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Instructor’s Resource Manual Australia • Brazil • Japan Korea • Mexico • Singapore • Spain • United Kingdom • United States Calculus Early Transcendental Functions SIXTH EDITION Ron Larson The Pennsylvania State University, The Behrend College Bruce H. Edwards University of Florida

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Page 1: SIXTH EDITION Ron Larson - testbankwizard.eutestbankwizard.eu/.../Solution-Manual-for-Calculus...Edition-by-Larson.pdf · Instructor’s Resource Manual Australia • Brazil • Japan

Instructor’s Resource Manual

Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States

Calculus Early Transcendental Functions

SIXTH EDITION

Ron Larson

The Pennsylvania State University, The Behrend College

Bruce H. Edwards University of Florida

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iii

Contents Chapter 1: Preparation for Calculus ..................................................................................... 1 Chapter 2: Limits and Their Properties ……....................................................................... 12 Chapter 3: Differentiation .................................................................................................... 20 Chapter 4: Applications of Differentiation ………….......…............................................... 33 Chapter 5: Integration ………….......................……………………………..……........….. 50 Chapter 6: Differential Equations ……….……......………………………………….......... 66 Chapter 7: Applications of Integration ..................................................................................74 Chapter 8: Integration Techniques, L’Hôpital’s Rule, and Improper Integrals …................ 85 Chapter 9: Infinite Series …………………….....................................................................101 Chapter 10: Conics, Parametric Equations, and Polar Coordinates ......................................119 Chapter 11: Vectors and the Geometry of Space .......................……………........................131 Chapter 12: Vector-Valued Functions....................................................................................142 Chapter 13: Functions of Several Variables …….....…..….....……………………..........151 Chapter 14: Multiple Integration .………………………......................................................168 Chapter 15: Vector Analysis .………………………..........................................................179 Project Answers .………………………...................................................................................... A1

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Chapter 1 Preparation for CalculusChapter CommentsChapter 1 is a review chapter and can be covered quickly. Spend about 3 or 4 days on this chapter,placing most of the emphasis on Section 1.3. Of course, you cannot cover every single item that isin this chapter in that time, so this is a good opportunity to encourage your students to read thebook. To convince your students of this, assign homework problems or give a quiz on some of thematerial that is in this chapter but that you do not go over in class. Although you will not hold yourstudents responsible for everything in all 16 chapters, the tools in this chapter need to be readily athand, that is, memorized.

Sections 1.1 and 1.2 can be covered in a day. Students at this level of mathematics have graphedequations before so let them read about that information on their own. Discuss intercepts,emphasizing that they are points, not numbers, and should be written as ordered pairs. Also dis-cuss symmetry with respect to the -axis, the -axis, and the origin. Be sure to do a problem likeExample 5 in Section 1.1. Students need to be able to find the points of intersection of graphs inorder to calculate the area between two curves in Chapter 7.

In Section 1.2, discuss the slope of a line, the point-slope form of a line, equations of vertical and horizontal lines, and the slopes of parallel and perpendicular lines. You need to emphasize thepoint-slope form of a line because this is needed to write the equation of a tangent line in Chapter 3.

Students need to know everything in Section 1.3, so carefully go over the definition of a function,domain and range, function notation, transformations, the terms algebraic and transcendental, andthe composition of functions. Note that the authors assume a knowledge of trigonometric functions.If necessary to review these functions, refer to Appendix C. Because students need practice handling

be sure to do an example calculating Your students should know the graphs of theeight basic functions in Figure 1.27. A knowledge of even and odd functions will be helpful withdefinite integrals.

Section 1.4 introduces the idea of fitting models to data. Because a basic premise of science is thatmuch of the physical world can be described mathematically, you and your students would benefitfrom looking at the three models presented in this section.

The authors assume students have a working knowledge of inequalities, the formula for the distance between two points, absolute value, and so forth. If needed, you can find a review ofthese concepts in Appendix C.

Section 1.1 Graphs and Models

Section Comments 1.1 Graphs and Models—Sketch the graph of an equation. Find the intercepts of a graph.

Test a graph for symmetry with respect to an axis and the origin. Find the points of intersection of two graphs. Interpret mathematical models for real-life data.

Teaching Tips

You may want to spend time reviewing factoring, solving equations involving square roots, andsolving polynomial equations. For further review, encourage students to study the following material in Precalculus, 9th edition, by Larson.

• Factoring: Appendix A.3

• Solving equations involving square roots: Appendix A.5

• Solving polynomial equations: Appendix A.5

f �x � �x�.�x,

yx

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Encourage students who have access to a graphing utility or computer algebra system to use thetechnology to check their answers.

Start class by having students practice finding intercepts of the graph of an equation. Considerdoing an in class example of finding the intercepts of the graph of an equation with a radical, suchas or

Students have a hard time deciding if certain graphs are symmetric to the -axis, -axis, and origin.To help their understanding, tell students that the word symmetric conveys balance. If you want toset a dinner table, you want to have matching plates, utensils, and everything in line. To further aidstudents’ understanding of symmetry, draw pictures of various symmetries. Suggested examplesare shown below:

When finding points of intersection, it is useful to have students find the points both algebraicallyand by using a graphing calculator. Use Example 5 on page 6 to find the points of intersectionusing a graphing calculator.

How Do You See It? Exercise

Page 9, Exercise 76 Use the graphs of the two equations to answer the questions below.

(a) What are the intercepts for each equation?

(b) Determine the symmetry for each equation.

(c) Determine the point of intersection of the two equations.

Solution

(a) Intercepts for

-intercept:

-intercepts:

Intercepts for

-intercept:

-intercepts:

None, cannot equal 0.y

0 � x2 � 2x

y � 0 � 2 � 2; �0, 2�y

y � x2 � 2:

�0, 0�, �1, 0�, ��1, 0�0 � x3 � x � x�x2 � 1� � x�x � 1��x � 1�;x

y � 03 � 0 � 0; �0, 0�y

y � x3 � x:

−2 2 4x

−4

2

4

6

y

y = x3 − xy = x2 + 2

x

b

y

−a a−b

(a, b)

(0, 0)

(−a, −b)

ax

b

y

−a

(a, b)(−a, b)

ax

b

−b

y

(a, b)

(a, −b)

yx

y � x�4 � x2.y � �x � 4

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(b) Symmetry with respect to the origin for because

Symmetry with respect to the -axis for because

(c)

Point of intersection:

Note: The polynomial has no real roots.

Suggested Homework Assignment

Pages 8–9: 1–15 odd, 19–25 odd, 29–37 odd, 41, 45, 47, 51, 53, 55, 59, 63, 65, 67, 77, and 79.

Section 1.2 Linear Models and Rates of Change

Section Comments1.2 Linear Models and Rates of Change—Find the slope of a line passing through two

points. Write the equation of a line with a given point and slope. Interpret slope as a ratio or as a rate in a real-life application. Sketch the graph of a linear equation in slope-intercept form. Write equations of lines that are parallel or perpendicular to a given line.

Teaching Tips

Spend time reviewing the following concepts: slope, writing equations of lines, and slope as a rateof change. For further review, encourage students to study the following material in Precalculus,9th edition, by Larson.

• Slope: Section 1.3

• Finding the slope of a line: Section 1.3

• Writing linear equations in two variables: Section 1.3

• Parallel and perpendicular lines: Section 1.3

• Slope as a rate of change: Section 1.3

Encourage students who have access to a graphing utility or computer algebra system to use thetechnology to check their answers.

Review with the class that slope measures the steepness of a line. In addition, the slope of a line isa rate of change. Rate of change is an important topic in calculus, and inform students that rate ofchange will come up later in the semester.

Consider doing an example in class to remind students how to rewrite an equation such asin slope-intercept form. Then show them how to identify the slope and -intercept.

Remind students that the slope of a vertical line is undefined and the slope of a horizontal line is0. (As mentioned before, you can also direct students to the appropriate material in Precalculus.)

yx � 3y � 12

x2 � x � 1

�2, 6�

x � 2 ⇒ y � 6

�x � 2��x2 � x � 1� � 0

x3 � x2 � x � 2 � 0

x3 � x � x2 � 2

y � ��x�2 � 2 � x2 � 2.

y � x2 � 2y

�y � ��x�3 � ��x� � �x3 � x.

y � x3 � x

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When finding equations of lines, be sure to write each solution in four ways: slope-intercept form,two equations in point-slope form (depending on which point is chosen for and in generalform. This way, students will see how to get from one form to the next.

How Do You See It? Exercise

Page 17, Exercise 74 Several lines (labeled a-f ) are shown below.

(a) Which lines have a positive slope?

(b) Which lines have a negative slope?

(c) Which lines appear parallel?

(d) Which lines appear perpendicular?

Solution

(a) Lines and have positive slopes.

(b) Lines and have negative slopes.

(c) Lines and appear parallel. Lines and appear parallel.

(d) Lines and appear perpendicular. Lines and appear perpendicular.

Suggested Homework Assignment

Pages 16–18: 11–23 odd, 25–65 odd, 73, 75, 81, 93, and 95.

Section 1.3 Functions and Their Graphs

Section Comments1.3 Functions and Their Graphs—Use function notation to represent and evaluate a

function. Find the domain and range of a function. Sketch the graph of a function.Identify different types of transformations of functions. Classify functions and recognizecombinations of functions.

dbfb

fdec

ba

fe,d,c,

x

a

c

d

e

f

b

−3 1 3

−4−5

−7−8

1

3

5678

y

�x1, y1),

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Teaching Tips

Spend time reviewing the following concepts: evaluating a function and the domain and range of a function. For further review, encourage students to study the following material in Precalculus,9th edition, by Larson.

• Evaluating a function: Section 1.4

• Domain and range of a function: Section 1.4

Encourage students who have access to a graphing utility or computer algebra system to use thetechnology to check their answers.

When evaluating a function, ask students what the function is doing with any value of Forexample, use the function:

Here, the function takes a value of multiplies it by 2 and adds 3. Have students quickly fillout and Follow up with asking what does with stuff, This will prove to be useful for students when evaluatingfunctions using the difference quotient.

Use the function to ask a series of questions building up to finding the differencequotient. Ask students to find and simplify:

1.

2.

3.

Also consider presenting the functions and to find

the difference quotient. For you may want to point out how Pascal’s Triangle can be used toexpand

Be sure to spend some time evaluating functions using the difference quotient. Some suggested

functions to use are: and Using will

test students’ ability to find a least common denominator and using will test students’ abilityto rationalize a numerator. Be sure to tell students rationalizing the numerator will become usefulin Calculus.

Often times, students have trouble visualizing how to graph piece-wise defined functions. You may

want to present how to graph and practice how to find the domain and

range. This problem can be used later on in Chapter 2 with finding one-sided limits and continuity.A good teaching strategy is to use colored markers to graph the entire graphs of and

Then erase the parts of each graph not satisfied by the domain restrictions.

When asking students to find the domain and range of functions, use different colored markers to help students visualize the domain and range. For example, graph and

After graphing these two functions by using the transformations, tell students that if you pick any point on either or and map the point back to the -axis and -axisusing different colors, students will be more likely to see what the domain and range are.Examples of these two functions are graphed below.

yxgfg�x� � �x � 1 � 4.

f �x� � �x � 2� � 3

f�x� � x � 1.f�x� � x2

f �x� � �x2,x � 1,

x < 0x � 0

h�x�g�x�h�x� � �x � 2.g�x� �

1x � 3

,f �x) � 2x2 � 5x � 1,

�x � �x�3.g�x�,

h�x� �3x

g�x� � x3 � 2x2,f�x� � 2x2 � 3x,

f�x � �x� � f�x��x

f �x � �x� � f �x�f�x � �x�

f�x� � 2x � 3

f �stuff� � 2 � stuff � 3.f �x�f �a student’s name�.f �b�,f �a�,f �0�,f ��3�,f �2�,f �1�,

x,f �x�

f �x� � 2x � 3.

x.

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When going over the Leading Coefficient Test, start by graphing the simplest monomial functionsand Describe the end behavior of each. Next, insert negatives and describe the

end behaviors. Summarizing the results, if the degree is even with a positive leading coefficient,both ends will be rising; if a negative leading coefficient, both sides will be falling. If the degree isodd with a positive leading coefficient, the right side will be rising and the left will be falling. Theopposite is true if the leading coefficient is negative. Expose students to the notation,

so that they will have a preview of Calculus once limits are reached.

How Do You See It? Exercise

Page 29, Exercise 94

Water runs into a vase of height 30 centimeters at a constant rate. The vase is full after 5 seconds.Use this information and the shape of the vase shown to answer the questions when is the depthof the water in centimeters and is the time in seconds (see figure).

(a) Explain why is a function of

(b) Determine the domain and range of the function.

(c) Sketch a possible graph of the function.

(d) Use the graph in part (c) to approximate What does this represent?

Solution

(a) For each time there corresponds a depth

(b) Domain:

Range:

(c)

(d) At time 4 seconds, the depth is approximately 18 cm.d�4� � 18.

t

d

5

10

15

20

25

30

1 2 3 4 5 6

0 � d � 30

0 � t � 5

d.t,

d�4�.

t.d

30 cm

d

td

f �x� → �x →,

g�x� � x3.f �x� � x2

x

y

(−1, −4)

(3, −2)

−2 1 2 4 5 6

−5−6

123

x

y

(−1, 6)

(2, 3)

−1−2 1 2 3 4 5 6

12345

7

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Suggested Homework Assignment

Pages 27–30: 3–31 odd, 37, 39, 43–51 odd, 57, 59, 61, 65, 67, 69, 73, 91, 104, and 105–119 odd.

Section 1.4 Fitting Models to Data

Section Comments1.4 Fitting Models to Data—Fit a linear model to a real-life data set. Fit a quadratic model

to a real-life data set. Fit a trigonometric model to a real-life data set.

Teaching Tips

Exercise 12 on page 36 is a good review of the following concepts:

• Fitting a linear model to a real-life data set

• Fitting a quadratic model to a real-life data set

• Fitting a cubic model to a real-life data set

Much of the physical world can be described mathematically, so students will benefit from reviewingthese models. Also, students need to know these concepts because they are used throughout the text.

Students should obtain models similar to the ones given in part (a) below. Use the transparency to show the models and the data together. It is clear from the graph that the cubic model is a betterfit than the linear model. Next, give the quadratic model and show its graph using the secondtransparency. From the graph, the quadratic model does not appear to be a good fit. Part (e) willgive you an opportunity to show students that a model can be a poor predictor using values outsidethe domain of the given data set. If there is another model you wish to investigate (e.g., higher-orderpolynomials), you can do part (f).

Solution

(a)

(b)

(c) The cubic model is better.

(d)

The model does not fit the data well.

(e) For 2014, and million and million. Neither seems accurate. Theestimates are too low.

(f) Answers will vary.

Use a graphing calculator to show how the Stat Plot and Table features work.

N3 � 115.5N1 � 92.2t � 24,

0 20

100

40

N2 � �0.414t2 � 11.00t � 4.4

0 20

100

40

N1

N2

N3 � 0.0485t3 � 2.015t2 � 27.00t � 42.3

N1 � 1.89t � 46.8

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How Do You See It? Exercise

Page 34, Exercise 6 Determine whether the data can be modeled by a linear function, a quadraticfunction, or a trigonometric function, or that there appears to be no relationship between and

(a) (b)

(c) (d)

Solution

(a) Trigonometric function

(b) Quadratic function

(c) No relationship

(d) Linear function

Suggested Homework Assignment

Pages 34–36: 1, 5, 9, 11, and 13.

Section 1.5 Inverse Functions

Section Comments1.5 Inverse Functions—Verify that one function is the inverse function of another function.

Determine whether a function has an inverse function. Find the derivative of an inversefunction.

Teaching Tips

Students should be familiar with inverse functions from their Algebra courses, so the first part ofSection 5 could be covered quickly. However, they probably are not familiar with the idea of aone-to-one function, so spend some time explaining this idea and use the terminology frequently.Inverse trigonometric functions might not appeal to your students. Be sure to do problems likethose in Example 7. The concept of using a right triangle is important for Chapter 5 and Chapter 8.

Consider doing an example with finding an inverse of a rational function such as:

Discuss how this function passes the Horizontal Line Test and review finding vertical and horizon-tal asymptotes.

f�x� �2x � 35 � 7x

.

x

y

x

y

x

y

x

y

y.x

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Another suggested problem to cover with students is:

The point is on the graph of Does lie on the graph of

If not, does this contradict the definition of inverse function?

This example is a good review of the following concepts:

• Definition of an inverse function

• Domain and range of the cosine and arccosine functions

• Properties of inverse trigonometric functions

Use the transparency graph of and Note that the point lies on the

graph of but does not lie on the graph of This does not contradict

the definition of an inverse function because the domain of is and its range is

So, Be sure to cover the material at the top of page 43 and the Properties of

Inverse Trigonometric Functions.

How Do You See It? Exercise

Page 46, Exercise 86 You use a graphing utility to graph and then use the drawinverse feature to graph (see figure). Is the inverse function of Why or why not?

Solution

No, is not the inverse of is not one-to-one. The graph of is not the graph of a function.

Suggested Homework Assignment

Pages 44–47: 1, 5, 9–12, 15, 23–39 odd, 45, 65–81 odd, 87–93 odd, 101–119 odd,and 141–145 odd.

Section 1.6 Exponential and Logarithmic Functions

Section Comments1.6 Exponential and Logarithmic Functions—Develop and use properties of exponential

functions. Understand the definition of the number Understand the definition of thenatural logarithmic function, and develop and use properties of the natural logarithmicfunction.

e.

gf�x� � sin xf.g

4

−4

2p−2p

f

g

f ?ggf�x� � sin x

arccos�0� �

2.�0, .

��1, 1y � arccos x

y � arccos x.0, 3

2 �y � cos x,

3

2, 0�y � arccos x.y � cos x

y � arcos x?0, 3

2 �y � cos x.3

2, 0�

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Teaching Tips

Although students should be familiar and comfortable with both the exponential and logarithmicfunctions, sometimes they are not. Therefore, in Section 1.6, present these functions as though theyare new ideas. Have your students commit to memory the graphs and properties of these functions.Your students will learn more about the number in Chapter 2 when limits are discussed.

Use Example 3 to show students how to find the decimal approximation of You may want to usethe following mnemonic device to help students remember the first six digits of “AndrewJackson served two terms as the 7th President of the United States. He was elected to his firstPresidency in 1828.”

Remind students that the natural logarithmic function is the logarithmic function to the base andis written as where

Try to show more examples of expanding logarithmic expressions than condensing since studentswill need this skill when performing derivatives using logarithms.

You may want to reference Figure 1.50 on page 51 to help students describe the relationshipbetween the graphs of and Because the natural logarithmic function and thenatural exponential function are inverses, the graphs of and are mirrorimages across the line

How Do You See It? Exercise

Page 55, Exercise 122

The figure below shows the graph of or Which graph is it? What are thedomains of and Does for all real values of Explain.

Solution

The graph is that of

The domain of is

The domain of is

No, for all real values of They are equal for

Suggested Homework Assignment

Pages 53–55: 7–21 odd, 25–37 odd, 51, 77, 79, 89–109 odd, and 125.

x > 0.x.ln ex � eln x

x > 0.y2 � eln x

��, �.y1 � ln�ex�

y2 � eln x.

3

−2

−3

2

x?ln ex � eln xy2?y1

y2 � eln x.y1 � ln ex

y � x.g�x� � exf�x� � ln x

g�x� � ex.f�x� � ln x

x > 0.loge x � ln x,e

e.e.

e

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Chapter 1 Project

Height of a Ferris Wheel CarThe Ferris wheel was designed by American engineer George Ferris (1859–1896). The first Ferris wheel was built for the 1893 World’s Columbian Exposition in Chicago, and later used atthe 1904 World’s Fair in St. Louis. It had a diameter of 250 feet, and each of its 36 cars could hold60 passengers.

Exercises

In Exercises 1–3, use the following information. A Ferris wheel with a diameter of 100 feetrotates at a constant rate of 4 revolutions per minute. Let the center of the Ferris wheel be at the origin.

1. Each of the Ferris wheel’s cars travels around a circle.

(a) Write an equation of the circle where and are measured in feet.

(b) Sketch a graph of the equation you wrote in part (a).

(c) Use the Vertical Line Test to determine whether is a function of

(d) What does your answer to part (c) mean in the context of the problem?

2. The height (in feet) of a Ferris wheel car located at the point is given by

where is related to the angle (in radians) by the equation

as shown in the figure.

(a) Write an equation of the height in terms of time (in minutes). (Hint: One revolution is radians.)

(b) Sketch a graph of the equation you wrote in part (a).

(c) Use the Vertical Line Test to determine whether is a function of

(d) What does your answer to part (c) mean in the context of the problem?

3. The model in Exercise 2 yields a height of 50 feet when Alter the model so that theheight of the car is 0 feet when Explain your reasoning.t � 0.

t � 0.

t.h

2t

h

y � 50 sin �

�y

h � 50 � y

�x, y�

10

20

30

40

x

y

θ

(x, y)

h

50 y = 50 sin

Ground

Height above ground ish = 50 + 50 sin .

θ

θh

x.y

yx

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Chapter 2 Limits and Their PropertiesChapter CommentsSection 2.1 gives a preview of calculus. On pages 63 and 64 of the textbook are examples of someof the concepts from precalculus extended to ideas that require the use of calculus. Review theseideas with your students to give them a feel for where the course is heading.

The idea of a limit is central to calculus. So you should take the time to discuss the tangent lineproblem and or the area problem in this section. Exercise 8 of Section 2.1 is yet another exampleof how limits will be used in calculus. A review of the formula for the distance between two pointscan be found in Appendix C.

The discussion of limits is difficult for most students the first time that they see it. For this reason,you should carefully go over the examples and the informal definition of a limit presented inSection 2.2. Stress to your students that a limit exists only if the answer is a real number.Otherwise, the limit fails to exist, as shown in Examples 3, 4, and 5 of Section 2.2. You mightwant to work Exercise 65 of Section 2.2 with your students in preparation for the definition of thenumber You may choose to omit the formal definition of a limit.

Carefully go over the properties of limits found in Section 2.3 to ensure that your students arecomfortable with the idea of a limit and also with the notation used for limits. By the time you getto Theorem 2.6, it should be obvious to your students that all of these properties amount to directsubstitution.

When direct substitution for the limit of a quotient yields the indeterminate form tell your students they must rewrite the fraction using legitimate algebra. Then do at least one problemusing dividing out techniques and another using rationalizing techniques. Exercises 62 and 64 ofSection 2.3 are examples of other algebraic techniques needed for the limit problems. You need togo over the Squeeze Theorem, Theorem 2.8, with your students so that you can use it to prove

The proof of Theorem 2.8 and many other theorems can be found in Appendix A.

Your students need to memorize both of the results in Theorem 2.9 as they will need these facts todo problems throughout the textbook. Most of your students will need help with Exercises 67–80in this section.

Continuity, which is discussed in Section 2.4, is another idea that often puzzles students. However,if you describe a continuous function as one in which you can draw the entire graph without liftingyour pencil, the idea seems to stay with them. Distinguishing between removable and nonremovablediscontinuities will help students determine vertical asymptotes.

To discuss infinite limits, in Section 2.5, remind your students of the graph of the function studied in Section 1.3. Be sure to make your students write a vertical asymptote as an equation, notjust a number. For example, for the function the vertical asymptote is

Section 2.1 A Preview of Calculus

Section Comments2.1 A Preview of Calculus—Understand what calculus is and how it compares with precalculus.

Understand that the tangent line problem is basic to calculus. Understand that the areaproblem is also basic to calculus.

x � 0.y � 1�x,

f�x� � 1�x

limx→0

sin x

x� 1.

00,

e.

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Teaching Tips

In this first section, students see how limits arise when attempting to find a tangent to a curve. Besure to say to students that they can think about the word tangent as meaning “touching” a curveat one particular point. Drawing a circle with a line that touches the circle at a specific point canillustrate tangency. Drawing a curve where a line intersects a curve twice illustrates a line that isnot tangent to a curve as it crosses the curve more than once.

How Do You See It? Exercise

Page 67, Exercise 8 How would you describe the instantaneous rate of change of an automobile’sposition on a highway?

Solution

Answers will vary. Sample answer: The instantaneous rate of change of an automobile’s positionis the velocity of the automobile, and can be determined by the speedometer.

Suggested Homework Assignment

Page 67: 1–9 odd.

Section 2.2 Finding Limits Graphically and Numerically

Section Comments2.2 Finding Limits Graphically and Numerically—Estimate a limit using a numerical or

graphical approach. Learn different ways that a limit can fail to exist. Study and use aformal definition of limit.

Teaching Tips

In this section, we turn our focus to limits in general. We consider how to find limits both

graphically and numerically. Ask students to consider and what is happening to

as tends to 2. This will lead into the definition of a limit.

Make sure students find limits analytically instead of using their graphing calculators, as they can

be misleading. For example, present to the class, Constructing a table of values of

0.1, and 0.01 leads to the assumption that However, this limit does not exist.

When introducing the definition of a limit, using graphs will help students understand the meaning

of the definition. The graph of shows that even though cannot equal 1, the limit of

as approaches 1 is 0.5.

1

0.5

x

y

y = x − 1x2 − 1

xf�x�xy �

x � 1x2 � 1

limx→0

sin �

x� 0.

14

,12

,13

,

x � 1,limx→0

sin �

x.

xf �x�

f �x� �x2 � 4x � 2

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How Do You See It? Exercise

Page 78, Exercise 68 Use the graph of to identify the values of for which exists.

(a) (b)

Solution

(a) exists for all

(b) exists for all

Suggested Homework Assignment

Pages 75–78: 1–49 odd, 53– 57 odd, 63, 66, 67–75 odd.

Section 2.3 Evaluating Limits Analytically

Section Comments2.3 Evaluating Limits Analytically—Evaluate a limit using properties of limits. Develop

and use a strategy for finding limits. Evaluate a limit using the dividing out and rationalizing techniques. Evaluate a limit using the Squeeze Theorem.

Teaching Tips

When starting this section, review how to factor using differences of two squares and two cubes,sum of two cubes, and rationalizing numerators. If you do not have time to review these concepts,encourage students to study the following material in Precalculus, 9th edition, by Larson.

State all limit properties as presented in this section and discuss the proper ways to write solutionswhen finding limits analytically. Give examples of direct substitution; some suggested examples

include and

When rationalizing the numerator, start with an example without limits. A suggestion is

After rationalizing the numerator, find the limit as approaches 0.

When evaluating trigonometric limits, show students examples directly related to Theorem 2.9 in

the text. Some examples are and

For a good review of trig, present Exercise 76. It is also recommended to present Exercise 77 usingsince this skill will be needed for Chapter 5.e,

lim�→0

cos � � 1

sin �.lim

x→0 sin 7x

4x

x

�x � 2 � �2x

.

limx→�3

4x3 � 5x � 1.limx→2

x2 � 3x � 4

x � 2

c � �2, 0.limx→c

f �x�

c � �3.limx→c

f �x�

y

x2−4 4 6

2

4

6

y

x2 4−2

−2

4

6

limx→c

f �x�cf

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How Do You See It? Exercise

Page 88, Exercise 104 Would you use the dividing out technique or the rationalizing technique tofind the limit of the function? Explain your reasoning.

(a) (b)

Solution

(a) Use the dividing out technique because the numerator and denominator have a common factor.

(b) Use the rationalizing technique because the numerator involves a radical expression.

Suggested Homework Assignment

Pages 87–89: 3, 5, 13, 17, 19, 23–27 odd, 33, 37, 41, 47, 55–65 odd, 71–79 odd,81, 91–97 odd, 107, and 109.

Section 2.4 Continuity and One-Sided Limits

Section Comments2.4 Continuity and One-Sided Limits—Determine continuity at a point and continuity on

an open interval. Determine one-sided limits and continuity on a closed interval. Useproperties of continuity. Understand and use the Intermediate Value Theorem.

Teaching Tips

If students are having trouble with limits involving trigonometric functions, rationalizing denominators and numerators, or rational functions, encourage them to sharpen their algebra skillsby studying the following material in the appendix or in Precalculus, 9th edition, by Larson.

• Trigonometric functions: Appendix C.3

• Rationalizing numerators: Precalculus Appendix A.2

• Simplifying rational expressions: Precalculus Appendix A.4

� limx→0

1

�x � 4 � 2�

1�4 � 2

�1

4

� limx→0

�x � 4� � 4

x��x � 4 � 2�

limx→0

�x � 4 � 2

x� lim

x→0 �x � 4 � 2

x�

�x � 4 � 2�x � 4 � 2

� limx→�2

�x � 1� � �2 � 1 � �3

limx→�2

x2 � x � 2

x � 2� lim

x→�2 �x � 2��x � 1�

x � 2

x

y

−2 −1 1−3−4

1.00

0.75

0.50x

y

−1−2−3 1 2 3

−3

−4

1

2

limx→0

�x � 4 � 2

xlim

x→�2 x2 � x � 2

x � 2

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Address the following for this section:

• Describe the difference between a removable discontinuity and a nonremovable discontinuity.

• Write a function that has a removable discontinuity.

• Write a function that has a nonremovable discontinuity.

• Write a function that has a removable discontinuity and a nonremovable discontinuity.

Students will need to know these concepts when they study vertical asymptotes in Section 2.5,improper integrals in Section 8.8, and functions of several variables in Section 13.2.

Discuss the difference between a removable discontinuity and a nonremovable discontinuity.

Use the function to illustrate a function with a nonremovable discontinuity.

Note that it is not possible to remove the discontinuity by redefining the function. When you

illustrate a removable discontinuity, as in the function note that the function

can be redefined to remove the discontinuity. Ask students how they would do this. Listen for themto tell you to let

Ask students to consider the following function and its graph. Ask the students if there are anyremovable or nonremovable discontinuities. Once they have identified as removable and as nonremovable, ask them to redefine to remove the discontinuity at Let

You may want to use a rational function with both a vertical asymptote and a hole to show the differences between removable and nonremovable discontinuities. A suggested example

is

How Do You See It? Exercise

Page 101, Exercise 112 Every day you dissolve 28 ounces of chlorine in a swimming pool. Thegraph shows the amount of chlorine in the pool after days. Estimate and interpret and

y

t6 754321

140

112

84

56

28

limt→4�

f �t�.lim

t→4� f �t�tf �t�

f �x� �8x2 � 26x � 152x2 � x � 15

.

−2−4−6 2 4 6−1

−2

1

2

3

4

x

y

f �x� � �0,1,0,1,

x < �4x � �4�4 < x < 4x � 4

f ��4� � 0.x � �4.fx � 4

x � �4

f ��4� � 1.

f �x� �sin�x � 4�

x � 4,

f �x� ��x � 4�x � 4

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Solution

At the end of day 3, the amount of chlorine in the pool has decreased to about 28 ounces. At the beginning of day 4, more chlorine was added, and the amount is now about 56 ounces.

Suggested Homework Assignment

Pages 99–102: 1–71 odd, 77–83 odd, 99, 101, 107–113 odd, and 120.

Section 2.5 Infinite Limits

Section Comments2.5 Infinite Limits—Determine infinite limits from the left and from the right. Find and

sketch the vertical asymptotes of the graph of a function.

Teaching Tips

It is vital for students to know asymptotes of a rational function for this section. Encourage students to review material on asymptotes by studying Section 2.6 in Precalculus, 9th edition,by Larson.

Students need to be able to decipher when vertical asymptotes versus holes will occur in a rational function. A suggested problem for students to consider is to determine if is a

vertical asymptote of

For students who believe that the graph of has a vertical asymptote at have them provide aproof. For those who think this is not always true, ask them to provide a counterexample. A sam-ple answer is given in the solution, and you can use a transparency to illustrate a counterexample. If desired, you could also ask students the following:

(a) At which -values (if any) is not continuous?

(b) Which of the discontinuities are removable?

Solution

No, it is not always true. Consider The function

has a hole at not a vertical asymptote.�1, 2�,

f �x� �x2 � 1x � 1

�p�x�

x � 1

p�x� � x2 � 1.

fx

x � 1,f

f �x� �p�x�

x � 1.

x � 1

limt→4�

f �t� � 56

limt→4�

f �t� � 28

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How Do You See It? Exercise

Page 109, Exercise 66 For a quantity of gas at a constant temperature, the pressure is inverselyproportional to the volume What is the limit of as approaches 0 from the right? Explainwhat this means in the context of the problem.

Solution

As the volume of the gas decreases, the pressure increases.

Suggested Homework Assignment

Pages 108–110: 1–57 odd, 71, and 73.

limV→0�

P �

Volume

Pres

sure

V

P

VPV.P

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Chapter 2 Project

Medicine in the BloodstreamA patient’s kidneys purify 25% of the blood in her body in 4 hours.

Exercises

In Exercises 1–3, a patient takes one 16-milliliter dose of a medication.

1. Determine the amount of medication left in the patient’s body after 4, 8, 12, and 16 hours.

2. Notice that after the first 4-hour period, of the 16 milliliters of medication is left in the body,

after the second 4-hour period, of the 16 milliliters of medication is left in the body, and soon. Use this information to write an equation that represents the amount of medication left inthe patient’s body after 4-hour periods.

3. Can you find a value of for which equals 0? Explain.

In Exercises 4–9, the patient takes an additional 16-milliliter dose every 4 hours.

4. Determine the amount of medication in the patient’s body immediately after taking the seconddose.

5. Determine the amount of medication in the patient’s body immediately after taking the thirdand fourth doses. What is happening to the amount of medication in the patient’s body overtime?

6. The medication is eliminated from the patient’s body at a constant rate. Sketch a graph that shows the amount of medication in the patient’s body during the first 16 hours. Let represent the number of hours and represent the amount of medication in the patient’s body in milliliters.

7. Use the graph in Exercise 6 to find the limits.

(a)

(b)

(c)

(d)

8. Discuss the continuity of the function represented by the graph in Exercise 6. Interpret any discontinuities in the context of the problem.

9. The amount of medication in the patient’s body remains constant when the amount eliminatedin 4 hours is equal to the additional dose taken at the end of the 4-hour period. Write and solvean equation to find this amount.

limx→12�

f �x�

limx→12�

f �x�

limx→4�

f �x�

limx→4�

f �x�

yx

an

na

916

34

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