametrizations. For example, we already saw that the unit circle can be parametrized byg(t) = (cosat, sinat) such thatt ∈ [0, 2π/a).

Generally, iff is one parametrization of a curve, andg is another, with

f(t0) = g(s0)

there is a functionu(t) such thatu(t0)=s0, andg'(u(t)) = f(t) neart0.

This means, in a sense, the functionu(t) "speeds up" the curve, but keeps the curve's shape.

Surfaces

A surface in space can be described by the image of a functionf : R2→ Rn. f is said to be

the parametrization of that surface.

For example, consider the function

f(α, β) = α(2,1,3)+β(-1,2,0)

This describes an infinite plane inR3. If we restrictα andβ to some domain, we get a paral-

lelogram-shaped surface inR3.

Surfaces can also be described explicitly, as the graph of a functionz = f(x, y) which has astandard parametrization asf(x,y)=(x, y, f(x,y)), or implictly, in the formf(x, y, z)=c.

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Level sets

The concept of thelevel set(or contour) is an important one. If you have a functionf(x, y,

z), a level set inR3 is a set of the form {(x,y,z)|f(x,y,z)=c}. Each of these level sets is a surface.

Level sets can be similarly defined in anyRn

Level sets in two dimensions may be familiar from maps, or weather charts. Each line repre-sents a level set. For example, on a map, each contour represents all the points where the heightis the same. On a weather chart, the contours represent all the points where the air pressure is thesame.

Limits and continuity

Before we can look at derivatives of multivariate functions, we need to look at how limitswork with functions of several variables first, just like in the single variable case.

If we have a functionf : Rm → Rn, we say thatf(x) approachesb (in Rn) asx approachesa

(in Rm) if, for all positiveε, there is a corresponding positive numberδ, |f(x)-b| < ε whenever |x-a| < δ, with x ≠ a.

This means that by making the difference betweenx anda smaller, we can make the differ-ence betweenf(x) andb as small as we want.

If the above is true, we say

• f(x) haslimit b ata•

• f(x) approachesb asx approachesa• f(x) → b asx → a

These four statements are all equivalent.

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Rules

Since this is an almost identical formulation of limits in the single variable case, many ofthe limit rules in the one variable case are the same as in the multivariate case.

For f andg, mappingRm to Rn, andh(x) a scalar function mappingRm to R, with

• f(x) → b asx → a• g(x) → c asx → a• h(x) → H asx → a

then:

•

•

and consequently

•

•

when H≠0

•

Continuity

Again, we can use a similar definition to the one variable case to formulate a definition ofcontinuity for multiple variables.

If f : Rm → Rn, f is continuous at a pointa in Rm if f(a) is defined and

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Just as for functions of one dimension, iff, g are both continuous atp, f+g, λf (for a scalar

λ), f·g, andf×g are continuous also. Ifφ : Rm → R is continus atp, φf, f/φ are too ifφ is neverzero.

From these facts we also have that ifA is some matrix which isn×m in size, withx in Rm,a functionf(x)=A x is continuous in that the function can be expanded in the formx1a1+...+xmam,

which can be easily verified from the points above.

If f : Rm → Rn which is in the formf(x) = (f1(x),...,fn(x) is continuous if and only if each of

its component functions are a polynomial or rational function, whenever they are defined.

Finally, if f is continuous atp, g is continuous atf(p), g(f(x)) is continuous atp.

Special note about limits

It is important to note that we can approach a pointin more than one direction, and thus, thedirection that we approach that point counts in our evaluation of the limit. It may be the case thata limit may exist moving in one direction, but not in another.

Differentiable functions

We will start from the one-variable definition of the derivative at a pointp, namely

Let's change above to equivalent form of

which achieved after pulling f'(p) inside and putting it over a common denominator.

We can't divide by vectors, so this definition can't be immediately extended to the multiplevariable case. Nonetheless, we don't have to: the thing we took interest in was the quotient oftwo small distances (magnitudes), not their other properties (like sign). It's worth noting that

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'other' property of vector neglected is its direction. Now we can divide by the absolute value ofa vector, so lets rewrite this definition in terms of absolute values

Another form of formula above is obtained by letingh = x − p we havex = p + h and if

, the

, so

,

whereh can be thought of as a 'small change'.

So, how can we use this for the several-variable case?

If we switch all the variables over to vectors and replace the constant (which performs alinear map in one dimension) with a matrix (which denotes also a linear map), we have

or

If this limit exists for somef : Rm → Rn, and there is a linear mapA : Rm → Rn (denoted bymatrixA which ism×n), we refer to this map as being the derivative and we write it as Dp f.

A point on terminology - in referring to the action of taking the derivative (giving the linearmapA), we write Dp f, but in referring to the matrixA itself, it is known as theJacobianmatrix

and is also writtenJp f. More on the Jacobian later.

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Properties

There are a number of important properties of this formulation of the derivative.

Affine approximations

If f is differentiable atp for x close top, |f(x)-(f(p)+A(x-p))| is small compared to |x-p|, whichmeans thatf(x) is approximately equal tof(p)+A(x-p).

We call an expression of the formg(x)+c affine, wheng(x) is linear andc is a constant.f(p)+A(x-p) is an affine approximation tof(x).

Jacobian matrix and partial derivatives

The Jacobian matrix of a function is in the form

for a f : Rm → Rn, Jp f' is am×n matrix.

The consequence of this is that iff is differentiable atp, all the partial derivatives off existatp.

However, it is possible that all the partial derivatives of a function exist at some point yetthat function is not differentiable there, so it's very important not to mix derivative (linear map)with the Jacobian (matrix) especially when cited situation arised.

Continuity and differentiability

Furthermore, if all the partial derivatives exist, and are continuous in some neighbourhoodof a pointp, thenf is differentiable atp. This has the consequence that for a functionf which hasits component functions built from continuous functions (such as rational functions, differentiablefunctions or otherwise),f is differentiable everywheref is defined.

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We use the terminologycontinuously differentiablefor a function differentiable atp whichhas all its partial derivatives existing and are continuous in some neighbourhood atp.

Rules of taking Jacobians

If f : Rm → Rn, andh(x) : Rm → R are differentiable at 'p' :

•••

Important: make sure the order is right - matrix multiplication is not commutative!

Chain rule

The chain rule for functions of several variables is as follows. Forf : Rm → Rn andg : Rn

→ Rp, andg o f differentiable atp, then the Jacobian is given by

Again, we have matrix multiplication, so one must preserve this exact order. Compositionsin one order may be defined, but not necessarily in the other way.

Alternate notations

For simplicity, we will often use various standard abbreviations, so we can write most of theformulae on one line. This can make it easier to see the important details.

We can abbreviate partial differentials with a subscript, e.g,

When we are using a subscript this way we will generally use the HeavisideD rather than∂,

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Mostly, to make the formulae even more compact, we will put the subscript on the functionitself.

If we are using subscripts to label the axes,x1, x2 …, then, rather than having two layers of

subscripts, we will use the number as the subscript.

We can also use subscripts for the components of a vector function,u=(ux, uy, uy) or

u=(u1,u2…un)

If we are using subscripts for both the components of a vector and for partial derivatives wewill separate them with a comma.

The most widely used notation is hx. Both h1 and∂1h are also quite widely used whenever

the axes are numbered. The notation∂xh is used least frequently.

We will use whichever notation best suits the equation we are working with.

Directional derivatives

Normally, a partial derivative of a function with respect to one of its variables, say,xj, takes

the derivative of that "slice" of that function parallel to thexj'th axis.

More precisely, we can think of cutting a functionf(x1,...,xn) in space along thexj'th axis,

with keeping everything but thexj variable constant.

From the definition, we have the partial derivative at a pointp of the function along this sliceas

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provided this limit exists.

Instead of the basis vector, which corresponds to taking the derivative along that axis, wecan pick a vector in any direction (which we usually take as being a unit vector), and we take thedirectional derivativeof a function as

whered is the direction vector.

If we want to calculate directional derivatives, calculating them from the limit definition is

rather painful, but, we have the following: iff : Rn → R is differentiable at a pointp, |p|=1,

There is a closely related formulation which we'll look at in the next section.

Gradient vectors

The partial derivatives of a scalar tell us how much it changes if we move along one of theaxes. What if we move in a different direction?

We'll call the scalarf, and consider what happens if we move an infintesimal directiondr=(dx,dy,dz), using the chain rule.

This is the dot product ofdr with a vector whose components are the partial derivatives off, called the gradient off

We can form directional derivatives at a pointp, in the directiond then by taking the dotproduct of the gradient withd

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.

Notice that gradf looks like a vector multiplied by a scalar. This particular combination ofpartial derivatives is commonplace, so we abbreviate it to

We can write the action of taking the gradient vector by writing this as anoperator. Recallthat in the one-variable case we can writed/dx for the action of taking the derivative with respectto x. This case is similar, but∇ acts like a vector.

We can also write the action of taking the gradient vector as:

Properties of the gradient vector

Geometry

• Gradf(p) is a vector pointing in the direction of steepest slope off. |gradf(p)| is the rateof change of that slope at that point.

For example, if we consider h(x, y)=x2+y2. The level sets ofh are concentric circles, centredon the origin, and

gradh points directly away from the origin, at right angles to the contours.

• Along a level set, (∇f)(p) is perpendicular to the level set {x|f(x)=f(p) atx=p}.

If dr points along the contours off, where the function is constant, thendf will be zero. Sincedf is a dot product, that means that the two vectors,df and gradf, must be at right angles, i.e thegradient is at right angles to the contours.

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Algebraicproperties

Like d/dx, ∇ is linear. For any pair of constants,a andb, and any pair of scalar functions,fandg

Since it's a vector, we can try taking its dot and cross product with other vectors, and withitself.

Divergence

If the vector functionu mapsRn to itself, then we can take the dot product ofu and∇. Thisdot product is called the divergence.

If we look at a vector function likev=(1+x2,xy) we can see that to the left of the origin allthev vectors are converging towards the origin, but on the right they are diverging away fromit.

Div u tells us how muchu is converging or diverging. It is positive when the vector is diverg-ing from some point, and negative when the vector is converging on that point.

Example:

Forv=(1+x2, xy), div v=3x, which is positive to the right of the origin, wherev is diverging,and negative to the left of the origin, wherev is converging.

Like grad, div is linear.

Later in this chapter we will see how the divergence of a vector function can be integratedto tell us more about the behaviour of that function.

To find the divergence we took the dot product of∇ and a vector with∇ on the left. If wereverse the order we get

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To see what this means consideri·∇ This is Dx, the partial differential in thei direction.

Similarly, u·∇ is the the partial differential in theu direction, multiplied by |u|

Curl

If u is a three-dimensional vector function onR3 then we can take its cross product with∇.This cross product is called thecurl.

Curl u tells us if the vectoru is rotating round a point. The direction of curlu is the axis ofrotation.

We can treat vectors in two dimensions as a special case of three dimensions, withuz=0 and

Dzu=0. We can then extend the definition of curlu to two-dimensional vectors

This two dimensional curl is a scalar. In four, or more, dimensions there is no vector equiva-lent to the curl.

Example:Consideru=(-y, x). These vectors are tangent to circles centred on the origin, so appear to

be rotating around it anticlockwise.

ExampleConsideru=(-y, x-z, y), which is similar to the previous example.

Thisu is rotating round the axisi+k

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Later in this chapter we will see how the curl of a vector function can be integrated to tellus more about the behaviour of that function.

Product and chain rules

Just as with ordinary differentiation, there are product rules for grad, div and curl.

• If g is a scalar andv is a vector, then

the divergence ofgv is

the curl ofgv is

• If u andv are both vectors then

the gradient of their dot product is

the divergence of their cross product is

the curl of their cross product is

We can also write chain rules. In the general case, when both functions are vectors and thecomposition is defined, we can use the Jacobian defined earlier.

whereJu is the Jacobian ofu at the pointv.

Normally J is a matrix but if either the range or the domain ofu is R1 then it becomes avector. In these special cases we can compactly write the chain rule using only vector notation.

• If g is a scalar function of a vector andh is a scalar function ofg then

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• If g is a scalar function of a vector then

This substitution can be made in any of the equations containing∇

Second order differentials

We can also consider dot and cross products of∇ with itself, whenever they can be defined.Once we know how to simplify products of two∇'s we'll know out to simplify products withthree or more.

The divergence of the gradient of a scalarf is

This combination of derivatives is theLaplacian of f. It is commmonplace in physics andmultidimensional calculus because of its simplicity and symmetry.

We can also take the Laplacian of a vector,

The Laplacian of a vector is not the same as the divergence of its gradient

Both the curl of the gradient and the divergence of the curl are always zero.

This pair of rules will prove useful.

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Integration

We have already considered differentiation of functions of more than one variable, whichleads us to consider how we can meaningfully look at integration.

In the single variable case, we interpret the definite integral of a function to mean the areaunder the function. There is a similar interpretation in the multiple variable case: for example, if

we have a paraboloid inR3, we may want to look at the integral of that paraboloid over someregion of thexyplane, which will be thevolumeunder that curve and inside that region.

Riemann sums

When looking at these forms of integrals, we look at the Riemann sum. Recall in the one-variable case we divide the interval we are integrating over into rectangles and summing the areasof these rectangles as their widths get smaller and smaller. For the multiple-variable case, we

need to do something similar, but the problem arises how to split upR2, or R3, for instance.

To do this, we extend the concept of the interval, and consider what we call an-interval. Ann-interval is a set of points in some rectangular region with sides of some fixed width in each

dimension, that is, a set in the form {x∈Rn|ai ≤ xi ≤ bi with i = 0,...,n}, and its area/size/volume

(which we simply call itsmeasureto avoid confusion) is the product of the lengths of all its sides.

So, ann-interval inR2 could be some rectangular partition of the plane, such as {(x,y) | x ∈[0,1] andy ∈ [0, 2]|}. Its measure is 2.

If we are to consider the Riemann sum now in terms of sub-n-intervals of a regionΩ, it is

wherem(Si) is the measure of the division ofΩ into k sub-n-intervalsSi, andx*i is a point

in Si. The index is important - we only perform the sum whereSi falls completely withinΩ - any

Si that is not completely contained inΩ we ignore.

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As we take the limit ask goes to infinity, that is, we divide upΩ into finer and finer sub-n-intervals, and this sum is the same no matter how we divide upΩ, we get theintegral of f overΩ which we write

f∫

Ω

For two dimensions, we may write

and likewise forn dimensions.

Iterated integrals

Thankfully, we need not always work with Riemann sums every time we want to calculatean integral in more than one variable. There are some results that make life a bit easier for us.

For R2, if we have some region bounded between two functions of the other variable (sotwo functions in the formf(x) = y, or f(y) = x), between a constant boundary (so, betweenx = aandx =b or y = a andy = b), we have

An important theorem (calledFubini's theorem) assures us that this integral is the same as

.

Order of integration

In some cases the first integral of the entire iterated integral is difficult or impossible tosolve, therefore, it can be to our advantage to change the order of integration.

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As of the writing of this, there is no set method to change an order of integration from dxdyto dydx or some other variable. Although, it is possible to change the order of integration in anx and y simple integration by simply switching the limits of integration around also, in non-simple x and y integrations the best method as of yet is to recreate the limits of the integrationfrom the graph of the limits of integration.

In higher order integration that can't be graphed, the process can be very tedious. For exam-ple, dxdydz can be written into dzdydx, but first dxdydz must be switched to dydxdz and then todydzdx and then to dzdydx (but since 3-dimensional cases can be graphed, doing this would beseemingly idiotic).

Parametric integrals

If we have a vector function,u, of a scalar parameter,s, we can integrate with respect tossimply by integrating each component ofu seperately.

Similarly, if u is given a function of vector of parameters,s, lying in Rn, integration withrespect to the parameters reduces to a multiple integral of each component.

Line integrals


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http://en.wikibooks.org/wiki/File:Line-Integral.gif

In one dimension, saying we are integrating froma to b uniquely specifies the integral.

In higher dimensions, saying we are integrating froma to b is not sufficient. In general, wemust also specify the path taken betweena andb.

We can then write the integrand as a function of the arclength along the curve, and integrateby components.

E.g, given a scalar functionh(r ) we write

whereC is the curve being integrated along, andt is the unit vector tangent to the curve.

There are some particularly natural ways to integrate a vector function,u, along a curve,

where the third possibility only applies in 3 dimensions.

Again, these integrals can all be written as integrals with respect to the arclength,s.

If the curve is planar andu a vector lieing in the same plane, the second integral can beusefully rewritten. Say,

wheret, n, andb are the tangent, normal, and binormal vectors uniquely defined by thecurve.

Then

For the 2-d curves specifiedb is the constant unit vector normal to their plane, andub is al-

ways zero.

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Therefore, for such curves,

Green's Theorem


Let C be a piecewise smooth, simple closed curve that bounds a region S on the Cartesianplane. If two function M(x,y) and N(x,y) are continuous and their partial derivatives are continu-ous, then

In order for Green's theorem to work there must be no singularities in the vector field withinthe boundaries of the curve.

Green's theorem works by summing the circulation in each infinitesimal segment of areaenclosed within the curve.

Inverting differentials

We can use line integrals to calculate functions with specified divergence, gradient, or curl.

• If gradV = u

whereh is any function of zero gradient and curlu must be zero.

• If div u = V

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http://en.wikibooks.org/wiki/File:Green%27s-theorem-simple-region.svg

wherew is any function of zero divergence.

• If curl u = v

wherew is any function of zero curl.

For example, if V=r2 then

and

so this line integral of the gradient gives the original function.

Similarly, if v=k then

Consider any curve from0 to p=(x, y', z), given byr=r (s) with r (0)=0 andr (S)=p for someS, and do the above integral along that curve.

and curlu is

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as expected.

We will soon see that these three integrals do not depend on the path, apart from a constant.

Surface and Volume Integrals

Just as with curves, it is possible to parameterise surfaces then integrate over those parame-ters without regard to geometry of the surface.

That is, to integrate a scalar functionV over a surfaceA parameterised byr ands we calcu-late

whereJ is the Jacobian of the tranformation to the parameters.

To integrate a vector this way, we integrate each component seperately.

However, in three dimensions, every surface has an associated normal vectorn, which canbe used in integration. We writedS=ndS.

For a scalar function,V, and a vector function,v, this gives us the integrals

These integrals can be reduced to parametric integrals but, written this way, it is clear thatthey reflect more of the geometry of the surface.

When working in three dimensions,dV is a scalar, so there is only one option for integralsover volumes.

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Gauss's divergence theorem


We know that, in one dimension,

Integration is the inverse of differentiation, so integrating the differential of a function returnsthe original function.

This can be extended to two or more dimensions in a natural way, drawing on the analogiesbetween single variable and multivariable calculus.

The analog ofD is ∇, so we should consider cases where the integrand is a divergence.

Instead of integrating over a one-dimensional interval, we need to integrate over an-dimen-sional volume.

In one dimension, the integral depends on the values at the edges of the interval, so we ex-pect the result to be connected with values on the boundary.

This suggests a theorem of the form,

This is indeed true, for vector fields in any number of dimensions.

This is calledGauss's theorem.

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http://en.wikibooks.org/wiki/File:Divergence_theorem.svg

There are two other, closely related, theorems for grad and curl:

•

,

•

,

with the last theorem only being valid where curl is defined.

Stokes' curl theorem


These theorems also hold in two dimensions, where they relate surface and line integrals.Gauss's divergence theorem becomes

wheres is arclength along the boundary curve and the vectorn is the unit normal to the curvethat lies in the surfaceS, i.e in the tangent plane of the surface at its boundary, which is not neces-sarily the same as the unit normal associated with the boundary curve itself.

Similarly, we get

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http://en.wikibooks.org/wiki/File:Stokes%27_Theorem.svg

,

whereC is the boundary ofS

In this case the integral does not depend on the surfaceS.

To see this, suppose we have different surfaces,S1 andS2, spanning the same curveC, then

by switching the direction of the normal on one of the surfaces we can write

The left hand side is an integral over a closed surface bounding some volumeV so we canuse Gauss's divergence theorem.

but we know this integrand is always zero so the right hand side of (2) must always be zero,i.e the integral is independent of the surface.

This means we can choose the surface so that the normal to the curve lying in the surface isthe same as the curves intrinsic normal.

Then, ifu itself lies in the surface, we can write

just as we did for line integrals in the plane earlier, and substitute this into (1) to get

This isStokes' curl theorem

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Ordinary differential equations

Partial differential equations→Calculus← Multivariable calculus

Ordinary differential equa-tions

Ordinary differential equations involve equations containing:

• variables• functions• their derivatives

and their solutions.

In studying integration, youalreadyhave considered solutions to very simple differentialequations. For example, when you look to solving

for g(x), you are really solving the differential equation

Notations and terminology

The notations we use for solving differential equations will be crucial in the ease of solubilityfor these equations.

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This document will be usingthree notations primarily:

• f' to denote the derivative of f• D f to denote the derivative off•

to denote the derivative off (for separable equations).

Terminology

Consider the differential equation

Since the equation's highest derivative is 2, we say that the differential equation is oforder2.

Some simple differential equations

A key idea in solving differential equations will be that ofintegration.

Let us consider the second order differential equation (remember that a function acts on avalue).

How would we go about solving this? It tells us that on differentiating twice, we obtain theconstant 2 so, if we integrate twice, we should obtain our result.

Integrating once first of all:

We have transformed the apparently difficult second order differential equation into a rathersimpler one, viz.

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This equation tells us that if we differentiate a function once, we get 2x + C1. If we integrate

once more, we should find the solution.

This is thesolutionto the differential equation. We will get

for all values ofC1 andC2.

The valuesC1 andC2 are related to quantities known asinitial conditions.

Why are initial conditions useful? ODEs (ordinary differential equations) are useful inmodeling physical conditions. We may wish to model a certain physical system which is initiallyat rest (so one initial condition may be zero), or wound up to some point (so an initial conditionmay be nonzero, say 5 for instance) and we may wish to see how the system reacts under suchan initial condition.

When we solve a system with given initial conditions, we substitute them after our processof integration.

Example

When we solved

say we had the initial conditions

and

. (Note, initial conditions need not occur at f(0)).

After we integrate we make substitutions:

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Without initial conditions, the answer we obtain is known as thegeneral solutionor the solu-tion to thefamily of equations. With them, our solution is known as aspecific solution.

Basic first order DEs

In this section we will considerfour main types of differential equations:

• separable• homogeneous• linear• exact

There are many other forms of differential equation, however, and these will be dealt within the next section

Separable equations

A separableequation is in the form (using dy/dx notation which will serve us greatly here)

Previously we have only dealt with simple differential equations with g(y)=1. How do wesolve such a separable equation as above?

We groupx anddx terms together, andy anddy terms together as well.

Integrating both sides with respect to y on the left hand side and x on the right hand side:

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we will obtain the solution.

Worked example

Here is a worked example illustrating the process.

We are asked to solve

Separating

Integrating

Lettingk = eC where k is a constant we obtain

which is the general solution.

Verification

This step does not need to be part of your work, but if you want to check your solution, youcan verify your answer by differentiation.

We obtained

as the solution to

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Differentiating our solution with respect to x,

And since

, we can write

We see that we obtain our original differential equation, thus our work must be correct.

Homogeneous equations

A homogeneousequation is in the form

This looks difficult as it stands, however we can utilize the substitution

so that we are now dealing with F(v) rather than F(y/x).

Now we can express y in terms of v, asy=xvand use the product rule.

The equation above then becomes, using the product rule

Then

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which is a separable equation and can be solved as above.

However let's look at a worked equation to see how homogeneous equations are solved.

Worked example

We have the equation

This does not appear to be immediately separable, but let us expand to get

Substitutingy=xvwhich is the same as substitutingv=y/x:

Now

Cancelingv from both sides

Separating

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Integrating both sides

which is our desired solution.

Linear equations

A linear first order differential equation is a differential equation in the form

Multiplying or dividing this equation by any non-zero function ofx makes no difference toits solutions so we could always divide bya(x) to make the coefficient of the differential 1, butwriting the equation in this more general form may offer insights.

At first glance, it is not possible to integrate the left hand side, but there is one special case.If b happens to be the differential ofa then we can write

and integration is now straightforward.

Since we can freely multiply by any function, lets see if we can use this freedom to writethe left hand side in this special form.

We multiply the entire equation by an arbitrary,I(x), getting

then impose the condition

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If this is satisfied the new left hand side will have the special form. Note that multiplyingIby any constant will leave this condition still satisfied.

Rearranging this condition gives

We can integrate this to get

We can set the constantk to be 1, since this makes no difference.

Next we useI on the original differential equation, getting

Because we've chosenI to put the left hand side in the special form we can rewrite this as

Integrating both sides and dividing byI we obtain the final result

We call I an integrating factor. Similar techniques can be used on some other calclulusproblems.

Example

Consider

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First we calculate the integrating factor.

Multiplying the equation by this gives

or

We can now integrate

Exact equations

An exact equation is in the form

f(x, y) dx + g(x, y) dy = 0

and, has the property that

Dx f = Dy g

(If the differential equation does not have this property then we can't proceed any further).

As a result of this, if we have an exact equation then there exists a function h(x, y) such that

Dy h = f and Dx h = g

So then the solutions are in the form

h(x, y) = c

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by using the fact of the total differential. We can find then h(x, y) by integration

Basic second and higher order ODE's

The generic solution of anth order ODE will containn constants of integration. To calculatethem we needn more equations. Most often, we have either

boundary conditions, the values ofy and its derivatives take for two different values ofx

or

initial conditions, the values ofy and its firstn-1derivatives take for one particular valueof x.

Reducible ODE's

1. If the independent variable,x, does not occur in the differential equation then its ordercan be lowered by one. This will reduce a second order ODE to first order.

Consider the equation:

Define

Then

Substitute these two expression into the equation and we get

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=0

which is a first order ODE

Example

Solve

if at x=0, y=Dy=1

First, we make the substitution, getting

This is a first order ODE. By rearranging terms we can separate the variables

Integrating this gives

u2 / 2 = c + 1 / 2y

We know the values ofy andu whenx=0 so we can findc

Next, we reverse the substitution

and take the square root

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To find out which sign of the square root to keep, we use the initial condition, Dy=1 atx=0,again, and rule out the negative square root. We now have another separable first order ODE,

Its solution is

Sincey=1 whenx=0, d=2/3, and

2. If the dependent variable,y, does not occur in the differential equation then it may alsobe reduced to a first order equation.

Consider the equation:

Define

Then

Substitute these two expressions into the first equation and we get

=0

which is a first order ODE

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Linear ODEs

An ODE of the form

is calledlinear. Such equations are much simpler to solve than typical non-linear ODEs.Though only a few special cases can be solved exactly in terms of elementary functions, there ismuch that can be said about the solution of a generic linear ODE. A full account would be beyondthe scope of this book

If F(x)=0 for all x the ODE is calledhomogeneous

Two useful properties of generic linear equations are

1. Any linear combination of solutions of an homogeneous linear equation is also a solu-tion.

2. If we have a solution of a nonhomogeneous linear equation and we add any solutionof the corresponding homogenous linear equation we get another solution of the nonho-mogeneous linear equation

Variation of constants

Suppose we have a linear ODE,

and we know one solution,y=w(x)

The other solutions can always be written asy=wz. This substitution in the ODE will give

us terms involving every differential ofzupto thenth, no higher, so we'll end up with annth orderlinear ODE forz.

We know thatz is constant is one solution, so the ODE forzmust not contain az term, which

means it will effectively be ann-1th order linear ODE. We will have reduced the order by one.

Lets see how this works in practice.

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Example

Consider

One solution of this isy=x2, so substitutey=zx2 into this equation.

Rearrange and simplify.

x2D2z + 6xDz= 0

This is first order for Dz. We can solve it to get

Since the equation is linear we can add this to any multiple of the other solution to get thegeneral solution,

y = Ax - 3 + Bx2

Linear homogeneous ODE's with constant coefficients

Suppose we have a ODE

(Dn + a1Dn - 1 + ... +an - 1D + a0)y = 0

we can take an inspired guess at a solution (motivate this)

y = epx

For this function Dny=pny so the ODE becomes

(pn + a1pn - 1 + ... +an - 1p + a0)y = 0

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y=0 is a trivial solution of the ODE so we can discard it. We are then left with the equation

pn + a1pn - 1 + ... +an - 1p + a0) = 0

This is called thecharacteristicequation of the ODE.

It can have up ton roots, p1, p2 … pn, each root giving us a different solution of the ODE.

Because the ODE is linear, we can add all those solution together in any linear combinationto get a general solution

To see how this works in practice we will look at the second order case. Solving equationslike this of higher order uses the exact same principles; only the algebra is more complex.

Secondorder

If the ODE is second order,

D2y + bDy+ cy= 0

then the characteristic equation is a quadratic,

p2 + bp+ c = 0

with roots

What these roots are like depends on the sign ofb2-4c, so we have three cases to consider.

1) b2 > 4c

In this case we have two different real roots, so we can write down the solution straightaway.

2) b2 < 4c

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In this case, both roots are imaginary. We could just put them directly in the formula, but ifwe are interested in real solutions it is more useful to write them another way.

Defining k2=4c-b2, then the solution is

For this to be real, theA's must be complex conjugates

Make this substitution and we can write,

y = Ae - bx / 2cos(kx+ a)

If b is positive, this is a damped oscillation.

3) b2 = 4c

In this case the characteristic equation only gives us one root,p=-b/2. We must use anothermethod to find the other solution.

We'll use the method of variation of constants. The ODE we need to solve is,

D2y - 2pDy+ p2y = 0

rewritingb andc in terms of the root. From the characteristic equation we know one solution

is y = epx so we make the substitutiony = zepx, giving

(epxD2z + 2pepxDz+ p2epxz) - 2p(epxDz+ pepxz) + p2epxz = 0

This simplifies to D2z=0, which is easily solved. We get

so the second solution is the first multiplied byx.

Higher order linear constant coefficient ODE's behave similarly: an exponential for everyreal root of the characteristic and a exponent multiplied by a trig factor for every complex conju-gate pair, both being multiplied by a polynomial if the root is repeated.

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E.g, if the characteristic equation factors to

(p - 1)4(p - 3)(p2 + 1)2 = 0

the general solution of the ODE will be

y = (A + Bx+ Cx2 + Dx3)ex + Ee3x + Fcos(x + a) + Gxcos(x + b)

The most difficult part is finding the roots of the characteristic equation.

Linear nonhomogeneous ODEs with constant coefficients

First, let's consider the ODE

Dy - y = x

a nonhomogeneous first order ODE which we know how to solve.

Using the integrating factore-x we find

y = ce - x + 1 - x

This is the sum of a solution of the corresponding homogeneous equation, and a polynomial.

Nonhomogeneous ODE's of higher order behave similarly.

If we have a single solution,yp of the nonhomogeneous ODE, called aparticular solution,

then the general solution isy=yp+yh, whereyh is the general solution of the homogeneous

ODE.

Findyp for an arbitraryF(x) requires methods beyond the scope of this chapter, but there are

some special cases where findingyp is straightforward.

Remember that in the first order problemyp for a polynomialF(x) was itself a polynomial

of the same order. We can extend this to higher orders.

Example:

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D2y + y = x3 - x + 1

Consider a particular solution

yp = b0 + b1x + b2x2 + x3

Substitute fory and collect coefficients

x3 + b2x2 + (6 + b1)x + (2b2 + b0) = x3 - x + 1

Sob2=0, b1=-7, b0=1, and the general solution is

y = asinx + bcosx + 1 - 7x + x3

This works because all the derivatives of a polynomial are themselves polynomials.

Two other special cases are

wherePn,Qn,An, andBn are all polynomials of degreen.

Making these substitutions will give a set of simultaneous linear equations for the coeffi-cients of the polynomials.

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Partial differential equations

Multivariable and differentialcalculus:Exercises→

Calculus← Ordinary differential equa-tions

Partial differential equations

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Introduction

First order

Any partial differential equation of the form

whereh1, h2 … hn, andb are all functions of bothu andRn can be reduced to a set of ordi-

nary differential equations.

To see how to do this, we will first consider some simpler problems.

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Special cases

We will start with the simple PDE

Becauseu is only differentiated with respect toz, for any fixedx andy we can treat this like

the ODE,du/dz=u. The solution of that ODE iscez, wherec is the value ofu whenz=0, for thefixed x and y

Therefore, the solution of the PDE is

u(x,y,z) = u(x,y,0)ez

Instead of just having a constant of integration, we have an arbitary function. This will betrue for any PDE.

Notice the shape of the solution, an arbitary function of points in thexy, plane, which isnormal to the 'z' axis, and the solution of an ODE in the 'z' direction.

Now consider the slightly more complex PDE

where h can be any function, and eacha is a real constant.

We recognize the left hand side as beinga·∇, so this equation says that the differential ofuin thea direction ish(u). Comparing this with the first equation suggests that the solution can bewritten as an arbitary function on the plane normal toa combined with the solution of an ODE.

Remembering fromCalculus/Vectorsthat any vectorr can be split up into components paral-lel and perpendicular toa,

we will use this to split the components ofr in a way suggested by the analogy with (1).

Let's write

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and substitute this into (2), using the chain rule. Because we are only differentiating in thea direction, adding any function of the perpendicular vector tos will make no difference.

First we calculate grads, for use in the chain rule,

On making the substitution into (2), we get,

which is an ordinary differential equation with the solution

The constantccan depend on the perpendicular components, but not upon the parallel coordi-nate. Replacings with a monotonic scalar function ofs multiplies the ODE by a function ofs,which doesn't affect the solution.

Example:

u(x,t)x = u(x,t)t

For this equation,a is (1, -1),s=x-t, and the perpendicular vector is (x+t)(1, 1). The reducedODE isdu/ds=0so the solution is

u=f(x+t)

To find f we need initial conditions onu. Are there any constraints on what initial conditionsare suitable?

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Consider, if we are given

• u(x,0), this is exactlyf(x),• u(3t,t), this isf(4t) andf(t) follows immediately• u(t3+2t,t), this isf(t3+3t) andf(t) follows, on solving the cubic.• u(-t,t), then this isf(0), so if the given function isn't constant we have a inconsistency,

and if it is the solution isn't specified off the initial line.

Similarly, if we are givenu on any curve which the linesx+t=c intersect only once, and towhich they are not tangent, we can deducef.

For any first order PDE with constant coefficients, the same will be true. We will have a setof lines, parallel tor=at, along which the solution is gained by integrating an ODE with initialconditions specified on some surface to which the lines aren't tangent.

If we look at how this works, we'll see we haven't actually used the constancy ofa, so let'sdrop that assumption and look for a similar solution.

The important point was that the solution was of the formu=f(x(s),y(s)), where (x(s),y(s)) isthe curve we integrated along -- a straight line in the previous case. We can add constant func-tions of integration tos without changing this form.

Consider a PDE,

a(x,y)ux + b(x,y)uy = c(x,y,u)

For the suggested solution,u=f(x(s),y(s)), the chain rule gives

Comparing coefficients then gives

so we've reduced our original PDE to a set of simultaneous ODE's. This procedure can bereversed.

The curves (x(s),y(s)) are calledcharacteristicsof the equation.

Example:Solveyux = xuy givenu=f(x) for x≥0 The ODE's are

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subject to the initial conditions ats=0,

This ODE is easily solved, giving

so the characteristics are concentric circles round the origin, and in polar coordinatesu(r,θ)=f(r)

Considering the logic of this method, we see that the independence ofa andb from u hasnot been used either, so that assumption too can be dropped, giving the general method forequations of thisquasilinearform.

Quasilinear

Summarising the conclusions of the last section, to solve a PDE

subject to the initial condition that on the surface, (x1(r1,…,rn-1, …xn(r1,…,rn-1), u=f(r1,…,rn-

1) --this being an arbitary paremetrisation of the initial surface--

• we transform the equation to the equivalent set of ODEs,

subject to the initial conditions

• Solve the ODE's, givingxi as a function ofs and ther i.• Invert this to gets and ther i as functions of thexi.• Substitute these inverse functions into the expression foru as a function ofsand ther i

obtained in the second step.

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Both the second and third steps may be troublesome.

The set of ODEs is generally non-linear and without analytical solution. It may even beeasier to work with the PDE than with the ODEs.

In the third step, ther i together withs form a coordinate system adapted for the PDE. We

can only make the inversion at all if the Jacobian of the transformation to Cartesian coordinatesis not zero,

This is equivalent to saying that the vector (a1, &hellip:, an) is never in the tangent plane to

a surface of constants.

If this condition is not false whens=0 it may become so as the equations are integrated. Wewill soon consider ways of dealing with the problems this can cause.

Even when it is technically possible to invert the algebraic equations it is obviously inconve-nient to do so.

Example

To see how this works in practice, we willa/ consider the PDE,

uux + uy + ut = 0

with generic initial condition,

u = f(x,y) on t = 0

Naming variables for future convenience, the corresponding ODE's are

subject to the initial conditions atτ=0

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These ODE's are easily solved to give

These are the parametric equations of a set of straight lines, the characteristics.

The determinant of the Jacobian of this coordinate transformation is

This determinant is 1 whent=0, but if fr is anywhere negative this determinant will eventual-

ly be zero, and this solution fails.

In this case, the failure is because the surfacesfr = - 1 is an envelope of the characteristics.

For arbitaryf we can invert the transformation and obtain an implicit expression foru

u = f(x - tu,y - x)

If f is given this can be solved foru.

1/ f(x,y) = ax, The implicit solution is

This is a line in theu-x plane, rotating clockwise ast increases. Ifa is negative, this lineeventually become vertical. Ifa is positive, this line tends towardsu=0, and the solution is validfor all t.

2/ f(x,y)=x2, The implicit solution is

This solution clearly fails when 1 + 4tx < 0, which is just whensfr = - 1 . For anyt>0 this

happens somewhere. Ast increases this point of failure moves toward the origin.

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Notice that the point whereu=0 stays fixed. This is true for any solution of this equation,whateverf is.

We will see later that we can find a solution after this time, if we consider discontinuoussolutions. We can think of this as a shockwave.

3/ f(x,y) = sin(xy)The implicit solution is

and we can not solve this explitly foru. The best we can manage is a numerical solution ofthis equation.

b/We can also consider the closely related PDE

uux + uy + ut = y

The corresponding ODE's are

subject to the initial conditions atτ=0

These ODE's are easily solved to give

Writing f in terms ofu, s, andτ, then substituting into the equation forx gives an implicitsolution

It is possible to solve this foru in some special cases, but in general we can only solve thisequation numerically. However, we can learn much about the global properties of the solutionfrom further analysis

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Characteristic initial value problems

What if initial conditions are given on a characteristic, on an envelope of characteristics, ona surface with characteristic tangents at isolated points?

Discontinuous solutions

So far, we've only considered smooth solutions of the PDE, but this is too restrictive. Wemay encounter initial conditions which aren't smooth, e.g.

If we were to simply use the general solution of this equation for smooth initial conditions,

we would get

which appears to be a solution to the original equation. However, since the partial differen-tials are undefined on the characteristicx+ct=0, so it becomes unclear what it means to say thatthe equation is true at that point.

We need to investigate further, starting by considering the possible types of discontinuities.

If we look at the derivations above, we see we've never use any second or higher orderderivatives so it doesn't matter if they aren't continuous, the results above will still apply.

The next simplest case is when the function is continuous, but the first derivative is not, e.g|x|. We'll initially restrict ourselves to the two-dimensional case,u(x, t) for the generic equation.

Typically, the discontinuity is not confined to a single point, but is shared by all points onsome curve, (x0(s), t0(s) )

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Then we have

We can then compareu and its derivatives on both sides of this curve.

It will prove useful to name thejumpsacross the discontinuity. We say

Now, since the equation (1) is true on both sides of the discontinuity, we can see that bothu+ andu-, being the limits of solutions, must themselves satisfy the equation. That is,

Subtracting then gives us an equation for the jumps in the differentials

We are considering the case whereu itself is continuous so we know that [u]=0. Differentiat-ing this with respect tos will give us a second equation in the differential jumps.

The last two equations can only be both true if one is a multiple of the other, but multiplyingsby a constant also multiplies the second equation by that same constant while leaving the curveof discontinuity unchanged, hence we can without loss of generality defines to be such that

But these are the equations for a characteristic, i.ediscontinuities propagate along charac-teristics. We could use this property as an alternative definition of characteristics.

We can deal similarly with discontinuous functions by first writing the equation inconserva-tion form, so called because conservation laws can always be written this way.

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Notice that the left hand side can be regarded as the divergence of (au, bu). Writing theequation this way allows us to use the theorems of vector calculus.

Consider a narrow strip with sides parallel to the discontinuity and widthh

We can integrate both sides of (1) over R, giving

Next we use Green's theorem to convert the left hand side into a line integral.

Now we let the width of the strip fall to zero. The right hand side also tends to zero but theleft hand side reduces to the difference between two integrals along the part of the boundary ofRparallel to the curve.

The integrals along the opposite sides ofRhave different signs because they are in oppositedirections.

For the last equation to always be true, the integrand must always be zero, i.e

Since, by assumption [u] isn't zero, the other factor must be, which immediately implies thecurve of discontinuity is a characteristic.

Once again,discontinuities propagate along characteristics.

Above, we only considered functions of two variables, but it is straightforward to extendthis to functions ofn variables.

The initial condition is given on ann-1 dimensional surface, which evolves along the charac-teristics. Typical discontinuities in the initial condition will lie on an-2 dimensional surfaceembedded within the initial surface. This surface of discontinuity will propagate along the charac-teristics that pass through the initial discontinuity.

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The jumps themselves obey ordinary differential equations, much asu itself does on a charac-teristic. In the two dimensional case, foru continuous but not smooth, a little algebra shows that

while u obeys the same equation as before,

We can integrate these equations to see how the discontinuity evolves as we move along thecharacteristic.

We may find that, for some futures, [ux] passes through zero. At such points, the discontinu-

ity has vanished, and we can treat the function as smooth at that characteristic from then on.

Conversely, we can expect that smooth functions may, under the righr circumstances, be-come discontinuous.

To see how all this works in practice we'll consider the solutions of the equation

for three different initial conditions.

The general solution, using the techniques outlined earlier, is

u is constant on the characteristics, which are straight lines with slope dependent onu.

First considerf such that

While u is continuous its derivative is discontinuous atx=0, whereu=0, and atx=a, whereu=1. The characteristics through these points divide the solution into three regions.

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All the characteristics to the right of the characteristic throughx=a, t=0 intersect thex-axisto the right ofx=1, whereu=1 sou is 1 on all those characteristics, i.e wheneverx-t>a.

Similarly the characteristic through the origin is the linex=0, to the left of whichu remainszero.

We could find the value ofu at a point in between those two characteristics either by findingwhich intermediate characteristic it lies on and tracing it back to the initial line, or via the generalsolution.

Either way, we get

At largert the solutionu is more spread out than att=0 but still the same shape.

We can also consider what happens whena tends to 0, so thatu itself is discontinuous atx=0.

If we write the PDE in conservation form then use Green's theorem, as we did above for thelinear case, we get

[u²] is the difference of two squares, so if we takes=t we get

In this case the discontinuity behaves as if the value ofu on it were the average of the limit-ing values on either side.

However, there is a caveat.

Since the limiting value to the left isu- the discontinuity must lie on that characteristic, and

similarly for u+; i.e the jump discontinuity must be on an intersection of characteristics, at a point

whereu would otherwise be multivalued.

For this PDE the characteristic can only intersect on the discontinuity if

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If this is not true the discontinuity can not propagate. Something else must happen.

The limit a=0 is an example of a jump discontinuity for which this condition is false, so wecan see what happens in such cases by studying it.

Taking the limit of the solution derived above gives

If we had taken the limit of any other sequence of initials conditions tending to the samelimit we would have obtained a trivially equivalent result.

Looking at the characteristics of this solution, we see that at the jump discontinuity character-istics on whichu takes every value betweeen 0 and 1 all intersect.

At later times, there are two slope discontinuities, atx=0 andx=t, but no jump discontinuity.

This behaviour is typical in such cases. The jump discontinuity becomes a pair of slope dis-continuities between which the solution takes all appropriate values.

Now, lets consider the same equation with the initial condition

This has slope discontinuities atx=0 andx=a, dividing the solution into three regions.

The boundaries between these regions are given by the characteristics through these initialpoints, namely the two lines

These characteristics intersect att=a, so the nature of the solution must change then.

In between these two discontinuities, the characteristic throughx=b at t=0 is clearly

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All these characteristics intersect at the same point, (x,t)=(a,a).

We can use these characteristics, or the general solution, to writeu for t<a

As t tends toa, this becomes a step function. Sinceu is greater to the left than the right ofthe discontinuity, it meets the condition for propagation deduced above, so fort>a u is a stepfunction moving at the average speed of the two sides.

This is the reverse of what we saw for the initial condition previously considered, two slopediscontinuities merging into a step discontinuity rather than vice versa. Which actually happensdepends entirely on the initial conditions. Indeed, examples could be given for which both process-es happen.

In the two examples above, we started with a discontinuity and investigated how it evolved.It is also possible for solutions which are initially smooth to become discontinuous.

For example, we saw earlier for this particular PDE that the solution with the initial conditionu=x² breaks down when 2xt+1=0. At these points the solution becomes discontinuous.

Typically, discontinuities in the solution of any partial differential equation, not merely onesof first order, arise when solutions break down in this way and propagate similarly, merging andsplitting in the same fashion.

Fully non-linear PDEs

It is possible to extend the approach of the previous sections to reduce any equation of theform

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to a set of ODE's, foranyfunction,F.

We will not prove this here, but the corresponding ODE's are

If u is given on a surface parameterized byr1…rn then we have, as before,n initial conditions

on then, xi

given by the parameterization andoneinitial condition onu itself,

but, because we have an extran ODEs for theui's, we need an extran initial conditions.

These are,n-1 consistency conditions,

which state that theui's are the partial derivatives ofu on the initial surface, andoneinitial

condition

stating that the PDE itself holds on the initial surface.

Thesen initial conditions for theui will be a set of algebraic equations, which may have

multiple solutions. Each solution will give a different solution of the PDE.

Example

Consider

The initial conditions atτ=0 are

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and the ODE's are

Note that the partial derivatives are constant on the characteristics. This always happen whenthe PDE contains only partial derivatives, simplifying the procedure.

These equations are readily solved to give

On eliminating the parameters we get the solution,

which can easily be checked. abc

Second order

Suppose we are given a second order linear PDE to solve

The natural approach, after our experience with ordinary differential equations and withsimple algebraic equations, is attempt a factorisation. Let's see how for this takes us.

We would expect factoring the left hand of (1) to give us an equivalent equation of the form

and we can immediately divide through bya. This suggests that those particular combina-tions of first order derivatives will play a special role.

Now, when studying first order PDE's we saw that such combinations were equivalent tothe derivatives along characteristic curves. Effectively, we changed to a coordinate system de-fined by the characteristic curve and the initial curve.

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Here, we have two combinations of first order derivatives each of which may define a differ-ent characteristic curve. If so, the two sets of characteristics will define a natural coordinate sys-tem for the problem, much as in the first order case.

In the new coordinates we will have

with each of the factors having become a differentiation along its respective characteristiccurve, and the left hand side will become simplyurs giving us an equation of the form

If A, B, andC all happen to be zero, the solution is obvious. If not, we can hope that thesimpler form of the left hand side will enable us to make progress.

However, before we can do all this, we must see if (1) can actually be factored.

Multiplying out the factors gives

On comparing coefficients, and solving for theα's we see that they are the roots of

Since we are discussing real functions, we are only interested in real roots, so the existenceof the desired factorization will depend on the discriminant of this quadratic equation.

• If b(x,y)2 > 4a(x,y)c(x,y)

then we have two factors, and can follow the procedure outlined above. Equations like thisare calledhyperbolic

• If b(x,y)2 = 4a(x,y)c(x,y)

then we have only factor, giving us a single characteristic curve. It will be natural to usedistance along these curves as one coordinate, but the second must be determined by otherconsiderations.

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The same line of argument as before shows that use the characteristic curve this way givesa second order term of the formurr , where we've only taken the second derivative with re-spect to one of the two coordinates. Equations like this are calledparabolic

• If b(x,y)2 < 4a(x,y)c(x,y)

then we have no real factors. In this case the best we can do is reduce the second orderterms to the simplest possible form satisfying this inequality, i.eurr+uss

It can be shown that this reduction is always possible. Equations like this are calledelliptic

It can be shown that, just as for first order PDEs, discontinuities propagate along characteris-tics. Since elliptic equations have no real characteristics, this implies that any discontinuities theymay have will be restricted to isolated points; i.e, that the solution is almost everywhere smooth.

This is not true for hyperbolic equations. Their behavior is largely controlled by the shapeof their characteristic curves.

These differences mean different methods are required to study the three types of secondequation. Fortunately, changing variables as indicated by the factorisation above lets us reduceany second order PDE to one in which the coefficients of the second order terms are constant,which means it is sufficient to consider only three standard equations.

We could also consider the cases where the right hand side of these equations is a givenfunction, or proportional tou or to one of its first order derivatives, but all the essential propertiesof hyperbolic, parabolic, and elliptic equations are demonstrated by these three standard forms.

While we've only demonstrated the reduction in two dimensions, a similar reduction appliesin higher dimensions, leading to a similar classification. We get, as the reduced form of the sec-ond order terms,

where each of theais is equal to either 0, +1, or -1.

If all theais have thesame signthe equation iselliptic

If anyof theais arezerothe equation isparabolic

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If exactly oneof theais has theopposite signto the rest the equation ishyperbolic

In 2 or 3 dimensions these are the only possibilities, but in 4 or more dimensions there is afourth possibility:at least twoof theais arepositive, andat least twoof theais arenegative.

Such equations are calledultrahyperbolic. They are less commonly encountered than theother three types, so will not be studied here.

When the coefficients are not constant, an equation can be hyperbolic in some regions ofthexy plane, and elliptic in others. If so, different methods must be used for the solutions in thetwo regions.

Elliptic

Standard form, Laplace's equation:

Quote equation in spherical and cylindrical coordinates, and give full solution for cartesianand cylindrical coordinates. Note averaging property Comment on physical significance, rotationinvariance of laplacian.

Hyperbolic

Standard form, wave equation:

Solution, any sum of functions of the form

These are waves. Compare with solution from separating variables. Domain of dependance,etc.

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Parabolic

The canonical parabolic equation is the diffusion equation:

Here, we will consider some simple solutions of the one-dimensional case.

The properties of this equation are in many respects intermediate between those of hyperbol-ic and elliptic equation.

As with hyperbolic equations but not elliptic, the solution is well behaved if the value isgiven on the initial surfacet=0.

However, the characteristic surfaces of this equation are the surfaces of constantt, thus thereis no way for discontinuities to propagate to positive t.

Therefore, as with elliptic equations but not hyberbolic, the solutions are typically smooth,even when the initial conditions aren't.

Furthermore, at a local maximum ofh, its Laplacian is negative, soh is decreasing witht,while at local minima, where the Laplacian will be positive,h will increase witht. Thus, initialvariations inh will be smoothed out ast increases.

In one dimension, we can learn more by integrating both sides,

Provided thathx tends to zero for largex, we can take the limit asa andb tend to infinity,

deducing

so the integral ofh over all space is constant.

This means this PDE can be thought of as describing some conserved quantity, initiallyconcentrated but spreading out, or diffusing, over time.

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This last result can be extended to two or more dimensions, using the theorems of vectorcalculus.

We can also differentiate any solution with respect to any coordinate to obtain another solu-tion. E.g ifh is a solution then

sohx also satisfies the diffusion equation.

Similarity solution

Looking at this equation, we might notice that if we make the change of variables

then the equation retains the same form. This suggests that the combination of variablesx²/t,which is unaffected by this variable change, may be significant.

We therefore assume this equation to have a solution of the special form

then

and substituting into the diffusion equation eventually gives

which is an ordinary differential equation.

Integrating once gives

Reverting toh, we find

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This last integral can not be written in terms of elementary functions, but its values are wellknown.

In particular the limiting values ofh at infinity are

taking the limit ast tends to zero gives

We see that the initial discontinuity is immediately smoothed out. The solution at later timesretains the same shape, but is more stretched out.

The derivative of this solution with respect tox

is itself a solution, withh spreading out from its initial peak, and plays a significant role inthe further analysis of this equation.

The same similiarity method can also be applied to some non-linear equations.

Separating variables

We can also obtain some solutions of this equation by separating variables.

giving us the two ordinary differential equations

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and solutions of the general form

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Multivariable and differentialcalculus:Exercises

Extensions→Calculus←Partial differential equations

Multivariable and differen-tial calculus:Exercises

Exercise

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Extensions

Systems of ordinary differen-tial equations→

Calculus←Partial differential equations

Extensions

Further Analysis

• Systems of Ordinary Differential Equations


Formal Theory of Calculus

• Real numbers


• Complex numbers

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Systems of ordinary differentialequations

We have already examined cases where we have a single differential equation and foundseveral methods to aid us in finding solutions to these equations. But what happens if we havetwo or more differential equations that depend on each other? For example, consider the casewhere

Dtx(t) = 3y(t)2 + x(t)t

and

Dty(t) = x(t) + y(t)

Such a set of differential equations is said to becoupled. Systems of ordinary differentialequations such as these are what we will look into in this section.

First order systems

A general system of differential equations can be written in the form

Instead of writing the set of equations in a vector, we can write out each equation explicitly,in the form:

If we have the system at the very beginning, we can write it as:

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where

and

or write each equation out as shown above.

Why are these forms important? Often, this arises as a single, higher order differentialequation that is changed into a simpler form in a system. For example, with the same example,

Dtx(t) = 3y(t)2 + x(t)t

Dty(t) = x(t) + y(t)

we can write this as a higher order differential equation by simple substitution.

Dty(t) - y(t) = x(t)

then

Dtx(t) = 3y(t)2 + (Dty(t) - y(t))t

Dtx(t) = 3y(t)2 + tDty(t) - ty(t)

Notice now that the vector form of the system is dependent ont since

the first component is dependent ont. However, if instead we had

notice the vector field is no longer dependent ont. We call such systemsautonomous. Theyappear in the form

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We can convert between an autonomous system and a non-autonomous one by simplymaking a substitution that involvest, such asy=(x, t), to get a system:

In vector form, we may be able to separateF in a linear fashion to get something that lookslike:

whereA(t) is a matrix andb is a vector. The matrix could contain functions or constants,clearly, depending on whether the matrix depends ont or not.

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Real numbers

Fields

You are probably already familiar with many different sets of numbers from your past experi-ence. Some of the commonly used sets of numbers are

• Natural numbers, usually denoted with anN, are the numbers 0,1,2,3,...• Integers, usually denoted with aZ, are the positive and negative natural numbers: ...-

3,-2,-1,0,1,2,3...• Rational numbers, denoted with aQ, are fractions of integers (excluding division by

zero): -1/3, 5/1, 0, 2/7. etc.• Real numbers, denoted with aR, are constructed and discussed below.

Note that different sets of numbers have different properties. In the set integers for example,any number always has anadditive inverse: for any integerx, there is another integert such thatx + t = 0 This should not be terribly surprising: from basic arithmetic we know thatt = − x. Tryto prove to yourself that not all natural numbers have an additive inverse.

In mathematics, it is useful to note the important properties of each of these sets of numbers.The rational numbers, which will be of primary concern in constructing the real numbers, havethe following properties:

There exists a number 0 such that for any other numbera, 0+a=a+0=a

For any two numbersa andb, a+b is another number

For any three numbersa,b, andc, a+(b+c)=(a+b)+c

For any numbera there is another number-a such thata+(-a)=0

For any two numbersa andb, a+b=b+a

For any two numbersa andb,a*b is another number

There is a number 1 such that for any numbera, a*1=1*a=a

For any two numbersa andb, a*b=b*a

For any three numbersa,bandc, a(bc)=(ab)c

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For any three numbersa,bandc, a(b+c)=ab+ac

For every numbera there is another numbera-1 such that aa-1=1

As presented above, these may seem quite intimidating. However, these properties arenothing more than basic facts from arithmetic. Any collection of numbers (and operations + and* on those numbers) which satisfies the above properties is called afield. The properties aboveare usually calledfield axioms.As an exercise, determine if the integers form a field, and if not,which field axiom(s) they violate.

Even though the list of field axioms is quite extensive, it does not fully explore the propertiesof rational numbers. Rational numbers also have anordering.' A total ordering must satisfyseveral properties: for any numbersa, b,andc

if a ≤ b andb ≤ a thena = b (antisymmetry)

if a ≤ b andb ≤ c thena ≤ c (transitivity)

a ≤ b or b ≤ a (totality)

To familiarize yourself with these properties, try to show that (a) natural numbers, integersand rational numbers are all totally ordered and more generally (b) convince yourself that anycollection of rational numbers are totally ordered (note that the integers and natural numbers areboth collections of rational numbers).

Finally, it is useful to recognize one more characterization of the rational numbers: everyrational number has a decimal expansion which is either repeating or terminating. The proof ofthis fact is omitted, however it follows from the definition of each rational number as a fraction.When performing long division, the remainder at any stage can only take on positive integervalues smaller than the denominator, of which there are finitely many.

Constructing the Real Numbers

There are two additional tools which are needed for the construction of the real numbers:the upper bound and the least upper bound.Definition A collection of numbersE is boundedabove if there exists a numbermsuch that for allx in E x≤m. Any numbermwhich satisfies thiscondition is called an upper bound of the setE.

Definition If a collection of numbersE is bounded above withm as an upper bound ofE,and all other upper bounds ofE are bigger thanm, we callm theleast upper boundor supremumof E, denoted by supE.

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Many collections of rational numbers do not have a least upper bound which is also rational,although some do. Suppose the the numbers 5 and 10/3 are, together, taken to beE. The number10/3 is not only an upper bound ofE, it is a least upper bound. In general, there are many upperbounds (12, for instance, is an upper bound of the collection above), but there can be at most oneleast upper bound.

Consider the collection of numbers

: You may recognize these decimals as the first few digits of pi. Since each decimal termi-nates, each number in this collection is a rational number. This collection has infinitely manyupper bounds. The number 4, for instance, is an upper bound. There is no least upper bound, atleast not in the rational numbers. Try to convince yourself of this fact by attempting to constructsuch a least upper bound: (a) why does pi not work as a least upper bound (hint: pi does not havea repeating or terminating decimal expansion), (b) what happens if the proposed supremum isequal to pi up to some decimal place, and zeros after (c) if the proposed supremum is bigger thanpi, can you find a smaller upper bound which will work?

In fact, there are infinitely many collections of rational numbers which do not have a rationalleast upper bound. We define a real number to be any number that is the least upper bound ofsome collection of rational numbers.

Properties of Real Numbers

The reals are well ordered.

For all reals;a, b, c

Eitherb>a, b=a, or b<a.

If a<b andb<c thena<c

Also

b>a impliesb+c>a+c

b>a andc>0 impliesbc>ac

b>a implies-a>-b

Upper bound axiom

Every non-empty set of real numbers which is bounded above has a supremum.

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The upper bound axiom is necessary for calculus. It is not true for rational numbers.

We can also define lower bounds in the same way.

Definition A setE is bounded below if there exists a real M such that for allx∈E x≥M AnyM which satisfies this condition is called an lower bound of the setE

Definition If a set,E, is bounded below,M is an lower bound ofE, and all other lowerbounds ofE are less thanM, we callM thegreatest lower boundor inifimumof E, denoted byinf E

The supremum and infimum of finite sets are the same as their maximum and minimum.

Theorem

Every non-empty set of real numbers which is bounded below has an infimum.

Proof:

Let E be a non-empty set of of real numbers, bounded below

Let L be the set of all lower bounds of E

L is not empty, by definition of bounded below

Every element of E is an upper bound to the set L, by definition

Therefore, L is a non empty set which is bounded above

L has a supremum, by the upper bound axiom

1/ Every lower bound of E is≤sup L, by definition of supremum

Suppose there were ane∈E such that e<sup L

Every element of L is≤e, by definition

Therefore e is an upper bound of L and e<sup L

This contradicts the definition of supremum, so there can be no such e.

If e∈E then e≥sup L, proved by contradiction

2/ Therefore, sup L is a lower bound of E

inf E exists, and is equal to sup L, on comparing definition of infinum to lines 1 & 2

Bounds and inequalities, theorems:

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Theorem: (The triangle inequality)

Proofby considering cases

If a≤b≤c then|a-c|+|c-b| = (c-a)+(c-b)= 2(c-b)+(b-a)>b-a= |b-a|

Exercise:Prove the other five cases.

This theorem is a special case of the triangle inequality theorem from geometry: The sumof two sides of a triangle is greater than or equal to the third side. It is useful whenever we needto manipulate inequalities and absolute values.

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Complex numbers


Complex NumbersIn mathematics, acomplex number is a number of the form

wherea andb are real numbers, andi is the imaginary unit, with the propertyi 2 = −1. Thereal numbera is called thereal partof the complex number, and the real numberb is theimagi-nary part. Real numbers may be considered to be complex numbers with an imaginary part ofzero; that is, the real numbera is equivalent to the complex numbera+0i.

For example, 3 + 2i is acomplex number, with real part 3 and imaginary part 2. Ifz = a +bi, the real part (a) is denoted Re(z), orℜ(z), and the imaginary part (b) is denoted Im(z), orℑ(z).

Complex numbers can be added, subtracted, multiplied, and divided like real numbers andhave other elegant properties. For example, real numbers alone do not provide a solution for ev-ery polynomial algebraic equation with real coefficients, while complex numbers do (this is thefundamental theorem of algebra).

Equality

Two complex numbers are equal if and only if their real parts are equalandtheir imaginaryparts are equal. That is,a + bi = c + di if and only if a = c andb = d.

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http://en.wikibooks.org/wiki/File:Wikiversity-logo-Snorky.svg

http://en.wikiversity.org/wiki/Complex_Numbers

Notation and operations

The set of all complex numbers is usually denoted byC, or in blackboard bold by

(Unicodeℂ). The real numbers,R, may be regarded as "lying in"C by considering everyreal number as a complex:a = a + 0i.

Complex numbers are added, subtracted, and multiplied by formally applying the associative,

commutative and distributive laws of algebra, together with the equationi2 = −1:

Division of complex numbers can also be defined (see below). Thus, the set of complexnumbers forms a field which, in contrast to the real numbers, is algebraically closed.

In mathematics, the adjective "complex" means that the field of complex numbers is theunderlying number field considered, for example complex analysis, complex matrix, complexpolynomial and complex Lie algebra.

The field of complex numbers

Formally, the complex numbers can be defined as ordered pairs of real numbers (a, b) togeth-er with the operations:

So defined, the complex numbers form a field, the complex number field, denoted byC (afield is an algebraic structure in which addition, subtraction, multiplication, and division are de-fined and satisfy certain algebraic laws. For example, the real numbers form a field).

The real numbera is identified with the complex number (a, 0), and in this way the field ofreal numbersR becomes a subfield ofC. The imaginary uniti can then be defined as the complexnumber (0, 1), which verifies

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In C, we have:

• additive identity ("zero"): (0, 0)• multiplicative identity ("one"): (1, 0)• additive inverse of (a,b): (−a, −b)• multiplicative inverse (reciprocal) of non-zero (a, b):

Since a complex numbera + bi is uniquely specified by an ordered pair (a, b) of real num-bers, the complex numbers are in one-to-one correspondence with points on a plane, called thecomplex plane.

The complex plane

A complex numberz can be viewed as a point or a position vector in a two-dimensionalCartesian coordinate system called thecomplex planeorArgand diagram . The point and hencethe complex numberz can be specified by Cartesian (rectangular) coordinates. The Cartesiancoordinates of the complex number are the real partx = Re(z) and the imaginary party = Im(z).The representation of a complex number by its Cartesian coordinates is called theCartesian formor rectangular formor algebraic formof that complex number.

Polar form

Alternatively, the complex numberzcan be specified by polar coordinates. The polar coordi-nates arer = |z| ≥ 0, called theabsolute valueor modulus, andφ = arg(z), called theargumentof z. For r = 0 any value ofφ describes the same number. To get a unique representation, a con-ventional choice is to set arg(0) = 0. Forr > 0 the argumentφ is unique modulo 2π; that is, ifany two values of the complex argument differ by an exact integer multiple of 2π, they are consid-ered equivalent. To get a unique representation, a conventional choice is to limitφ to the interval(-π,π], i.e. −π < φ ≤ π. The representation of a complex number by its polar coordinates is calledthepolar formof the complex number.

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Conversion from the polar form to the Cartesian form

Conversion from the Cartesian form to the polar form

The previous formula requires rather laborious case differentiations. However, many pro-gramming languages provide a variant of the arctangent function. A formula that uses the arccosfunction requires fewer case differentiations:

Notation of the polar form

The notation of the polar form as

is calledtrigonometric form. The notation cisφ is sometimes used as an abbreviation forcosφ + i sinφ. UsingEuler's formulait can also be written as

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http://en.wikipedia.org/wiki/Euler%27s_formula

which is calledexponential form.

Multiplication, division, exponentiation, and rootextraction in the polar form

Multiplication, division, exponentiation, and root extraction are much easier in the polarform than in the Cartesian form.

Usingsum and difference identitiesits possible to obtain that

and that

Exponentiation with integer exponents; according tode Moivre's formula,

Exponentiation with arbitrary complex exponents is discussed in the article onexponentia-tion.

The addition of two complex numbers is just the addition of two vectors, and multiplicationby a fixed complex number can be seen as a simultaneous rotation and stretching.

Multiplication by i corresponds to a counter-clockwise rotation by 90° (π/2 radians). The

geometric content of the equationi2 = −1 is that a sequence of two 90 degree rotations results ina 180 degree (π radians) rotation. Even the fact (−1) · (−1) = +1 from arithmetic can be under-stood geometrically as the combination of two 180 degree turns.

All the roots of any number, real or complex, may be found with a simple algorithm. Thenth roots are given by

for k = 0, 1, 2, …,n−1, where

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http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Angle_sum_and_difference_identities

http://en.wikipedia.org/wiki/De_Moivre%27s_formula

http://en.wikibooks.org/w/index.php?title=Exponentiation&action=edit&redlink=1

http://en.wikibooks.org/w/index.php?title=Exponentiation&action=edit&redlink=1

represents the principalnth root ofr.

Absolute value, conjugation and distance

Theabsolute value(or modulusor magnitude) of a complex numberz = r eiφ is defined as|z| = r. Algebraically, ifz = a + bi, then

One can check readily that the absolute value has three important properties:

if and only if

(triangle inequality)

for all complex numberszandw. It then follows, for example, that | 1 | = 1 and |z / w | = |z| / | w | . By defining thedistancefunctiond(z, w) = |z − w| we turn the set of complex numbersinto ametric spaceand we can therefore talk about limits and continuity.

Thecomplex conjugateof the complex numberz= a + bi is defined to bea − bi, written as

or

. As seen in the figure,

is the "reflection" ofz about the real axis. The following can be checked:

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http://en.wikipedia.org/wiki/metric_space

if and only if z is real

if z is non-zero.

The latter formula is the method of choice to compute the inverse of a complex number if itis given in rectangular coordinates.

That conjugation commutes with all the algebraic operations (and many functions;e.g.

) is rooted in the ambiguity in choice ofi (−1 has two square roots). It is important to note,however, that the function

is not complex-differentiable.

Complex fractions

We can divide a complex number (a + bi) by another complex number (c + di) ≠ 0 in twoways. The first way has already been implied: to convert both complex numbers into exponentialform, from which their quotient is easily derived. The second way is to express the division as afraction, then to multiply both numerator and denominator by the complex conjugate of the de-nominator. The new denominator is a real number.

Matrix representation of complex numbers

While usually not useful, alternative representations of the complex field can give some in-sight into its nature. One particularly elegant representation interprets each complex number as

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a 2×2 matrix with real entries which stretches and rotates the points of the plane. Every suchmatrix has the form

wherea andb are real numbers. The sum and product of two such matrices is again of thisform. Every non-zero matrix of this form is invertible, and its inverse is again of this form.Therefore, the matrices of this form are a field. In fact, this is exactly the field of complex num-bers. Every such matrix can be written as

which suggests that we should identify the real number 1 with the identity matrix

and the imaginary uniti with

a counter-clockwise rotation by 90 degrees. Note that the square of this latter matrix is indeedequal to the 2×2 matrix that represents −1.

The square of the absolute value of a complex number expressed as a matrix is equal to thedeterminant of that matrix.

If the matrix is viewed as a transformation of the plane, then the transformation rotates pointsthrough an angle equal to the argument of the complex number and scales by a factor equal tothe complex number's absolute value. The conjugate of the complex numberzcorresponds to thetransformation which rotates through the same angle aszbut in the opposite direction, and scalesin the same manner asz; this can be represented by the transpose of the matrix corresponding toz.

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If the matrix elements are themselves complex numbers, the resulting algebra is that of thequaternions. In other words, this matrix representation is one way of expressing the Cayley-Dickson construction of algebras.

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References

Table of Trigonometry→Calculus← Complex numbers

References

• Table of Trigonometry• Summation notation• Tables of Derivatives• Tables of Integrals

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Table of Trigonometry


SeePic-

Trigonometric_identity


Definitions

•

•

•

•

Pythagorean Identities

•••

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- 516-



http://en.wikipedia.org/wiki/Special:Search/Trigonometric_identity?go=Go

http://en.wikibooks.org/wiki/File:Wikipedia-logo.png


Double Angle Identities

•••

•

•

Angle Sum Identities

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- 517 -


Product-to-sum identities

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- 518-


Summation notation

Summation notation allows an expression that contains a sum to be expressed in a simple,compact manner. The uppercase Greek letter sigma,Σ, is used to denote the sum of a set ofnumbers.

Example

Let f be a function andN,M are integers withN < M. Then

We sayN is the lower limit andM is the upper limit of the sum.

We can replace the letteri with any other variable. For this reasoni is referred to as adummyvariable. So

Conventionally we use the lettersi, j, k, m for dummy variables.

Example

Here, thedummy variableis i, thelower limit of summation is 1, and theupper limit is 5.

ExampleSometimes, you will see summation signs with no dummy variable specified, e.g.,

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- 519 -


In such cases the correct dummy variable should be clear from the context.

You may also see cases where the limits are unspecified. Here too, they must be deducedfrom the context.

Common summations


SeePic-

Summation#Capital-sigma notation


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- 520-



http://en.wikipedia.org/wiki/Special:Search/Summation?go=Go#Capital-sigma_notation



Tables of Derivatives


SeePic-

Table_of_derivatives


General Rules

Powers and Polynomials

•

•

•

•

•

•

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- 521 -



http://en.wikipedia.org/wiki/Special:Search/Table_of_derivatives?go=Go



Trigonometric Functions

Exponential and Logarithmic Functions

•

•

•

•

•

•


•

•

•

•

•

•

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- 522-


Hyperbolic and Inverse Hyperbolic Functions

Calculus

- 523 -


Tables of Integrals


SeePic-

Lists of integrals


Rules

•

•

•

•

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http://en.wikipedia.org/wiki/Special:Search/Lists_of_integrals?go=Go



Powers

•

•

•

•

•

•

Trigonometric Functions

Basic Trigonometric Functions

•

•

•

•

•

•

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- 525 -


Reciprocal Trigonometric Functions

•

•

•

•

•

•

•

•

•


•

•

•

•

Calculus

- 526-


Exponential and Logarithmic Functions

•

•

•

•


•

•

•

Calculus

- 527 -


Acknowledgements

Acknowledgements

Portions of this book have been copied from relevantWikipediaarticles.

Contributors

In alphabetical order (by surname or display name):

• Aaron Paul (AKAGrimm)• "Professor M." (no user page available)• Chaotic llama• User:Cronholm144• User:Fephisto• User:Juliusross• User:Stranger104• User:Whiteknight

Further Reading

The following books listCalculusas a prerequisite:

• Electrodynamics• Clock and data recovery• Signal Processing

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http://en.wikipedia.org/

http://en.wikibooks.org/wiki/User:The_Grimm_Ripper

http://en.wikibooks.org/wiki/User:Cronholm144

http://en.wikibooks.org/wiki/User:Fephisto

http://en.wikibooks.org/wiki/User:Juliusross

http://en.wikibooks.org/wiki/User:Stranger104

http://en.wikibooks.org/wiki/User:Whiteknight

http://en.wikibooks.org/wiki/Electrodynamics

http://en.wikibooks.org/wiki/Clock_and_data_recovery

http://en.wikibooks.org/wiki/Signal_Processing

Other Calculus Textbooks

Other calculus textbooks available online:

• Calculus Refresherby Paul Garrett, notes on first-year calculus (PDF/TeX).• Difference Equations to Differential Equations: An Introduction to Calculusby Dan

Sloughter, available under a Creative Commons license (PDF).• The Calculus of Functions of Several Variablesby Dan Sloughter, available under a

Creative Commons license (PDF).• Lecture Notes for Applied Calculus(PDF) by Karl Heinz Dovermann, first-semester

calculus without using limits.• Elements of the Differential and Integral Calculusby William Granville (1911), a

classic calculus textbook now available online in various forms. (It is also partiallyavailable atWikisource.)

• Calculus(3rd Ed., 1994) byMichael Spivak, is a more rigorous introductory calculustextbook.

Using infinitesimals

• Elementary Calculus: An Approach Using Infinitesimals(2nd Ed., 1986) by H. JeromeKeisler, an out-of-print nonstandard calculus textbook now available online under aCreative Commons license (PDF).

• Yet Another Calculus Textby Dan Sloughter, an introduction to calculus using infinitesi-mals available under a Creative Commons license (PDF).

• Calculusby Benjamin Crowell, an introduction to calculus available under a CreativeCommons license (PDF). Also see Crowell's"Five Free Calculus Textbooks"(2004)review on Slashdot.

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http://www.math.umn.edu/garrett/calculus/

http://synechism.org/drupal/de2de/

http://synechism.org/drupal/cfsv/

http://www.math.hawaii.edu/heiner/calculus.pdf

http://www.archive.org/details/elementsofdiffer00granuoft

http://en.wikisource.org/wiki/Elements_of_the_Differential_and_Integral_Calculus

http://en.wikipedia.org/wiki/Calculus_%28book%29

http://en.wikipedia.org/wiki/Michael_Spivak

http://www.math.wisc.edu/keisler/calc.html

http://synechism.org/drupal/yact/

http://www.lightandmatter.com/calc/

http://books.slashdot.org/books/04/03/04/028253.shtml

Choosing delta

Recall the definition of a limit:

A numberL is the limit of a functionf(x) asx approachesc if and only if for all numbersε> 0 there exists a numberδ > 0 such that

whenever

.

In other words, given a numberε we must construct a numberδ such that assuming

we can prove

;moreover, this proof must work forall values ofε > 0.

Note: this definition is not constructive -- it does not tell you how tofind the limit L, onlyhow to check whether a particular value is indeed the limit. We use the informal definition of thelimit, experience with similar problems, or theorems (L'Hopital's rule, for example), to determinethe value, and then can prove the correctness of this value using the formal definition.

Example 1: Suppose we want to find the limit off(x) = x + 5 asx approachesc = 9. We knowthat the limitL is 9+5=14, and desire to prove this.

We chooseδ = ε (this will be explained later). Then, since we assume

we can show

,which is what we wanted to prove.

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We choseδ by working backwards from the formula we are trying to prove:

.In this case, we desire to prove

,given

,so the easiest way to prove it is by choosingδ = ε. This example, however, is too easy to

adequately explain how to chooseδ in general. Lets try something harder:

Example 2: Prove that the limit off(x) = x² - 9 asx approaches 2 isL = -5.

We want to prove that

given

.

We chooseδ by working backwards. First, we need to rewrite the equation we want to proveusingδ instead ofx:

Note: we used the fact that |x + 2| <δ + 4, which can be proven with the triangle inequality.

Word of caution: the above series of equations is not a logical series of steps, and is not partof any proof, but is an informal technique used to help write the proof. We will select a value ofδ so that the last equation is true, and then use the last equation to prove the equations above itin turn (which is what was meant earlier byworking backwards).

Note: in the equations above, whenδ was substituted forx, the sign < was replaced with =.This can be done (but is not necessary) because we are not told that |x-2| =δ, but rather |x-2| <δ. The justification for this becomes clear when the above equations are used in backwards orderin the proof.

We can solve this last equation forδ using the quadratic formula:

Note:δ is alwaysin terms ofε. A constant value ofδ (e.g.,δ = 0.5) will never work.

Now, we have a value ofδ, and we can do our proof:

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given

,

.

Here a few more examples of choosingδ; try to figure them out before reading the explana-tion.

Example 3: Prove that the limit off(x) = sin(x)/x asx approaches 0 isL = 1.

Explanation:

Example 4: Prove thatf(x) = 1/x has no limit asx approaches 0.

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More Differentation Rules

Higher Order Derivatives→Calculus← Chain Rule


External links

• Online interactive exercises on derivatives• Visual Calculus - Interactive Tutorial on Derivatives, Differentiation, and Integration

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http://wims.unice.fr/wims/wims.cgi?module=home&search_keywords=derivative&search_category=X

http://archives.math.utk.edu/visual.calculus/2/index.html

Mean Value Theorem

Integration/Contents→Calculus← Rolle's Theorem

Mean Value Theorem

Mean Value TheoremIf

is continuous on the closed interval


, there exists a number,


such that

.

Examples


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http://en.wikibooks.org/w/index.php?title=File:Lagrange_mean_value_theorem.svg&filetimestamp=20060628113529

What does this mean? As usual, let us utilize an example to grasp the concept. Visualize (or

graph) the functionf(x) = x3. Choose an interval (anything will work), but for the sake of simplici-ty, [0,2]. Draw a line going from point (0,0) to (2,8). Between the pointsx = 0 andx = 2 exists anumberx = c, where the derivative off at pointc is equal to the slope of the line you drew.

Solution:

1: Using the definition of the mean value theorem

insert values. Our chosen interval is [0,2]. So, we have

2: By the definition of the mean value theorem, we know that somewhere in the intervalexists a point that has the same slope as that point. Thus, let us take the derivative to find thispointx = c.

Now, we know that the slope of the point is 4. So, the derivative at this pointc is 4. Thus, 4

= 3x2. The square root of 4/3 is the point.

Example 2:Find the point that satisifes the mean value theorem on the functionf(x) = sin(x)and the interval [0,π].

Solution:

1: Always start with the definition:

so,

(Remember, sin(π) and sin(0) are both 0.)

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2: Now that we have the slope of the line, we must find the pointx = c that has the sameslope. We must now get the derivative!

The cosine function is 0 atπ / 2 + πn (wheren is an integer). Remember, we are bound bythe interval [0,π], soπ / 2 is the pointc that satisfies the Mean Value Theorem.

Differentials

Assume a functiony = f(x) that is differentiable in the open interval (a,b) that contains x.Δy=

The "Differential of x" is theΔx. This is an approximate change in x and can be considered"equivalent" todx. The same holds true for y. What is this saying? One can approximate a changein y by knowing a change in x and a change in x at a point very nearby. Let us view an example.

Example: A schoolteacher has asked her students to discover what 4.12 is. The students,bereft of their calculators, are too lazy to multiply this out by hand or in their head and desire toutilize calculus. How can they approximate this?

1: Set up a function that mimics the procedure. What are they doing? They are taking anumber (Call it x) and they are squaring it to get a new number (call it y). Thus, y = x^2 Writeyourself a small chart. Make notes of values for x, y,Δx, Δy, and

. We are seeking what y really is, but we need the change in y first.

2: Choose a number close by that is easy to work with. Four is very close to 4.1, so writethat down as x. Yourδx is .1 (This is the "change" in x from the approximation point to the pointyou chose.)

3: Take the derivative of your function.

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= 2x. Now, "split" this up (This is not really what is happening, but to keep things simple, assumeyou are "multiplying"dxover.)

3b. Now you havedy

. We are assumingdyanddxare approximately the same as the change in x, thus we can useΔx and y.

3c. Insert values:dy

. Thus,dy= .8.

4: To findF(4.1), takeF(4) +dy to get an approximation. 16 + .8 = 16.8; This approximationis nearly exact (The real answer is 16.81. This is only one hundredth off!)

Definition of Derivative

The exact value of the derivative at a point is the rate of change over an infinitely smalldistance, approaching zero. Therefore, if h approaches 0 and the function is f(x):

If h approaches 0, then:

Cauchy's Mean Value Theorem

Cauchy's Mean Value TheoremIf

,

are continuous on the closed interval

Calculus

- 537 -



,

and

, then there exists a number,


such that

.

Calculus

- 538-


Infinite series

Sequences→Calculus← Polar Integration


Basics


Series and calculus


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GNU Free Documentation License

Version 1.2, November 2002

Copyright (C) 2000,2001,2002 Free Software Foundation, Inc.51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USAEveryone is permitted to copy and distribute verbatim copiesof this license document, but changing it is not allowed.

0. PREAMBLE

The purpose of this License is to make a manual, textbook, or other functional and usefuldocument "free" in the sense of freedom: to assure everyone the effective freedom to copy andredistribute it, with or without modifying it, either commercially or noncommercially.Secondarily, this License preserves for the author and publisher a way to get credit for their work,while not being considered responsible for modifications made by others.

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A "Modified Version" of the Document means any work containing the Document or aportion of it, either copied verbatim, or with modifications and/or translated into anotherlanguage.

A "Secondary Section" is a named appendix or a front-matter section of the Document thatdeals exclusively with the relationship of the publishers or authors of the Document to theDocument's overall subject (or to related matters) and contains nothing that could fall directlywithin that overall subject. (Thus, if the Document is in part a textbook of mathematics, aSecondary Section may not explain any mathematics.) The relationship could be a matter ofhistorical connection with the subject or with related matters, or of legal, commercial,philosophical, ethical or political position regarding them.

The "Invariant Sections" are certain Secondary Sections whose titles are designated, as beingthose of Invariant Sections, in the notice that says that the Document is released under thisLicense. If a section does not fit the above definition of Secondary then it is not allowed to bedesignated as Invariant. The Document may contain zero Invariant Sections. If the Documentdoes not identify any Invariant Sections then there are none.

The "Cover Texts" are certain short passages of text that are listed, as Front-Cover Texts orBack-Cover Texts, in the notice that says that the Document is released under this License. AFront-Cover Text may be at most 5 words, and a Back-Cover Text may be at most 25 words.

A "Transparent" copy of the Document means a machine-readable copy, represented in aformat whose specification is available to the general public, that is suitable for revising thedocument straightforwardly with generic text editors or (for images composed of pixels) genericpaint programs or (for drawings) some widely available drawing editor, and that is suitable forinput to text formatters or for automatic translation to a variety of formats suitable for input totext formatters. A copy made in an otherwise Transparent file format whose markup, or absenceof markup, has been arranged to thwart or discourage subsequent modification by readers is notTransparent. An image format is not Transparent if used for any substantial amount of text. Acopy that is not "Transparent" is called "Opaque".

Examples of suitable formats for Transparent copies include plain ASCII without markup,Texinfo input format, LaTeX input format, SGML or XML using a publicly available DTD, andstandard-conforming simple HTML, PostScript or PDF designed for human modification.Examples of transparent image formats include PNG, XCF and JPG. Opaque formats includeproprietary formats that can be read and edited only by proprietary word processors, SGML orXML for which the DTD and/or processing tools are not generally available, and themachine-generated HTML, PostScript or PDF produced by some word processors for outputpurposes only.

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The "Title Page" means, for a printed book, the title page itself, plus such following pagesas are needed to hold, legibly, the material this License requires to appear in the title page. Forworks in formats which do not have any title page as such, "Title Page" means the text near themost prominent appearance of the work's title, preceding the beginning of the body of the text.

A section "Entitled XYZ" means a named subunit of the Document whose title either isprecisely XYZ or contains XYZ in parentheses following text that translates XYZ in anotherlanguage. (Here XYZ stands for a specific section name mentioned below, such as"Acknowledgements", "Dedications", "Endorsements", or "History".) To "Preserve the Title" ofsuch a section when you modify the Document means that it remains a section "Entitled XYZ"according to this definition.

The Document may include Warranty Disclaimers next to the notice which states that thisLicense applies to the Document. These Warranty Disclaimers are considered to be included byreference in this License, but only as regards disclaiming warranties: any other implication thatthese Warranty Disclaimers may have is void and has no effect on the meaning of this License.

2. VERBATIM COPYING

You may copy and distribute the Document in any medium, either commercially ornoncommercially, provided that this License, the copyright notices, and the license notice sayingthis License applies to the Document are reproduced in all copies, and that you add no otherconditions whatsoever to those of this License. You may not use technical measures to obstructor control the reading or further copying of the copies you make or distribute. However, you mayaccept compensation in exchange for copies. If you distribute a large enough number of copiesyou must also follow the conditions in section 3.

You may also lend copies, under the same conditions stated above, and you may publiclydisplay copies.

3. COPYING IN QUANTITY

If you publish printed copies (or copies in media that commonly have printed covers) of theDocument, numbering more than 100, and the Document's license notice requires Cover Texts,you must enclose the copies in covers that carry, clearly and legibly, all these Cover Texts:Front-Cover Texts on the front cover, and Back-Cover Texts on the back cover. Both covers mustalso clearly and legibly identify you as the publisher of these copies. The front cover must presentthe full title with all words of the title equally prominent and visible. You may add other materialon the covers in addition. Copying with changes limited to the covers, as long as they preservethe title of the Document and satisfy these conditions, can be treated as verbatim copying in otherrespects.

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If the required texts for either cover are too voluminous to fit legibly, you should put thefirst ones listed (as many as fit reasonably) on the actual cover, and continue the rest ontoadjacent pages.

If you publish or distribute Opaque copies of the Document numbering more than 100, youmust either include a machine-readable Transparent copy along with each Opaque copy, or statein or with each Opaque copy a computer-network location from which the general network-usingpublic has access to download using public-standard network protocols a complete Transparentcopy of the Document, free of added material. If you use the latter option, you must takereasonably prudent steps, when you begin distribution of Opaque copies in quantity, to ensurethat this Transparent copy will remain thus accessible at the stated location until at least one yearafter the last time you distribute an Opaque copy (directly or through your agents or retailers) ofthat edition to the public.

It is requested, but not required, that you contact the authors of the Document well beforeredistributing any large number of copies, to give them a chance to provide you with an updatedversion of the Document.

4. MODIFICATIONS

You may copy and distribute a Modified Version of the Document under the conditions ofsections 2 and 3 above, provided that you release the Modified Version under precisely thisLicense, with the Modified Version filling the role of the Document, thus licensing distributionand modification of the Modified Version to whoever possesses a copy of it. In addition, youmust do these things in the Modified Version:

A. Use in the Title Page (and on the covers, if any) a title distinct from that of theDocument, and from those of previous versions (which should, if there were any, be listedin the History section of the Document). You may use the same title as a previous versionif the original publisher of that version gives permission.

B. List on the Title Page, as authors, one or more persons or entities responsible forauthorship of the modifications in the Modified Version, together with at least five of theprincipal authors of the Document (all of its principal authors, if it has fewer than five),unless they release you from this requirement.

C. State on the Title page the name of the publisher of the Modified Version, as thepublisher.

D. Preserve all the copyright notices of the Document.

E. Add an appropriate copyright notice for your modifications adjacent to the othercopyright notices.

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F. Include, immediately after the copyright notices, a license notice giving the publicpermission to use the Modified Version under the terms of this License, in the form shownin the Addendum below.

G. Preserve in that license notice the full lists of Invariant Sections and required CoverTexts given in the Document's license notice.

H. Include an unaltered copy of this License.

I. Preserve the section Entitled "History", Preserve its Title, and add to it an item statingat least the title, year, new authors, and publisher of the Modified Version as given on theTitle Page. If there is no section Entitled "History" in the Document, create one stating thetitle, year, authors, and publisher of the Document as given on its Title Page, then add anitem describing the Modified Version as stated in the previous sentence.

J. Preserve the network location, if any, given in the Document for public access to aTransparent copy of the Document, and likewise the network locations given in theDocument for previous versions it was based on. These may be placed in the "History"section. You may omit a network location for a work that was published at least four yearsbefore the Document itself, or if the original publisher of the version it refers to givespermission.

K. For any section Entitled "Acknowledgements" or "Dedications", Preserve the Title ofthe section, and preserve in the section all the substance and tone of each of the contributoracknowledgements and/or dedications given therein.

L. Preserve all the Invariant Sections of the Document, unaltered in their text and in theirtitles. Section numbers or the equivalent are not considered part of the section titles.

M. Delete any section Entitled "Endorsements". Such a section may not be included in theModified Version.

N. Do not retitle any existing section to be Entitled "Endorsements" or to conflict in titlewith any Invariant Section.

O. Preserve any Warranty Disclaimers.

If the Modified Version includes new front-matter sections or appendices that qualify asSecondary Sections and contain no material copied from the Document, you may at your optiondesignate some or all of these sections as invariant. To do this, add their titles to the list ofInvariant Sections in the Modified Version's license notice. These titles must be distinct fromany other section titles.

You may add a section Entitled "Endorsements", provided it contains nothing butendorsements of your Modified Version by various parties--for example, statements of peerreview or that the text has been approved by an organization as the authoritative definition of astandard.

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You may add a passage of up to five words as a Front-Cover Text, and a passage of up to25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version.Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or througharrangements made by) any one entity. If the Document already includes a cover text for thesame cover, previously added by you or by arrangement made by the same entity you are actingon behalf of, you may not add another; but you may replace the old one, on explicit permissionfrom the previous publisher that added the old one.

The author(s) and publisher(s) of the Document do not by this License give permission touse their names for publicity for or to assert or imply endorsement of any Modified Version.

5. COMBINING DOCUMENTS

You may combine the Document with other documents released under this License, underthe terms defined in section 4 above for modified versions, provided that you include in thecombination all of the Invariant Sections of all of the original documents, unmodified, and listthem all as Invariant Sections of your combined work in its license notice, and that you preserveall their Warranty Disclaimers.

The combined work need only contain one copy of this License, and multiple identicalInvariant Sections may be replaced with a single copy. If there are multiple Invariant Sectionswith the same name but different contents, make the title of each such section unique by addingat the end of it, in parentheses, the name of the original author or publisher of that section ifknown, or else a unique number. Make the same adjustment to the section titles in the list ofInvariant Sections in the license notice of the combined work.

In the combination, you must combine any sections Entitled "History" in the various originaldocuments, forming one section Entitled "History"; likewise combine any sections Entitled"Acknowledgements", and any sections Entitled "Dedications". You must delete all sectionsEntitled "Endorsements."

6. COLLECTIONS OF DOCUMENTS

You may make a collection consisting of the Document and other documents released underthis License, and replace the individual copies of this License in the various documents with asingle copy that is included in the collection, provided that you follow the rules of this Licensefor verbatim copying of each of the documents in all other respects.

You may extract a single document from such a collection, and distribute it individuallyunder this License, provided you insert a copy of this License into the extracted document, andfollow this License in all other respects regarding verbatim copying of that document.

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7. AGGREGATION WITH INDEPENDENT WORKS

A compilation of the Document or its derivatives with other separate and independentdocuments or works, in or on a volume of a storage or distribution medium, is called an"aggregate" if the copyright resulting from the compilation is not used to limit the legal rights ofthe compilation's users beyond what the individual works permit. When the Document is includedin an aggregate, this License does not apply to the other works in the aggregate which are notthemselves derivative works of the Document.

If the Cover Text requirement of section 3 is applicable to these copies of the Document,then if the Document is less than one half of the entire aggregate, the Document's Cover Textsmay be placed on covers that bracket the Document within the aggregate, or the electronicequivalent of covers if the Document is in electronic form. Otherwise they must appear on printedcovers that bracket the whole aggregate.

8. TRANSLATION

Translation is considered a kind of modification, so you may distribute translations of theDocument under the terms of section 4. Replacing Invariant Sections with translations requiresspecial permission from their copyright holders, but you may include translations of some or allInvariant Sections in addition to the original versions of these Invariant Sections. You mayinclude a translation of this License, and all the license notices in the Document, and anyWarranty Disclaimers, provided that you also include the original English version of this Licenseand the original versions of those notices and disclaimers. In case of a disagreement between thetranslation and the original version of this License or a notice or disclaimer, the original versionwill prevail.

If a section in the Document is Entitled "Acknowledgements", "Dedications", or "History",the requirement (section 4) to Preserve its Title (section 1) will typically require changing theactual title.

9. TERMINATION

You may not copy, modify, sublicense, or distribute the Document except as expresslyprovided for under this License. Any other attempt to copy, modify, sublicense or distribute theDocument is void, and will automatically terminate your rights under this License. However,parties who have received copies, or rights, from you under this License will not have theirlicenses terminated so long as such parties remain in full compliance.

10. FUTURE REVISIONS OF THIS LICENSE

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The Free Software Foundation may publish new, revised versions of the GNU FreeDocumentation License from time to time. Such new versions will be similar in spirit to thepresent version, but may differ in detail to address new problems or concerns. Seehttp://www.gnu.org/copyleft/.

Each version of the License is given a distinguishing version number. If the Documentspecifies that a particular numbered version of this License "or any later version" applies to it,you have the option of following the terms and conditions either of that specified version or ofany later version that has been published (not as a draft) by the Free Software Foundation. If theDocument does not specify a version number of this License, you may choose any version everpublished (not as a draft) by the Free Software Foundation.

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http://www.gnu.org/copyleft/

http://www.gnu.org/copyleft/

List of Contributors

Cronholm144The Grimm RipperDysprosiaJuliusrossTiledWhiteknightSwiftMrwojoIamunknownAdrignolaRoadrunnerThenub314TechnochefCarandolXiaodaiRobert HorningPseudohanSBJohnnyGwilmKarl WickZginderZondorW3asalWnealSbroolsThomas.haslwanterNumberTheoristRam einsteinMike.lifeguardMknMetricMihoshiLarge and in chargeLeonusIntManJgukHerbythymeHerraoticGraemeb1967HagindazEtscrivnerGeoffreyDcljrEric119DavidMcKenzieDavidmanheim

Darklama

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http://en.wikibooks.org/wiki/User:Cronholm144

http://en.wikibooks.org/wiki/User:The_Grimm_Ripper

http://en.wikibooks.org/wiki/User:Dysprosia

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http://en.wikibooks.org/wiki/User:Swift

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http://en.wikibooks.org/wiki/User:Eric119

http://en.wikibooks.org/wiki/User:DavidMcKenzie

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http://en.wikibooks.org/wiki/User:Darklama