our textbookmawjj.people.ust.hk/math1013_course/sec1-1.pdf · •chapter 2: limits and derivatives...

Our textbook:

Contents covered:

• Chapter 1: Functions and Models

• Chapter 2: Limits and Derivatives

• Chapter 3: Differentiation rules

• Chapter 4: Applications of Differentiation

• Chapter 5: Integrals

• Chapter 6: Applications of Integration

Copyright © 2008 Pearson Education, Inc.

Publishing as Pearson Addison-Wesley

Chapter 1 Sec1.1. Function

• Important！Functions are the major objects we deal with in calculus, because they are key to describing the real world in mathematical terms.

Examples

The temperature at which water boils depends on the elevation above sea

level

The interest paid on a cash investment depends on the length of time the

investment is held.

The area of a circle depends on the radius of the circle

Definition of Function

• The letter x, called the independent variable, represents the input value of ƒ

• y, the dependent variable, represents the corresponding output value of ƒ at x.

• The set D of all possible input values is called the domain of the function.

• The set of all values of ƒ(x) as x varies throughout D is called the range of the function.





Function: a kind of machine

• Example:

( )f x x



Arrow Diagram

Graphs of Functions

• If ƒ is a function with domain D, its graph consists of the points in the Cartesian plane whose coordinates are the input-output pairs for ƒ.



{( , ( )) | }x f x x D

Piecewise-Defined Functions

• Sometimes a function is described by using different formulas on different parts of its domain.

Absolute Value Function



The Greatest Integer Function

The function whose value at any number x is the greatest integer less than or equal to x is called the greatest integer function or the integer floor function. Denoted by .



x

The Least Integer Function

The function whose value at any number x is the smallest integer greater than or equal to x is called the least integer function or the integer ceiling function. Denoted by .



x

13. Increasing Versus Decreasing Functions

• Definition:

• We say that a function is monotone increasing (Decreasing) in Domain D, if

whenever



1 2 1 2( ) ( )( ( ) ( )),f x f x f x f x

1 2 1 2, , .x x D x x

14. Even Function, Odd Function

Remark:

• The graph of an even function is symmetric about the y-axis;

• The graph of an odd function is symmetric about the origin.



our textbookmawjj.people.ust.hk/math1013_course/sec1-1.pdf · •chapter 2: limits and derivatives...

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