Class Notes for

MATH 3A Calculus

Spring 2013

Prepared by

Stephanie Sorenson

Table of Contents 1.1 Limits (An Intuitive Approach) .......................................................................................................................... 1

1.2 Computing Limits ............................................................................................................................................. 6

1.3 Limits at Infinity; End Behavior of a Function ................................................................................................ 15

1.5 Continuity ....................................................................................................................................................... 21

1.6 Continuity of Trigonometric Functions .......................................................................................................... 30

2.1 Tangent Lines and Rates of Change ............................................................................................................... 34

2.2 The Derivative Function ................................................................................................................................. 42

2.3 Introduction to Techniques of Differentiation ............................................................................................... 48

2.4 The Product and Quotient Rules .................................................................................................................... 53

2.5 Derivatives of Trigonometric Functions ......................................................................................................... 56

2.6 The Chain Rule ................................................................................................................................................ 61

2.7 Implicit Differentiation ................................................................................................................................... 66

2.8 Related Rates .................................................................................................................................................. 71

2.9 Local Linear Approximation; Differentials ...................................................................................................... 76

3.1 Analysis of Functions I: Increase, Decrease, and Concavity ........................................................................... 86

3.2 Analysis of Functions II: Relative Extrema ...................................................................................................... 94

3.3 Analysis of Functions III: Rational Functions, Cusps and Vertical Tangents ................................................. 101

3.4 Absolute Maxima and Minima ..................................................................................................................... 107

3.5 Applied Maximum and Minimum Problems ................................................................................................ 110

3.6 Rectilinear Motion ........................................................................................................................................ 118

3.7 Newton’s Method ........................................................................................................................................ 122

3.8 Rolle’s Theorem; Mean-Value Theorem ...................................................................................................... 125

4.1 An Overview of the Area Problem ............................................................................................................... 128

4.2 The Indefinite Integral .................................................................................................................................. 133

4.3 Integration by Substitution .......................................................................................................................... 140

4.4 The Definition of Area as a Limit; Sigma Notation ....................................................................................... 145

4.5 The Definite Integral ..................................................................................................................................... 152

4.6 The Fundamental Theorem of Calculus ....................................................................................................... 160

4.7 Rectilinear Motion Revisited Using Integration ........................................................................................... 170

4.8 Average Value of a Function and its Applications ........................................................................................ 178

4.9 Evaluating Definite Integrals by Substitution ............................................................................................... 180

5.1 Area Between Two Curves ........................................................................................................................... 182

5.2 Volumes by Slicing; Disks and Washers ........................................................................................................ 187

5.3 Volumes by Cylindrical Shells ....................................................................................................................... 196

5.4 Length of a Plane Curve ................................................................................................................................ 201

5.5 Area of a Surface of Revolution .................................................................................................................... 204

5.6 Work ............................................................................................................................................................. 207

5.8 Fluid Pressure and Force .............................................................................................................................. 214

6.1 Exponential and Logarithmic Functions ....................................................................................................... 217

6.2 Derivatives and Integrals Involving Logarithmic Functions .......................................................................... 222

6.3 Derivatives of Inverse Functions & Derivatives and Integrals involving Exponential Functions .................. 228

Section 1.1 1

1.1 Limits (An Intuitive Approach)

The concept of a “limit” is the fundamental building block on which all calculus concepts are based. In this section we will study limits informally, with the goal of developing an intuitive feel for the basic ideas. In the next few sections we will focus on computational methods.

Limits Let us examine the behavior of the function for -values closer and closer to 2.

1 1.5 1.9 1.99 1.999 2 2.001 2.01 2.1 2.5 3 1 2.25 3.61 3.9601 3.996001 4.004001 4.0401 4.41 6.25 9

Since the values of can be made as close as we like to ________ by taking values of sufficiently close to ________ (but not equal to ________), then we write:

Limits (An Informal View) (A.K.A. “Two-Sided Limits”) If the values of can be made as close as we like to by taking values of sufficiently close to (but not equal to ), then we write

approaching 2 from the LEFT side

approaching 2 from the RIGHT side

2 Section 1.1

One-Sided Limits (An Informal View) If the values of can be made as close as we like to by taking values of sufficiently close to (but greater than ), then we write

If the values of can be made as close as we like to by taking values of sufficiently close to (but less than ), then we write

The Relationship between One-Sided and Two-Sided Limits The two-sided limit of a function exists at if and only if both of the one-sided limits exist at and have the same value; that is,

Example 1 Complete the table and make a guess about the value of the limit. Assume is in radians. (Note: You would get a different answer if was in degrees You should check this for yourself!)

CAUTION: Numerical evidence can sometimes lead to incorrect conclusions about limits because of roundoff error or because the sample values chosen do not reveal the true limiting behavior.

Section 1.1 3

Example 2 For the function graphed below, find

Example 3 For the function graphed below, find

4 Section 1.1

Infinite Limits Sometimes one-sided or two-sided limits fail to exist because the values of the function increase or decrease without bound. In such cases, we use the symbols and to describe the particular way in which the limit fails to exist.

The line is called a vertical asymptote of the curve .

Section 1.1 5

Example 4 For the function graphed below, find

The vertical asymptotes of the graph of Example 5 For the function graphed below (refer to Exercise #10 in the textbook), find

The vertical asymptotes of the graph of

6 Section 1.2

1.2 Computing Limits

Some Basic Limits Theorem* Let be a real number, and suppose that

That is, the limits exist and have values and , respectively. Then:

*These statements are also true for the one-sided limits as or as .

(a)

(b)

(c)

(d)

(e)

Section 1.2 7

Let’s look at a special case of part (c):

Consider the constant function .

The theorem actually holds for any finite number of functions (not just two functions!)…

In general,

Consider then the case when .

We summarize our findings: Special Cases

8 Section 1.2

Example 1

Section 1.2 9

Example 2

Theorem

10 Section 1.2

Example 3

Recall that a rational function is a ratio of two polynomials. Example 4

Theorem

Section 1.2 11

CASE 1:

For polynomials and ,

12 Section 1.2

CASE 2*:

*Note: You must determine the zeros of both the numerator AND the denominator to perform the sign analysis necessary for Case 2.

Section 1.2 13

Example 5 Find

******************************************************************************

A quotient

in which the numerator and denominator both have a limit of zero as is called

_____________________________________________________. We have discussed what happens

when and are polynomials, but when the functions are not polynomials, it is difficult to tell

by inspection whether the limit exists. Why?

We will be studying methods for evaluating general limits of indeterminate form of type 0/0 in later sections.

14 Section 1.2

Limits involving Radicals Sometimes, limits of indeterminate form of type 0/0 can be found by clever algebraic simplification. For limits involving radicals, try rationalizing the denominator! Example 6

Limits of Piecewise-defined Functions For functions that are defined piecewise, a two-sided limit at a point where the formula changes must be obtained by first finding the one-sided limits at that point. Example 7

Find

Section 1.3 15

1.3 Limits at Infinity; End Behavior of a Function

Limits at Infinity and Horizontal Asymptotes The behavior of a function as increases without bound ( ) or decreases without

bound ( ) is called ____________________________ of the function.

If the values of eventually get as close as we like to a number as increases without bound, then we write If the values of eventually get as close as we like to a number as decreases without bound, then we write If either limit holds, we call the line a _______________________________________ for the graph of . ****************************************************************************** Infinite Limits at Infinity Note: It is possible that the values of may increase or decrease without bound as or . Thus, the four cases are possible:

16 Section 1.3

Limit Laws for Limits at Infinity All the limit laws still apply for limits at infinity, provided the limits exist (as finite numbers). Example 1

Limits of as Recall the graphs of the basic power functions, : is odd is even

( (

Example 2

Section 1.3 17

Limits of Polynomials as Recall, the end behavior of a polynomial matches the end behavior of its highest degree term. Example 3

Limits of Rational Functions as To determine the end behavior of a rational function, divide each term in the numerator and denominator by the highest power of that occurs in the denominator. Example 4

18 Section 1.3

A Quick Method for Finding Limits of Rational Functions as Example 5

End Behavior of Trigonometric Functions

Section 1.3 19

Limits Involving Radicals To determine the end behavior of a function involving radicals, one strategy is to divide each term in

the numerator and denominator by , where is the degree of the polynomial under the radical expression. (We will only be considering the case when is even). It may help to recall the following:

Example 6

20 Section 1.3

Sometimes it is necessary to rationalize the numerator when the denominator is 1, as in the next example. Example 7

Section 1.5 21

1.5 Continuity An “Intuitive Approach” to Continuity The following functions have a discontinuity at : Definition of Continuity A function is said to be continuous at provided the following conditions are satisfied:

1.

2.

3.

22 Section 1.5

Example 1 Determine whether the following functions are continuous at . If not, determine whether it is a removable discontinuity.

Section 1.5 23

Definition of Continuity on an Interval

A function is continuous on an open interval if

. In the case where is continuous on we say that

.

is continuous from the left at if: is continuous from the right at if:

A function is continuous on a closed interval if the following conditions are satisfied:

1.

2.

3.

24 Section 1.5

Example 2

What can you say about the continuity of the function ?

Continuity Theorem 1 If the functions and are each continuous at , then the following are also continuous at :

(a) (b) (c)

(d)

****************************************************************************** Proof of (a):

Section 1.5 25

Continuity Theorem 2

(a) A polynomial

(b) A rational function

****************************************************************************** Proof of (a): Example 3 Find values of , if any, at which is not continuous. Determine whether each such value is a removable discontinuity.

26 Section 1.5

Example 4 Show that is continuous everywhere. Limit of Compositions Theorem

Section 1.5 27

*Special Case* - Limit of the Absolute Value of a Function Let . By the Limit of Compositions Theorem:

Example 5

Continuity of Compositions Theorem

******************************************************************************* Proof: Note: If and are continuous everywhere, then is continuous everywhere.

28 Section 1.5

Special Cases: Let . Then is continuous everywhere (see Example 4). By the Continuity of Compositions

Theorem, is continuous wherever is continuous.

Let , then is continuous for (see Example 2). By the Continuity of Compositions

Theorem, is continuous wherever is continuous and .

Example 6 Where is the function continuous?

Continuity of Inverse Functions Theorem The inverse of a continuous function is continuous.

Section 1.5 29

The Intermediate Value Theorem

30 Section 1.6

1.6 Continuity of Trigonometric Functions

Recall the domains of the six basic trigonometric functions: Domain

Theorem If is any number in the natural domain of the stated trigonometric function, then

Corollary The six basic trigonometric functions are continuous on their domains. Example 1 Find the limit.

Section 1.6 31

Example 2 Find the discontinuities, if any.

The Squeezing Theorem Let be functions satisfying

for all near . If and have the same limit as approaches , say

then also has this limit as approaches , that is,

Example 3 Find the limit. Use the Squeezing Theorem.

32 Section 1.6

Theorem

******************************************************************************** Note: The proof of (a) uses the Squeezing Theorem and is provided in the textbook. It is a difficult proof, and you will not be required to learn it in this class. Proof of (b): Example 4 Find the limits.

Section 1.6 33

Example 5

34 Section 2.1

2.1 Tangent Lines and Rates of Change Suppose is a point on the graph of at . Let represent any other point.

The slope of the secant line is given by:

Or alternatively,

The tangent line to the graph of at is

The slope of the tangent line at (or at is given by:

Or alternatively,

Section 2.1 35

Recall, the equation of the line through the point with slope is given by:

****************************************************************************** Example 1

(a) Find the slope of the secant line to the parabola through and .

(b) Find the slope of the tangent line to the parabola at .

(c) Find an equation for the tangent line to the parabola at .

36 Section 2.1

Rates of Change If is a function of , that is , then the average rate of change of with respect to over the interval is given by:

The instantaneous rate of change of with respect to at is found by taking the limit of the average rates of change over intervals as .

Section 2.1 37

Example 2 Find the average rate of change of with respect to over the interval .

Example 3 Find the instantaneous rate of change of with respect to at .

38 Section 2.1

Example 4 Find the instantaneous rate of change of with respect to at an arbitrary value of .

Velocity Velocity is the rate of change of position with respect to time elapsed. We will only consider for now an object moving along a line (this is called rectilinear motion). Thus, the object can have positive or negative velocity depending on the direction of movement.

Section 2.1 39

If an object in rectilinear motion moves along an -axis so that it’s position as a function of time elapsed is

then is called the position function of the object. The graph of is called the position versus time curve for the object. The average velocity of an object over a time interval is the average rate of change of the object’s position, , with respect to time elapsed, . Thus,

The instantaneous velocity of an object at time is the limit of its average velocities over time intervals between and as . Thus,

Time

Position

40 Section 2.1

Example 5 A particle starts at on the real number line. It’s position vs. time curve is given.

(a) Find the values of at which the instantaneous velocity is positive.

In which direction is the particle moving during this time?

(b) Find the values of at which the instantaneous velocity is negative.

In which direction is the particle moving during this time?

(c) Find the instantaneous velocity when .

What does this mean for the particle?

(d) Find the average velocity over the interval .

(e) Estimate the value of when the object is moving to the left most rapidly.

Section 2.1 41

Example 6 A ball is released from rest at a height of feet above the ground, and falls straight down. The height of the object is given by the position function

where is given in seconds.

(a) How high above the ground is the ball after 1 s?

(b) What is the average velocity of the ball during the first 1 s of falling?

(c) What is the instantaneous velocity of the ball at the end of 1 s?

42 Section 2.2

2.2 The Derivative Function

Recall, the slope of the tangent line of at , or the instantaneous rate of change of with respect to at is given by:

We can think of defining a function whose input is and whose output is the number that represents either the slope of the tangent line to at or the instantaneous rate of change of with respect to at . To emphasize this function point of view, we will replace by :

The function defined by the formula

is called . The domain of consists of all in the domain of for which .

The process of finding a derivative is called ________________________________. You can think of differentiation as an operation on the function . Here are other notations used to represent the derivative of :

Section 2.2 43

Example 1 Use the graph of to estimate the values.

Example 2

If , find

. Then sketch the graphs of and

on the same grid.

44 Section 2.2

Example 3 If , find and interpret the result.

Section 2.2 45

Example 4 Given that and , find an equation for the tangent line to the graph of at . Instantaneous Velocity If is the position function of an object in rectilinear motion, then the instantaneous velocity at an arbitrary time is given by:

Example 5 A particle’s position on the real number line at time in seconds is given by:

Find the velocity function of the particle. Interpret the meaning.

46 Section 2.2

Differentiability A function is said to be differentiable at if If is differentiable at each point of the open interval , then we say that it is differentiable on . In the case that is differentiable on , we say that is differentiable everywhere. Cases where is not differentiable at the point : Differentiability/Continuity Theorem:

Section 2.2 47

Example 6 Show that

is continuous and differentiable at .

48 Section 2.3

2.3 Introduction to Techniques of Differentiation

Suppose , for some constant . Then,

Derivative of a Constant Suppose is any real number. Then

Example 1

Derivative of Power Functions Observe the following, which can all be found using the definition of the derivative:

The Power Rule Suppose is any real number. Then

Section 2.3 49

Example 2 Find .

Derivative of a Constant Times a Function Suppose is a constant, and is differentiable at . Then,

Constant Multiple Rule If is differentiable at and is any real number, then is also differentiable at and

50 Section 2.3

Example 3 Find

Derivative of a Sum or Difference Suppose and are differentiable at . Then,

Sum and Difference Rules If and are differentiable at , then so are and and

*These rules can be extended to any finite number of functions.

Section 2.3 51

Example 4

Example 5

At what points, if any, does the graph of

have a tangent line with slope ?

Higher Derivatives The derivative of a function is itself a function and hence may have a derivative of its own. The derivative of is denoted and is called the second derivative of :

As long as each successive function is differentiable, we can continue this process of differentiating to obtain third, fourth, fifth and even higher derivatives of . The number of times that is differentiated is called the order of the derivative.

Alternate notation:

52 Section 2.3

Example 6 If , find the following:

Section 2.4 53

2.4 The Product and Quotient Rules

Consider the two functions and .

Compute the derivative of the product:

Now, let’s compute each derivative separately:

Thus taking the product of the derivatives:

What does this tell us?

The Product Rule If and are differentiable at , then so is the product , and

Proof:

54 Section 2.4

Example 1 Compute by:

(a) multiplying and then differentiating (b) using the product rule.

The Quotient Rule If and are differentiable at , and if , then is differentiable at and

Proof: The proof involves a similar strategy as the proof of the Product Rule, and is provided in the book.

Section 2.4 55

Example 2 Consider the function:

(a) Compute .

(b) Find all values of at which the tangent line to the curve is horizontal.

56 Section 2.5

2.5 Derivatives of Trigonometric Functions

Recall the following formulas for in radians:

****************************************************************************** Let’s compute:

Section 2.5 57

Derivatives of the Sine and Cosine Functions ( in radians)

Let’s compute:

Derivatives of the Other Trigonometric Functions*

*All of these can be derived by applying the Quotient Rule and using the derivatives of the sine and cosine function.

58 Section 2.5

Example 1 Find .

(a) (b)

(c)

Section 2.5 59

Example 2 Find if .

Example 3 Show that is a solution of the equation

60 Section 2.5

Example 4 An airplane is flying on a horizontal path at a height of 3800 ft, as shown. At what rate is the distance between the airplane and the fixed point changing with respect to when ?

3800 ft

Section 2.6 61

2.6 The Chain Rule Suppose . Then we can think of as a composition of two functions , where

_____________ and ______________ Let’s compute using techniques we’ve studied so far in the course: The Chain Rule If is differentiable at and is differentiable at , then the composition is differentiable at and

If we let , the chain rule can be expressed as:

62 Section 2.6

Example 1 Find .

Section 2.6 63

Example 2 Find .

In application problems, there is a more convenient way to represent the chain rule.

64 Section 2.6

Suppose: time (in hours) distance (in miles) amount of gas consumed (in gallons)

1. Which quantity represents how fast distance is changing with respect to time? What are the units?

2. Which quantity represents how fast the amount of gas consumed is changing with respect to distance? What are the units?

3. Which quantity represents how fast the amount of gas consumed is changing with respect to time? What are the units?

Section 2.6 65

Example 3 The following figure shows the graph of gas consumed (gal) versus a car’s distance (mi) for a single trip. The tangent line at is also shown on the graph.

If a car reaches a speed of 60 mph at the instant when it has traveled a distance of 12 miles in a single trip, at what rate is gas being consumed in gal/hr. at this instant?

Gas Consumed (gal)

Distance (mi)

6 12 18 24

2

1

3

0 0

30

4

66 Section 2.7

2.7 Implicit Differentiation Consider the equation:

Solving the equation for yields: Thus, does the original equation describe as a function of ? YES NO We say that the equation defines ____________________ as a function of . Whereas the equation defines ____________________ as a function of . Now, consider the equation:

Solving the equation for yields: Thus, does the original equation describe as a function of ? YES NO We say that the equation ____________________ defines the functions

and . Definition: An equation in and defines a function implicitly if the graph of coincides with a portion of the graph of the equation. Note: It is difficult or impossible to solve some equations explicitly for in terms of . For example,

Section 2.7 67

Implicit Differentiation Technique

Step 1: Differentiate both sides of the equation before solving for in terms of , treating as a differentiable function of (ie. must apply chain rule to ).

Step 2: Solve for . Step 3: (Optional) If possible, solve the original equation for in terms of , and then

substitute into the formula for . This expresses the derivative in terms of alone. (Note: Although it is usually more desirable to have a formula for expressed in terms of alone, having the formula in terms of and is not an impediment to finding slopes and equations of tangent lines.)

Example 1

(a) Solve the equation for as a function of , and find from that equation.

(b) Find by differentiating implicitly.

68 Section 2.7

Example 2 Find by implicit differentiation.

(a)

(b)

Section 2.7 69

Example 3 Find by implicit differentiation.

70 Section 2.7

Example 4

Find the slope of the tangent line to the curve at the points

and

.

Section 2.8 71

2.8 Related Rates Example 1 Suppose that and are both differentiable functions of and are related by the equation

Given that , find when .

Strategy for Solving Related Rates Problems Step 1) Draw a picture. Label quantities that vary with time with a variable, and label quantities that are fixed with a constant. Step 2) Identify all given quantities and the rate that is to be found (& when to find it). Step 3) Write an equation that relates the variables whose rates of change were identified in Step 2. Step 4) Differentiate both side of the equation from step 3 with respect to time. Step 5) Substitute all known values, and then solve for the unknown rate of change.

72 Section 2.8

Example 2 (Exercise #14) A spherical balloon is inflated so that its volume is increasing at the rate of 3 . How fast is the diameter of the balloon increasing when the radius is 1 ft?

Section 2.8 73

Example 3 (Exercise #30) A boat is pulled into a dock by means of a rope attached to a pulley on the dock. The rope is attached to the bow of the boat at a point 10 ft below the pulley. If the rope is pulled through the pulley at a rate of 20 ft/min, at what rate will the boat be approaching the dock when 125 ft of rope is out?

Pulley

74 Section 2.8

Example 4 Water is being drained from a conical water tank at a rate of . The water tank has a radius of 12 at the top and is 24 high. How fast is the depth of the water changing when the water is 10 deep.

Section 2.8 75

Example 5 (Exercise #34) An aircraft is flying at a constant altitude with a constant speed of 600 mi/h. An antiaircraft missile is fired on a straight line perpendicular to the flight path of the aircraft so that it will hit the aircraft at a point . At the instant the aircraft is 2 mi from the impact point the missile is 4 mi from and flying at 1200 mi/h. At that instant, how rapidly is the distance between missile and aircraft decreasing?

76 Section 2.9

2.9 Local Linear Approximation; Differentials If a function is differentiable at , then the tangent line exists at (and is non-vertical). The equation of the tangent line at is: Thus, the -value of any point along the tangent line at an arbitrary is given by: For values of near , we may approximate the value of the function, , by the value of the tangent line. (Note: The tangent line is tangent to the curve at )

This is called the _______________________________________________________________. If we let ___________, then ___________. So this formula can be expressed as Error in Local Linear Approximations: The accuracy of the local linear approximation to at will deteriorate as gets progressively farther from .

Section 2.9 77

Example 1 (a) Find the local linear approximation of at . (Use Formula 1)

(b) Use the local linear approximation in part (a) to approximate and without a calculator. Compare your approximation to the exact values obtained on a calculator.

(c) Graph and its tangent line at together, and use the graphs to illustrate the relationship between the exact values and the approximations of and .

78 Section 2.9

Example 2

Use an appropriate local linear approximation to estimate the value of . Compare your answer to the actual value. ****************************************************************************** Recall, there are two equivalent notations for the derivative of with respect to :

Up to now, we have interpreted as a single entity. What happens if we treat as an actual ratio of two quantities, and ? Multiplying both sides of the equality by yields: If we define to be an independent variable that can have any real value, then the above formula is how we will define . The symbols “ ” and “ ” are called ____________________________________.

Section 2.9 79

Geometrically, what does the differential represent? And how is it different from ? To summarize: Example 3

(a) Find a formula for .

(b) Find a formula for .

80 Section 2.9

Strategy: If , then to find the differential : First find the derivative , then multiply both sides through by . Example 4

Find the differential . ******************************************************************************

For close to 0, the slope of the secant line through and may be

approximated by the slope of the tangent line at . (Refer to the graph on Page 3.) That is,

Multiplying both sides by yields:

Since we may assume , we get the following result:

Hence, for close to 0,

******************************************************************************

Section 2.9 81

Example 5

(a) Find when changes from to . (Requires a calculator)

(b) Use the differential to approximate when changes from to . (Does not require a calculator! )

Error Propagation In scientific research and statistics, it is very important to be able to compute the “maximum error” of a calculation or measurement. In fact, all scientific measurements come with measurement errors included! Since researchers often must use these inexactly measured quantities to compute other quantities, the error is propagated (spread) from the measured quantities to the computed quantities. If , and the value of is to be computed from a measured value of , we define:

Measurement Error of (Measured Value, ) (Exact Value, )

Propagated Error of (Computed Value, ) (Exact Value, )

82 Section 2.9

Example 6 The side of a cube is measured to be 25 cm, with a possible error of cm.

(a) Use differentials to estimate the error in the calculated volume.

Section 2.9 83

If the exact value of a quantity is , and a measurement or calculation produces an error , then the relative error (or percentage error) is given by: Since the exact value of is usually unknown, the relative error is approximated by: The side of a cube is measured to be 25 cm, with a possible error of cm.

(b) Estimate the percentage errors in the side and volume of the cube.

84 Section 2.9

Example 7 The side of a square is measured with a possible percentage error of . Use differentials to estimate the percentage error in the area.

Section 2.9 85

Example 8 One side of a right triangle is known to be 25 cm exactly. The angle opposite to this side is measured to be , with a possible error of . Use differentials to estimate the error in the hypotenuse.

86 Section 3.1

3.1 Analysis of Functions I: Increase, Decrease, and Concavity

Increase/Decrease Increasing Decreasing Constant Recall the following definition from pre-calculus: Definition is increasing on an interval if ______________________ whenever . is decreasing on an interval if ______________________ whenever . is constant on an interval if ______________________ for all points and . Concavity Concave up _______________ water. Concave down ________________ water. Definition If is continuous on an open interval containing , and if changes concavity at , then we call

the point on the graph of an _________________________________________.

Section 3.1 87

(Closed) Intervals on which is increasing: (Closed) Intervals on which is decreasing: Theorem Let be a function that is ____________________________ on the closed interval and ____________________________ on the open interval .

1. is increasing on if _______________________ for all in .

2. is decreasing on if _______________________ for all in .

3. is constant on if _______________________ for all in . Strategy: To find intervals on which is increasing or decreasing:

1) Compute . 2) Determine where is zero or undefined in order to partition the real number line into

appropriate intervals. 3) Test a point from each interval in order to determine the sign of . 4) Wherever the sign of is positive, the graph of is increasing. Wherever the sign of is

negative, the graph of is decreasing.

88 Section 3.1

Example 1 Find the intervals on which is increasing and the intervals on which is decreasing.

(a)

Section 3.1 89

Find the intervals on which is increasing and the intervals on which is decreasing.

(b)

90 Section 3.1

(Open) intervals on which is concave up: (Open) intervals on which is concave down: -coordinates of all inflection points: Formal Definition for Concavity is concave up on an open interval if ___________________________________ on that interval. is concave down on an open interval if ___________________________________ on that interval. Theorem Let be twice differentiable on an open interval.

1. is concave up on if _______________________ for all in .

2. is concave down on if _______________________ for all in .

Strategy: To find intervals on which is concave up or concave down:

1) Compute . 2) Determine where is zero or undefined in order to partition the real number line into

appropriate intervals. 3) Test a point from each interval in order to determine the sign of . 4) Wherever the sign of is positive, the graph of is concave up. Wherever the sign of is

negative, the graph of is concave down.

Section 3.1 91

Example 2 Find the open intervals on which is concave up and the open intervals on which is concave down. Also, find the -coordinates of all inflection points.

(a)

92 Section 3.1

Find the open intervals on which is concave up and the open intervals on which is concave down. Also, find the -coordinates of all inflection points.

(b)

Section 3.1 93

Example 3 Find intervals of increasing, decreasing, concave up, and concave down. Also, find the -coordinates of all inflection points.

94 Section 3.2

3.2 Analysis of Functions II: Relative Extrema Definitions Suppose is in the domain of . has a relative maximum at if there is an open interval containing on which for all in the interval. has a relative minimum at if there is an open interval containing on which for all in the interval. is a critical point of if either _______________ or ________________________________.

(ie. either has a horizontal tangent line or is not differentiable at ). The critical point is a stationary point of if _______________. Identify all the relative extrema, critical points, and stationary points:

Section 3.2 95

Theorem Suppose that is a function defined on an open interval containing . If has a relative extremum at

, then ___________________________________________________.

Note: The reverse implication is not true in general. If is a critical point, this does NOT imply has a relative extremum at . (See the last graph on the previous page.) Important Implication of the Theorem: First Derivative Test Suppose that is continuous at a critical point .

1. If changes from positive to negative at , then has a _________________________ at .

2. If changes from negative to positive at , then has a _________________________ at .

3. If does not change sign at , then does not have a relative extremum at .

96 Section 3.2

Example 1 Use the First Derivative Test to locate the relative extrema of .

(a)

Section 3.2 97

Use the First Derivative Test to locate the relative extrema of .

(b)

98 Section 3.2

Second Derivative Test Suppose that is twice differentiable at . 1. If and is positive, then has a _______________________________________ at .

2. If and is negative, then has a ______________________________________ at .

3. If and , then ____________________________________________________.

Example 2 Use the Second Derivative Test to locate the relative extrema of .

(a)

,

Section 3.2 99

Use the Second Derivative Test to locate the relative extrema of .

(b)

100 Section 3.2

Example 3 Use any method to find the relative extrema of the function .

Section 3.3 101

3.3 Analysis of Functions III: Rational Functions, Cusps and Vertical Tangents

How to graph a rational function,

1) Factor the numerator and denominator, and cancel common factors.

(Recall, if the factor cancels, there is a hole at .)

2) Find - and - intercepts.

3) Find vertical asymptotes (where the denominator is zero after canceling common factors).

4) Determine where the graph is above or below the -axis by performing a sign analysis on . The -intercepts and vertical asymptotes partition the real line into test intervals.

5) Determine the end behavior of the graph. Case 1: If deg deg , is a H.A. Case 2: If deg deg , is a H.A., where is the ratio of

leading coefficients Case 3: If deg deg , there is a slant (oblique) asymptote or a

curvilinear asymptote. Use long division to rewrite the quotient and determine the end behavior.

Example of Slant Asymptote: Example of Curvilinear Asymptote:

6) Find , and perform sign analysis on in order to locate stationary points, intervals of increasing or decreasing, and relative extrema.

7) Find , and perform sign analysis on in order to determine concavity and locate inflection points.

102 Section 3.3

Example 1

Sketch the graph of

(You may use the next page to continue work on this problem, if needed.)

Section 3.3 103

104 Section 3.3

Example 2

Sketch the graph of

(You may use the next page to continue work on this problem, if needed.)

Section 3.3 105

106 Section 3.3

Graphs with Vertical Tangents and Cusps Example 3

Sketch the graph of

Section 3.4 107

3.4 Absolute Maxima and Minima Some graphs of absolute maxima and minima: The Extreme-Value Theorem If a function is continuous on a finite closed interval , then ________________________

_____________________________________________________________________________.

Specifically, the absolute extrema occur either at ______________________________________

or at _________________________________________________________________________.

Strategy to find absolute max./min. of a continuous function on the closed interval : 1. Find the critical points in . (where or is undefined) 2. Evaluate at all the critical points and at the endpoints and . The largest value is the

absolute maximum value. The smallest value is the absolute minimum value.

108 Section 3.4

Example 1 Find the absolute maximum and minimum values of on the given interval, and state where those values occur.

(a)

(b)

Section 3.4 109

When determining absolute extrema of a polynomial over the interval , one must consider the polynomial’s end-behavior. Some or none of the extrema may exist. Example 2 Find the absolute maximum and minimum values of , if any, on the given interval, and state where those values occur.

(a)

(b)

110 Section 3.5

3.5 Applied Maximum and Minimum Problems

Strategy for Solving Applied Max/Min Problems:

1) Draw a picture and label any given or unknown quantities appropriately.

2) Write a formula for the quantity to be maximized or minimized.

3) Express the quantity to be maximized or minimized from Step 2 as a function of one variable. You will need to use the conditions stated in the problem to eliminate variables.

4) Find the interval of possible values for the independent variable from the physical restrictions

in the problem.

5) Use the techniques from the previous section to obtain the maximum or minimum.

If a function is continuous on a restricted closed interval, the Extreme Value Theorem guarantees an absolute maximum and minimum on the interval.

If a continuous function has only one relative extremum on a finite or infinite interval, then that relative extremum must of necessity also be an absolute extremum. Use the 1st or 2nd Derivative Test to determine whether it is a maximum or a minimum.

Section 3.5 111

Example 1 (Exercise #6) A rectangle is to be inscribed in a right triangle having sides of length 6 in., 8 in., and 10 in. (See the figure below.) Find the dimensions of the rectangle with greatest area.

6 in.

8 in. 10 in.

112 Section 3.5

Example 2 (Exercise #22) A closed rectangular container with a square base is to have a volume of 2250 . The material for the top and bottom of the container will cost $2 per , and the material for the sides will cost $3 per . Find the dimensions of the container of least cost.

Section 3.5 113

Example 3 (Exercise #12) Show that among all rectangles with perimeter , the square has the maximum area.

114 Section 3.5

Example 4 (Exercise #27 applied to special case ) Find the dimensions of the right circular cylinder of largest volume that can be inscribed in a sphere of radius 1 cm.

Section 3.5 115

Example 5 (Exercise #48) ONLY IF TIME!! A drainage channel is to be made so that its cross section is a trapezoid with equally sloping sides. If the sides and bottom all have a length of 5 ft, how should the angle be chosen to yield the greatest cross-sectional area of the channel?

Continue your work on the next page…

116 Section 3.5

Economics If units are produced and sold,

where

Marginal Profit additional profit from producing and selling one additional unit after units have already been produced and sold. Marginal Revenue additional revenue from selling one additional unit after units have already been sold. Marginal Cost additional cost of producing one additional unit after units have already been produced.

Section 3.5 117

Example 6 A large bakery determines that it can sell cakes daily at a price per cake, where

The cost of producing cakes per day is

(a) Find the profit function.

(b) Determine how many cakes the bakery must produce and sell each day to maximize profit.

(c) The bakery finds that during peak season it is producing and selling 500 cakes per day. The owner wonders if it would be beneficial to expand the daily production. Use marginal analysis to approximate the effect on profit if daily production could be increased from 500 to 501 cakes.

118 Section 3.6

3.6 Rectilinear Motion Recall from Section 2.1, if an object in rectilinear motion moves along an -axis so that it’s position as a function of time elapsed is

then is called the position function of the object. The graph of is called the position versus time curve for the object.

The change in the object’s position as time elapses from to , indicated by

is called the ________________________ of the particle over the time interval .

We define the object’s velocity function to be Thus, the object can have positive or negative velocity depending on the direction of movement:

A positive value for means that is increasing with time, so the particle is moving in the positive direction.

A negative value for means that is decreasing with time, so the particle is moving in the negative direction.

We define the object’s speed function to be ****************************************************************************** Suppose the graph above represents an object moving on a horizontal line.

1) At time , the object is to the

of the origin and moving

.

2) At time , the object is to the

of the origin and moving

.

3) When is the object’s speed greater, at time or ? _____________________

Time,

Position

Section 3.6 119

The rate at which the velocity of an object changes with time is called acceleration. Thus, we define the object’s acceleration function to be Since , we can express the acceleration function in terms of the position function: Example 1 Let be the position function of a particle moving along an -axis, where is in meters and is in seconds.

(a) Sketch the graph of .

(b) Find the velocity function (c) Find the acceleration function and sketch the graph. and sketch the graph.

120 Section 3.6

Example 2 Let be the position function of a particle moving along an -axis where is in feet and is in seconds. Find the maximum speed of the particle during the time . Note: If an object has negative velocity but positive acceleration, this implies velocity is

___________________ from more negative to less negative, and so the object’s speed (the absolute

value of velocity) is actually ____________________. Thus, in this case a positive acceleration

corresponds to the object slowing down.

Speeding up and slowing down An object is speeding up (accelerating) when its velocity and acceleration have the same sign. This means the object’s position vs. time curve is either:

Increasing and concave up; or

Decreasing and concave down An object is slowing down (decelerating) when its velocity and acceleration have opposite signs. This means the object’s position vs. time curve is either:

Increasing and concave down; or

Decreasing and concave up

Section 3.6 121

Analysis of Particle Motion To analyze a particle’s motion, follow these steps:

1) Find the velocity and acceleration functions. 2) Perform sign analysis on both functions and to determine the object’s direction of

motion and intervals of speeding up or slowing down. 3) Evaluate the object’s position at all key times. 4) Draw a schematic picture of the object in motion. It is helpful to represent the object’s path

in one color when it is speeding up, and in another color when it is slowing down. Example 3 (Corresponds to Example 6 in the book) A position function of a particle moving along a coordinate line is given.

(a) Analyze the motion of the particle for , and give a schematic picture of the motion.

(b) Find the total distance traveled by the particle from time to .

122 Section 3.7

3.7 Newton’s Method Let’s solve the equation

to 4 decimal places using a technique called Newton’s Method.

(Note: This gives us a technique for approximating without ever using the !!) Step 1: Determine which two integer values the positive root must be between by applying the Intermediate Value Theorem. Step 2: Select a first approximation for the root of the equation. Find the equation of the tangent line of at , and determine the -intercept of this line. This gives a better second approximation for the root .

Section 3.7 123

Step 3: Repeat Step 2 using the approximation for the root of the equation. Continue the process of finding better and better approximations until the approximation is good to 4 decimal places.

124 Section 3.7

****************************************************************************** Newton’s Method If is the th approximation of the root of the equation , then the ( )st approximation is given by

Section 3.8 125

3.8 Rolle’s Theorem; Mean-Value Theorem

Rolle’s Theorem (A special case of the Mean-Value Theorem) Let be a function such that the following 3 conditions are satisfied:

(i) __________________________________________

(ii) __________________________________________

(iii) __________________________________________ Then Example 1 Verify that the hypotheses of Rolle’s Theorem are satisfied on the given interval, and find all values of in that interval that satisfy the conclusion of the theorem.

126 Section 3.8

Mean-Value Theorem Let be a function such that the following 2 conditions are satisfied:

(i) __________________________________________

(ii) __________________________________________ Then Example 2 Verify that the hypotheses of the Mean-Value Theorem are satisfied on the given interval, and find all values of in that interval that satisfy the conclusion of the theorem.

Section 3.8 127

Implication of the Mean-Value Theorem on Velocity Example 3 You are driving on a straight highway at which the speed limit is 60 mph. At 2:15 pm a highway patrol officer radars your car at 50 mph. At 2:45 pm a second highway patrol officer 40 miles down the road radars your car at 52 mph. A week later you get a traffic violation ticket in the mail for speeding! Why?

128 Section 4.1

4.1 An Overview of the Area Problem The Area Problem: Given a function that is continuous and nonnegative on an interval , find the area between the graph of and the interval on the -axis. We will address two methods of solving this problem, the 2nd of which was not discovered until the latter part of the seventeenth century by both Newton and Leibniz independently. Solution #1: The Rectangle Method Step 1: Subdivide the interval into equal parts. Step 2: Over each subinterval construct a rectangle that extends from the -axis to the point on

the curve that is above the right endpoint* of the subinterval. *Actually, you may use the midpoint, left endpoint or any other point on the subinterval. For simplicity and uniformity, we will choose to use the right endpoints.

Step 3: The total area of all the rectangles is an approximation of the exact area under the

curve. As increases, will the approximation get better or worse? ___________________

Step 4: If denotes the approximation to the exact area using rectangles, then:

Section 4.1 129

Example 1 Estimate the area between the graph of the function and the interval . Use an approximation scheme with rectangles. Estimate this area using the indicated number of rectangles.

(a) rectangles

(b) rectangles

130 Section 4.1

(c) rectangles

Section 4.1 131

Solution #2: The Antiderivative Method If is a nonnegative continuous function on the interval , and if denotes the area under the graph of over the interval , where is any point in the interval , then

The process of finding a function from its derivative is called antidifferentiation, and a procedure for finding areas via antidifferentiation is called the antiderivative method. Example 2 Use the antiderivative method to find the area between the graph of the function and the interval .

132 Section 4.1

Example 3 Graph each function over the specified interval. Then use simple area formulas from geometry to find the area function that gives the area between the graph of the specified function and the interval . Confirm that .

(a) ;

(b) ;

Section 4.2 133

4.2 The Indefinite Integral

A function is called an antiderivative of a function on a given open interval if for all in the interval. Example 1

(a) Verify that

is an antiderivative of on the interval .

(b) Verify that

is an antiderivative of on the interval .

(c) Verify that

, where is any constant, is an antiderivative of on

the interval . Theorem

If is an antiderivative of on an open interval, then , where is any constant, is an antiderivative on that interval.

Each antiderivative of on an interval can be expressed in the form by choosing the constant appropriately.

134 Section 4.2

Example 2 Find the general antiderivative of the following functions:

(a) (b)

(c) (d) The process of finding antiderivatives is called antidifferentiation or ________________________. Integral Notation Example 3 Write the antiderivatives found in Example 2 using integral notation.

(a) (b)

(c) (d)

Section 4.2 135

Basic Integration Formulas:

136 Section 4.2

Example 4 Evaluate the indefinite integrals.

Properties of the Indefinite Integral The following properties follow directly from the rules for derivatives:

(a) A constant factor can be moved through an integral sign.

(b) An antiderivative of a sum is the sum of the antiderivatives.

(c) An antiderivative of a difference is the difference of the antiderivatives.

Section 4.2 137

Example 5 Evaluate the indefinite integrals.

138 Section 4.2

Example 6 Evaluate the indefinite integrals.

Section 4.2 139

Velocity Recall, . Hence, Example 7 A particle moves along an -axis with position function and velocity function . Use the given information to find .

(This is called an initial-value problem, and the requirement that is called the initial condition for the problem.) Graphs of antiderivatives of a function are called integral curves of . Example 8 Find an equation of the curve such that at each point on the curve the slope is , and the curve passes through the point .

140 Section 4.3

4.3 Integration by Substitution

The method of -substitution may be used to integrate more complicated functions, as illustrated in the following examples. Example 1

Example 2

Section 4.3 141

Example 3

Example 4

142 Section 4.3

Example 5

Example 6

Section 4.3 143

Example 7

Example 8

144 Section 4.3

Example 9 Solve the initial-value problem.

Section 4.4 145

4.4 The Definition of Area as a Limit; Sigma Notation

Sigma Notation In general,

****************************************************************************** Example 1 Evaluate the following:

146 Section 4.4

Properties of Sums (a) A constant factor can be moved through a sigma sign.

(b) Sigma distributes across sums.

(c) Sigma distribute across differences.

Theorem (4.4.2) – Summation Formulas (These formulas will be provided on the exam)

Example 2 Evaluate

Section 4.4 147

Example 3 Express the sum in closed form.

****************************************************************************** Recall the Rectangle Method for finding the area under a curve from to (using right endpoints).

rectangle

148 Section 4.4

Now, let’s take a more general look at the Rectangle Method for finding the area under a curve from to (using an arbitrarily selected point

in each subinterval). Definition (4.4.3) of Area Under a Curve If the function is continuous on and if for all , then the area under the curve over the interval is defined by

Note: In practice, the values of

are chosen to be one of the following:

Left endpoints:

Midpoints:

Right endpoints:

Section 4.4 149

Example 4 Divide the specified interval into subintervals of equal length and then compute

with as (a) the left endpoint, (b) the midpoint, and (c) the right endpoint of each subinterval. Illustrate

each part with a graph of that includes the rectangles whose areas are represented in the sum.

(a) Left endpoints

(b) Midpoints

(c) Right endpoints

150 Section 4.4

Example 5 Find the area under the curve over the specified interval using right endpoints.

Section 4.4 151

Example 6 Find the area under the curve over the specified interval using left endpoints.

152 Section 4.5

4.5 The Definite Integral

Suppose is a continuous function defined on the interval , but that attains positive and negative values. Net Signed Area [Area of regions above the -axis] [Area of regions below the -axis] Example 1 Indicate the sign of each of the net areas shown: Example 2 Find the net signed area between and the interval .

Area Area

Area

Section 4.5 153

Recall the definition of area from Section 4.4:

Up until now, we have assumed that the rectangles formed in the definition of area under the curve

from to have had equal width

.

What if we allow the widths of the rectangles to vary? A partition of the interval is a collection of points

that divides into subintervals (possibly of unequal widths): Then the width of the th rectangle is given by ____________________________. The sum of the areas of the rectangles using this general partition is thus represented as follows: This sum is called a _______________________________________. The largest of the widths of the rectangles is denoted by ________________ and is called the mesh size of the partition.

154 Section 4.5

Example 3

Find the values of:

Section 4.5 155

The Definite Integral

Theorem 4.5.2 If a function is __________________________ on an interval , then

the _________________________________________between the graph of and the interval

is given by:

*When this limit exists and does not depend on the choice of partitions or on the choice of the points

in the subintervals, the function is called integrable. All continuous functions are integrable.

Net signed area,

156 Section 4.5

Example 4 Express the following limit as an integral. (Do not evaluate the integrals.)

Example 5 Express the integral as the limit of a Riemann sum. (Do not evaluate the integral.)

Section 4.5 157

Example 6 Sketch the region whose signed area is represented by the definite integral, and evaluate the integral using appropriate formulas from geometry.

Properties of the Definite Integral (Definition 4.5.3/Theorem 4.5.4)

158 Section 4.5

Example 7 Use properties of the definite integral (Theorem 4.5.4) and appropriate formulas from geometry to evaluate the integral.

Theorem 4.5.5 If is integrable on a closed interval containing the three points and , then

Section 4.5 159

Example 8

Theorem 4.5.6 Suppose and are integrable on .

(a) If for all in , then

(b) If for all in , then

(c) If for all in , then Example 9 Use Theorem 4.5.6 to determine whether the value of the integral is positive or negative.

160 Section 4.6

4.6 The Fundamental Theorem of Calculus

Recall from Section 4.1, if represents the area under the curve from to , then: This formula states that __________ is an antiderivative of _________. Hence, if is any other antiderivative of , then: Thus,

But we learned in Section 4.5, based on our work with Riemann sums, that we can represent area as follows: The Fundamental Theorem of Calculus (Part I) If is continuous on and is any antiderivative of on , then (The proof is provided in the textbook (pg. 310) and is left to the interested student to read on their own.)

Section 4.6 161

Example 1 Use the Fundamental Theorem of Calculus (FTC) to evaluate the following definite integrals.

The FTC can be applied without modification to definite integrals in which the lower limit of integration is greater than or equal to the upper limit of integration.

Example 2 Find the area under the curve over the interval .

162 Section 4.6

Warning! If is not continuous on the interval , then the FTC cannot be applied.

For example, suppose we incorrectly apply the FTC to the function

over the interval :

Why is this impossible? __________________________________________________________ To integrate a continuous function that is defined piecewise on an interval , split the interval into subintervals at the breakpoints of the function, and integrate separately over each subinterval in

accordance with Theorem 4.4.5:

Example 3

Section 4.6 163

To integrate the absolute value of a function , represent | as a piecewise-defined function.

Then integrate the piecewise-defined function as shown previously. Example 4 Evaluate the integral.

Let’s see what this integral represents graphically!

164 Section 4.6

Definition If is a continuous function on the interval , then we define the _____________________ between the curve and the interval to be: Example 5

Find the total area between the curve and the -axis over the interval

.

Section 4.6 165

The Mean-Value Theorem for Integrals (The proof is provided in the textbook (pg. 315) and is left to the interested student to read on their own.)

166 Section 4.6

Example 6 Find all values of in the stated interval that satisfy the Mean-Value Theorem for Integrals, and explain what these numbers represent.

Section 4.6 167

If is the area under the curve over the interval , then we can express as the definite integral: Differentiating both sides of this equation WRT yields: The Fundamental Theorem of Calculus (Part 2) If is continuous on an interval, then has an antiderivative on that interval. In particular, if is any point in the interval, then the function defined by

is an antiderivative of ; that is, for each in the interval, or in an alternative notation: (The proof is provided in the textbook (pg. 317) and is left to the interested student to read on their own.) Parts I and II of the Fundamental Theorem of Calculus taken together rank as one of the greatest discoveries in the history of science, and its formulation by Newton and Leibniz is generally regarded to be the “discovery of calculus.”

168 Section 4.6

Example 7 Find the derivative using Part 2 of the Fundamental Theorem of Calculus (FTC). Then check the result by performing the integration and differentiation.

Example 8

(a)

(b)

(c)

Section 4.6 169

Part I of the FTC states:

where is an antiderivative of . Since , we may write: Example 9

(a) If is the rate of change of the area of an oil spill measured in ., what does the

integral

represent, and what are its units?

(b) If is the marginal profit (dollars per unit) that results from producing and selling units

of a product, what does the integral

represent, and what are its units?

170 Section 4.7

4.7 Rectilinear Motion Revisited Using Integration

We have already seen that for a particle in rectilinear motion with position function , its instantaneous velocity and acceleration are given by

and Thus,

and

Example 1

A particle moves with acceleration along the -axis. The particle starts with an initial velocity of 2 at time . If , find the position function of the particle.

Section 4.7 171

Recall from the last section,

Thus,

and

Example 2 The figure shows the acceleration versus time curve for a particle moving along a coordinate line. If the initial velocity of the particle is 7 ft/s, find the velocity at time s.

( )

( )

172 Section 4.7

Displacement vs. Distance Traveled Displacement from to = Distance traveled from to = Example 3 The velocity versus time curve is given for a particle moving along a line. Use the curve to find

(a) the displacement of the particle over the time interval .

(b) the distance traveled by the particle over the time interval .

Velocity vs. Time Curve

Section 4.7 173

Example 4 A particle moves with a velocity of m/s along an -axis.

(a) Find the displacement of the particle from to .

(b) Find the distance traveled by the particle from to .

174 Section 4.7

One of the most important cases of rectilinear motion occurs when a particle has

______________________________________________. Suppose that a particle moves with constant acceleration along an -axis, with position and velocity at time given by:

Then

and

Constant Acceleration If a particle moves with constant acceleration along an -axis, and if the velocity and acceleration at time are and , respectively, then

Section 4.7 175

Example 5 While driving along a straight road, you spot a rabbit (400 ft. ahead) in the road and slam on your brakes, reducing your speed from 60 mi/h to 45 mi/h at a constant rate over a distance of 154 ft. (You will need the conversion 88 ft/s = 60 mi/h)

(a) Find your acceleration in .

(b) At this acceleration, how far will the car travel (after slamming on the breaks) before coming to a stop?

(c) What is the fate of the rabbit?

176 Section 4.7

Free-Fall Model

An object rising and/or falling along a vertical line near the surface of the earth is called free-fall

motion. It is possible that the object was imparted some initial velocity by an external force, but the

only force thereafter acting on the object is __________________.

It is a fact of physics that a particle with free-fall motion has ________________ acceleration.

The magnitude of this constant, or _______________, is called the acceleration due

to gravity.

We assume in this model at the surface of the Earth and the positive direction is up. For an object in free-fall motion (ignoring the size of the object and air resistance),

Example 6 A man standing on a bridge wonders how high up he is. So he drops a stone from the bridge. What is the height of the bridge if the stone hits the water 4 s later?

Section 4.7 177

Example 7 A rocket is launched upward from ground level with an initial speed of 100 ft/s.

(a) How long does it take for the rocket to reach its highest point?

(b) How high does the rocket go?

(c) What is the speed of the projectile when it hits the ground?

178 Section 4.8

4.8 Average Value of a Function and its Applications

Recall the Mean-Value Theorem for Integrals formula from Section 4.6: Definition If is continuous on , then the __________________________________ of on is defined to be Example 1

Consider the function

.

(a) Find the average value of the function, , over the interval .

(b) Find all points in the interval at which the value of is the same as the average. (That is, find all points in such that .) Then illustrate this with a picture.

Section 4.8 179

Example 2 The temperature (in ) of a 10 in. long metal bar at the point on the bar which is in. from the far left end is given by the formula:

What is the average temperature of the bar? Applied to Rectilinear Motion If we apply our definition of average value to the velocity function over a time interval , we get: Example 3 The velocity versus time curve is given for a particle moving along a line. Use the curve to find the average velocity of the particle over the time interval .

180 Section 4.9

4.9 Evaluating Definite Integrals by Substitution

Example 1 Evaluate

Method 1 (By -substitution in the corresponding indefinite integral) Method 2 (By -substitution in the definite integral) – MUST CHANGE LIMITS OF INTEGRATION!!

Section 4.9 181

Example 2 Evaluate the definite integral by expressing it in terms of and evaluating the resulting integral using a formula from geometry.

Example 3 Evaluate the integral by Method 2 (changing limits).

Example 4 Evaluate the integral by Method 1 (without changing limits).

182 Section 5.1

5.1 Area Between Two Curves Area Between Two Curves (Integrating with respect to ) If and are continuous functions on the interval , and if for all , then the area of the region bounded above by , below by , on the left by the line , and on the right by the line is Example 1

Find the area between the curves and .

Section 5.1 183

Example 2 Sketch the region enclosed by the curves and find its area.

, , ,

184 Section 5.1

Example 3 Find the area between curves and .

Section 5.1 185

Area Between Two Curves (Integrating with respect to ) If and are continuous functions and if for all in , then the area of the region bounded on the left by , on the right by , below by , and above by is Let’s repeat Example 3, but this time integrating with respect to . Example 4 Find the area between curves and .

186 Section 5.1

Example 5 Find the area of the region bounded by , , , and .

Section 5.2 187

5.2 Volumes by Slicing; Disks and Washers Volume (by Slicing) To find the volume of a solid, integrate the cross-sectional area from one end of the solid to the other. Cross-sections taken perpendicular to -axis. Cross-sections taken perpendicular to -axis.

188 Section 5.2

Example 1 Find the volume of the solid whose base is the region enclosed between the curve and the -axis and whose cross sections taken perpendicular to the -axis are squares.

Section 5.2 189

Example 2 Derive the formula for the volume of a sphere of radius using circular cross-sections perpendicular to the -axis.

190 Section 5.2

Volume (by Method of Disks) – Solids of Revolution Axis of revolution is the -axis. Axis of revolution is the -axis.

Section 5.2 191

Example 3

Find the volume of the solid obtained by rotating about the -axis the region under the curve from to .

192 Section 5.2

Example 4 Find the volume of the solid that results when the region enclosed by the given curves is revolved about the -axis.

Section 5.2 193

Volume (by Method of Washers) – Solids of Revolution Axis of revolution is the -axis. Axis of revolution is the -axis.

194 Section 5.2

Example 5 Find the volume of the solid that results when the region enclosed by the given curves is revolved about the -axis.

Section 5.2 195

It is possible to use the methods from this section to find the volume of a solid of revolution whose axis of revolution is a line other than one of the coordinate axes… Example 6 Find the volume of the solid that results when the region enclosed by and is revolved about the line .

196 Section 5.3

5.3 Volumes by Cylindrical Shells Volume (by Cylindrical Shells)

Section 5.3 197

Example 1 Use cylindrical shells to find the volume of the solid generated when the region enclosed between , , , and the -axis is revolved about the -axis.

198 Section 5.3

Example 2 Use cylindrical shells to find the volume of the solid generated when the region bounded by

, , and is revolved about the -axis.

Section 5.3 199

Why use the Cylindrical Shell Method? Some problems are impossible to solve by the disk or washer method, and so the shell method provides another technique for computing the volume of a solid of revolution.

Example 3 Find the volume of the solid formed by revolving the region bounded by and and about the line . Washer Method Shell Method

200 Section 5.3

Summary of Methods for Finding Volume of Solids of Revolution

Note: may be replaced with .

1) Disks – slice perpendicular to axis of revolution

2) Washers – slice perpendicular to axis of revolution

3) Shells – slice parallel to axis of revolution

Example 4

Find the volume of the solid formed by revolving the region bounded by , and about the -axis by

(a) Disk Method

(b) Shell Method

Section 5.4 201

5.4 Length of a Plane Curve Arc Length

202 Section 5.4

Example 1 Find the exact arc length of the curve over the interval

from to

(a) integrating with respect to .

Section 5.4 203

(b) Integrating with respect to .

204 Section 5.5

5.5 Area of a Surface of Revolution Area of a Surface of revolution

Section 5.5 205

Example 1 Find the area of the surface generated by revolving the given curve about the -axis.

from to

206 Section 5.5

Example 2 Find the area of the surface generated by revolving the given curve about the -axis.

from to

Section 5.6 207

5.6 Work Definition of Work (Constant Force) If a constant force of magnitude is applied in the direction of motion of an object, and if that object moves a distance , then we define the work performed by the force on the object to be

Common Units Force Distance Work newton (N) meter (m) N m joule (J) pound (lb) foot (ft) lb

Conversion Factors

Force N lb lb N Work lb lb

Note: An object’s weight is the force that gravity exerts on the object. Thus, when lifting an object, the weight of the object is the force exerted.

Example 1

(a) How much work is done in lifting a 20-lb weight 70 inches off the ground?

(b) An object moves 3 m along a line while subjected to a constant force of 60 lb in its direction of motion. What is the work done?

208 Section 5.6

What happens if the force acting on an object is variable? Let’s suppose that an object moves along the -axis in the positive direction, from to , and at each point between and a force acts on the object in the direction of motion, where is a continuous function. We divide the interval into subintervals with endpoints and equal width . We choose a sample point

in the th subinterval . Then the force at that point is ___________. Since the values of don’t change very much over the th interval, is almost _______________ on the interval , and thus the work done in moving the object from to is approximately:

Thus, we can approximate the total work done in moving the object from to by

This approximation becomes better as we make larger. Taking the limit as yields

Definition of Work (Variable Force) Suppose that an object moves in the positive direction along a coordinate line over the interval while subjected to a variable force that is applied in the direction of motion. Then we define the work performed by the force on the object to be

Section 5.6 209

Example 2 A variable force in the positive -direction is graphed in the figure. Find the work done by the force on a particle that moves from to . Hooke’s Law A spring that is stretched units beyond its natural length pulls back with a force

where is a positive constant (called the spring constant). Example 3 A spring exerts a force of 40 N when it is stretched from its natural length of 0.1 m to a length of 0.15 m. How much work is required to stretch the spring from 0.15 m to 0.18 m.

Force (N)

Position (m)

210 Section 5.6

To find the work done on a complicated object, we slice the object up in such a way that we can find the work done on each piece. We calculate the work for each piece using , and we sum these pieces to approximate the total work as a Riemann sum. Letting the size of each piece tend to zero (or ), we obtain a definite integral that represents the total work. Example 4 A 200-lb cable (assume weight is evenly distributed) is 100 ft long and hangs vertically from the top of a tall building. How much work is required to lift the cable to the top of the building? Example 5 An aquarium 2 m long, 1 m wide, and 1 m deep is two-thirds filled with water. Find the work needed to pump all the water over the upper rim of the aquarium.

(Use 9810 N/m3 as the weight density of water.)

Section 5.6 211

Example 6 A cone-shaped water reservoir is 20 ft in diameter across the top and 15 ft deep. If the reservoir is filled to a depth of 10 ft, how much work is required to pump all the water to the top of the reservoir. (Use 62.4 lb/ft3 as the weight density of water.)

212 Section 5.6

Newton’s Second Law of Motion If an object with mass is subjected to a force , then the object undergoes an acceleration that satisfies the equation

Assume that an object moves in the positive direction along a coordinate line over the interval while subjected to a force that is applied in the direction of motion. Let denote the mass of the object. Assume the object has initial velocity at time when the object is at position , and final velocity at time when the object is at position .

Then,

Section 5.6 213

Work-Energy Relationship In words:

The work done on an object is equal to _______________________________________________

_______________________________________________________________________________. *The units of kinetic energy are the same as the units of work. Example 7 Assume that a Mars probe of mass kg is subjected only to the force of its own engine. Starting at a time when the speed of the probe is m/s, the engine is fired continuously over a distance of m with a constant force of N in the direction of motion. Use the work-energy relationship to find the final speed of the probe.

214 Section 5.8

5.8 Fluid Pressure and Force

Pressure If a force of magnitude is applied to a surface of area , then we define the pressure exerted by the force on the surface to be

Common Units Force Area Pressure newton (N) sq. meter (m ) N m pascal (Pa) pound (lb) sq. foot ( ) lb pound (lb) sq. inch (in ) lb in (psi)

Conversion Factors

Pressure lb in a lb a

The Pressure-Depth Equation In a static fluid that has constant weight density (weight per unit volume), the fluid pressure at depth is given by

Example 1 A 3 ft by 5 ft rectangular plate is placed horizontally in a swimming pool. Find the fluid pressure and

force on the top of the plate when it is 6 ft. deep. (The weight density of water is lb 3)

Section 5.8 215

When a flat surface is submerged vertically in a fluid, the problem of finding the total force on the surface is more difficult because the pressure is not constant over the surface. We must (1) slice the surface into narrow horizontal strips, (2) approximate the fluid force on each strip, and then (3) the total force is given by taking the limit of the sum of the forces on each strip as . Example 2 Now suppose the 3 ft by 5 ft rectangular plate is submerged vertically in the pool with the 5 ft length parallel to the bottom of the pool. Find the force due to fluid pressure on the plate when the bottom

of the plate is 6 ft deep. (The weight density of water is lb 3)

216 Section 5.8

Example 3

A cylindrical tank contains oil having a weight density of 53 N m3. If the axis is horizontal and the

diameter is 8 m, find the force due to fluid pressure on one end of the tank if the oil in the tank is half-filled with oil.

Section 6.1 217

6.1 Exponential and Logarithmic Functions

*EXPONENTIAL FUNCTIONS*

Definitions/Properties of Exponents

A function of the form , where is called an exponential function with base .

Increasing function Decreasing function

Properties of the function 1. The graph passes through the point 2. If then is increasing, and if then is decreasing 3. The -axis is a horizontal asymptote of the graph 4. Domain: ; Range: 5. is one-to-one (passes the horizontal line test) 6. is continuous everywhere

218 Section 6.1

Observe the following values:

1 2

10 2.593742

1,000 2.716924

100,000 2.718268

1,000,000 2.718280

Substituting yields:

Definition of (Euler’s constant) The function is called the natural exponential function. Note that since , the graph is increasing. *LOGARITHMIC FUNCTIONS* Definition: If and , then for ,

means

In words: denotes the exponent to which must be raised to produce .

Common Logarithm: (Base 10) Natural Logarithm: (Base )

Section 6.1 219

MEMORY DEVICE: Note: and are inverse functions, and thus their graphs are mirror images about the line .

Properties of the function 1. The graph passes through the point 2. If then is increasing, and if then is decreasing 3. The -axis is a vertical asymptote of the graph 4. Domain: ; Range: 5. is one-to-one (passes the horizontal line test) 6. is continuous everywhere

220 Section 6.1

Properties of Logarithms

1. _______

2.

3. and (since and are inverse functions, their composition is the identity function)

4.

5.

6.

7. Change of Base Formula:

Example 1 Find the exact value of the expression without using a calculator.

Example 2 Expand the logarithm.

Section 6.1 221

Example 3 Rewrite the expression as a single logarithm.

Example 4 Solve for without using a calculator.

222 Section 6.2

6.2 Derivatives and Integrals Involving Logarithmic Functions

Recall the definition of the derivative:

We are now ready to compute the derivative of the natural logarithmic function .

Section 6.2 223

By applying the Change of Base Formula we can compute the derivative of the general logarithmic function:

If is a differentiable function of , and if , then by applying the chain rule,

Example 1 Find .

(a)

(b)

(c)

224 Section 6.2

Example 2 Find .

When possible, the properties of logarithms should be applied before differentiating a function involving logarithms.

Example 3 Find .

Section 6.2 225

Now, consider the function ,

Example 4 Find .

226 Section 6.2

We now consider a technique called logarithmic differentiation that is useful for differentiating complicated functions that are composed of products, quotients, and powers.

Step 1: Express the function in the form Step : Take “ ” of both sides. Step 3: Differentiate both sides (implicitly) with respect to . Step 4: Solve for .

Example 5 Find using logarithmic differentiation.

Example 6 Find using logarithmic differentiation.

Section 6.2 227

We showed earlier that for :

Thus, we have the following integration formula: _____________________________________ Example 7 Evaluate the integrals.

228 Section 6.3

6.3 Derivatives of Inverse Functions & Derivatives and Integrals involving Exponential Functions

Let’s compute the derivative of the exponential function .

Summary of Derivatives

Section 6.3 229

Example 1 Find .

Functions of the form in which and are nonconstant functions of are neither exponential nor power functions. They must be differentiated using logarithmic differentiation. Example 2 Use logarithmic differentiation to find .

(a)

(b)

230 Section 6.3

From the derivative formulas

we get the companion integration formulas:

Summary of Integrals

Example 3 Evaluate the integrals.

Section 6.3 231

Derivatives of Inverse Functions Recall, if is a one-to-one function, then it has an inverse function, . In fact,

means Let us now compute the derivative of the inverse function . Theorem If is a differentiable and one-to-one* function, then

Or alternatively, if then

*Note: If is always increasing ( ) on an open interval , then is one-to-one on that interval.

232 Section 6.3

If is always decreasing ( ) on an open interval , then is one-to-one on that interval. Example 4 Consider the function .

a) Show that is one-to-one, and confirm that .

b) Find using Formula 2. Example 5 Consider the function

a) Show that is one-to-one.

b) Find using two different methods:

Using Formula 3 Using implicit differentiation