Calculus - Concepts and Applications - Foerster

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Photograph credits appear on the last two pages of the book.

To people from the past, including James H. Marable of Oak Ridge National Laboratory, from whom I first understood the concepts of calculus; Edmund Eickenroht, my former student, whose desire it was to write his own calculus text; and my late wife, Jo Ann. To my wife, Peggy, who shares my zest for life and accomplishment.

Consultants to the First Edition

Donald J. Albers, Mathematical Association of America, Washington D.C. Judith Broadwin, Jericho High School, Jericho, New York

Joan Ferrini-Mundy, University of New Hampshire, Durham, New Hampshire Gregory D. Foley, Sam Houston State University, Huntsville, Texas

John Kenelly, Clemson University, Clemson, South Carolina Dan Kennedy, Baylor School, Chattanooga, Tennessee Deborah B. Preston, Keystone School, San Antonio, Texas

Field Testers of the First Edition

Betty Baker, Bogan High School, Chicago, Illinois

© 2005 Key Curriculum Press Glenn C. Ballard, William Henry Harrison High School, Evansville, Indiana

Bruce Cohen, Lick-Wilmerding High School, San Francisco, California Christine J. Comins, Pueblo County High School, Pueblo, Colorado Deborah Davies, University School of Nashville, Nashville, Tennessee Linda E. de Sola, Plano Senior High School, Plano, Texas

Paul A. Foerster, Alamo Heights High School, San Antonio, Texas

Joan M. Gell, Palos Verdes Peninsula High School, Rolling Hills Estates, California Valmore E. Guernon, Lincoln Junior/Senior High School, Lincoln, Rhode Island David S. Heckman, Monmouth Academy, Monmouth, Maine

Don W. Hight, Pittsburg State University, Pittsburg, Kansas Edgar Hood, Dawson High School, Dawson, Texas

John G. Kelly, Arroyo High School, San Lorenzo, California Linda Klett, San Domenico School, San Anselmo, California

George Lai, George Washington High School, San Francisco, California Katherine P. Layton, Beverly Hills High School, Beverly Hills, California Debbie Lindow, Reynolds High School, Troutdale, Oregon

Robert Maass, International Studies Academy, San Francisco, California Guy R. Mauldin, Science Hill High School, Johnson City, Tennessee Windle McKenzie, Brookstone School, Columbus, Georgia

Bill Medigovich, Redwood High School, Larkspur, California Sandy Minkler, Redlands High School, Redlands, California Deborah B. Preston, Keystone School, San Antonio, Texas Sanford Siegel, School of the Arts, San Francisco, California Susan M. Smith, Ysleta Independent School District, El Paso, Texas Gary D. Starr, Girard High School, Girard, Kansas

Tom Swartz, George Washington High School, San Francisco, California Tim Trapp, Mountain View High School, Mesa, Arizona

Dixie Trollinger, Mainland High School, Daytona Beach, Florida David Weinreich, Queen Anne School, Upper Marlboro, Maryland John P. Wojtowicz, Saint Joseph’s High School, South Bend, Indiana Tim Yee, Malibu High School, Malibu, California

Author’s Acknowledgments

This text was written during the period when graphing calculator technology was making radical changes in the teaching and learning of calculus. The fundamental differences embodied in the text have arisen from teaching my own students using this technology. In addition, the text has been thoroughly revised to incorporate comments and suggestions from the many consultants and field testers listed on the previous page.

Thanks in particular to the original field test people—Betty Baker, Chris Comins, Debbie Davies, Val Guernon, David Heckman, Don Hight, Kathy Layton,

Guy Mauldin, Windle McKenzie, Debbie Preston, Gary Starr, and John Wojtowicz. These instructors were enterprising enough to venture into a new approach to teaching calculus and to put up with the difficulties of receiving materials at the last minute.

Special thanks to Bill Medigovich for editing the first edition, coordinating the field test program, and organizing the first two summer institutes for instructors. Special thanks also to Debbie Preston for drafting the major part of the Instructor’s Guide and parts of the Solutions Manual, and for working with the summer institutes for instructors. By serving as both instructors and consultants, these two have given this text an added dimension of clarity and teachability.

Thanks also to my students for enduring all those handouts, and for finding things to be changed! Special thanks to my students Craig Browning,

Meredith Fast, William Fisher, Brad Wier, and Matthew Willis for taking good class notes so that the text materials could include classroom-tested examples. Thanks to the late Richard V. Andree and his wife, Josephine, for allowing their children, Phoebe Small and Calvin Butterball, to make occasional appearances in my texts.

Finally, thanks to Chris Sollars, Debbie Davies, and Debbie Preston for their ideas and encouragement as I worked on the second edition of Calculus.

Paul A. Foerster

About the Author

Paul Foerster enjoys teaching mathematics at Alamo Heights High School in San Antonio, Texas, which he has done since 1961. After earning a bachelor’s degree in chemical engineering, he served four years in the U.S. Navy. Following his first five years at Alamo Heights, he earned a master’s degree in

mathematics. He has published five textbooks, based on problems he wrote for his own students to let them see more realistically how mathematics is applied in the real world. In 1983 he received the Presidential Award for Excellence in Mathematics Teaching, the first year of the award. He raised three children with the late Jo Ann Foerster, and he also has two grown stepchildren through his wife Peggy Foerster, as well as three grandchildren. Paul plans to continue teaching for the foreseeable future, relishing the excitement of the ever-changing content of the evolving mathematics curriculum.

Foreword

by John Kenelly, Clemson University

In the explosion of the information age and the resulting instructional reforms, we have all had to deal repeatedly with the question: “When machines do mathematics, what do mathematicians do?” Many feel that our historical role has not changed, but that the emphasis is now clearly on selection and interpretation rather than manipulation and methods. As teachers, we continue to sense the need for a major shift in the instructional means we employ to impart mathematical understanding to our students. At the same time, we recognize that behind any technology there must be human insight.

In a world of change, we must build on the past and take advantage of the future. Applications and carefully chosen examples still guide us through what works. Challenges and orderly investigations still develop mature thinking and insights. As much as the instructional environment might change, quality education remains our goal. What we need are authors and texts that bridge the transition. It is in this regard that Paul Foerster and his texts provide outstanding answers. In Calculus: Concepts and Applications, Second Edition, Paul is again at his famous best. The material is presented in an easily understood fashion with ample technology-based examples and exercises. The applications are intimately connected with the topic and amplify the key elements in the section. The material is a wealth of both fresh items and ancient insights that have stood the test of time. For example, alongside Escalante’s “cross hatch” method of repeated integration by parts, you’ll find Heaviside’s thumb trick for solving partial fractions! The students are repeatedly sent to their “graphers.” Early on, when differentiation is introduced, Paul discusses local linearity, and later he utilizes the zoom features of calculators in the coverage of l’Hospital’s rule—that’s fresh. Later still, he presents the logistic curve and slope fields in differential equations. All of these are beautiful examples of how computing technology has changed the calculus course.

The changes and additions found in this second edition exhibit the timeliness of the text. Exponentials and logarithms have been given an even more prominent role that reflects their greater emphasis in today’s calculus instruction. The narrative, problem sets, Explorations, and tests all support the position that the

choice between technology and traditional methods is not exclusively “one or the other” but correctly both. Rich, substantive, in-depth questions bring to mind superb Advanced Placement free response questions, or it might be that many AP questions remind you of Foerster’s style!

Throughout, you see how comprehensive Paul is in his study of the historical role of calculus and the currency of his understanding of the AP community and collegiate “calculus reform.” Brilliant, timely, solid, and loaded with tons of novel applications—your typical Foerster!

John Kenelly has been involved with the Advanced Placement Calculus program for over 30 years. He was Chief Reader and later Chair of the AP Calculus Committee when Paul Foerster was grading the AP exams in the 1970s. He is a leader in development of the graphing calculator and in pioneering its use in college and school classrooms. He served as president of the IMO 2001 USA, the organization that acts as host when the International Mathematical Olympiad (IMO) comes to the United States.

Contents

A Note to the Student from the Author

C H A P T E

R

1

Limits, Derivatives, Integrals, and Integrals

1-1 The Concept of Instantaneous Rate 3

1-2 Rate of Change by Equation, Graph, or Table 6

1-3 One Type of Integral of a Function 14

1-4 Definite Integrals by Trapezoids, from Equations and Data 18

1-5 Calculus Journal 24

1-6 Chapter Review and Test 25

C H A P T E

R

2

Properties of Limits

2-1 Numerical Approach to the Definition of Limit 33

2-2 Graphical and Algebraic Approaches to the Definition of Limit 34

2-3 The Limit

Theorems

40

2-4 Continuity and Discontinuity 45

2-5 Limits Involving Infinity 52

2-6 The Intermediate Value Theorem and Its Consequences 60

2-7 Chapter Review and Test 64

C H A P T E

R

3

Derivatives, Antiderivatives, and Indefinite Integrals

3-1 Graphical Interpretation of Derivative 73

3-2 Difference Quotients and One Definition of Derivative 74

3-3 Derivative Functions, Numerically and Graphically 78

3-4 Derivative of the Power Function and Another

Definition of Derivative 85

3-5 Displacement, Velocity, and Acceleration 92

3-6 Introduction to Sine, Cosine, and Composite Functions 100

3-7 Derivatives of Composite Functions—The Chain Rule

102

3-8 Proof and Application of Sine and Cosine Derivatives 107

3-9 Exponential and Logarithmic Functions 115

3-10 Chapter Review and Test 122

C H A P T E

R

4

Products, Quotients, and Parametric Functions

4-2 Derivative of a Product of Two Functions 132

4-3 Derivative of a Quotient of Two Functions 137

4-4 Derivatives of the Other Trigonometric Functions 142

4-5 Derivatives of Inverse Trigonometric Functions 146

4-6 Differentiability and Continuity 153

4-7 Derivatives of a Parametric Function 160

4-8 Graphs and Derivatives of Implicit Relations 169

4-9 Related Rates 174

4-10 Chapter Review and Test 180

C H A P T E

R

5

Definite and Indefinite Integrals

5-2 Linear Approximations and Differentials 190

5-3 Formal Definition of Antiderivative and Indefinite Integral 197

5-4 Riemann Sums and the Definition of Definite Integral 204

5-5 The Mean Value Theorem and Rolle's Theorem 211

5-6 The Fundamental Theorem of Calculus 221

5-7 Definite Integral Properties and Practice 227

5-8 Definite Integrals Applied to Area and Other Problems 233

5-9 Volume of a Solid by Plane Slicing 242

5-10 Definite Integrals Numerically by Grapher and

by Simpson's Rule 252

5-11 Chapter Review and Test 259

C H A P T E

R

6

The Calculus of Exponential and Logarithmic Functions

Form of the Fundamental Theorem 270

6-3 The Uniqueness Theorem and Properties of Logarithmic Functions 280 6-4

288

6-5 Limits of Indeterminate Forms: l'Hospital's Rule 295

6-6 Derivative and Integral Practice for Transcendental Functions 301

6-7 Chapter Review and Test 306

6-8 Cumulative Review: Chapters 1–6 311

C H A P T E

R

7

The Calculus of Growth and Decay

7-2 Exponential Growth and Decay 318

7-3 Other Differential Equations for Real-World Applications 324 7-4 Graphical Solution of Differential Equations by Using Slope Fields 333 7-5 Numerical Solution of Differential Equations by

Using Euler's Method 341

7-6 The Logistic Function, and Predator-Prey Population Problems 348 x © 2005 Key Curriculum Press

Logarithmic Differentiation

© 2005 Key Curriculum Press xi

the Elementary Functions

9

Algebraic Calculus Techniques for

7-7 Chapter Review and Test 359

7-8 Cumulative Review: Chapters 1–7 365

C H A P T E

R

8

The Calculus of Plane and Solid Figures

8-2 Critical Points and Points of Inflection 372

8-3 Maxima and Minima in Plane and Solid Figures 385

8-4 Volume of a Solid of Revolution by Cylindrical Shells 395

8-5 Length of a Plane Curve—Arc Length

401

8-6 Area of a Surface of Revolution 407

8-7 Lengths and Areas for Polar Coordinates 414

8-8 Chapter Review and Test 423

C H A P T E

R 431

9-1 Introduction to the Integral of a Product of Two Functions 433

9-2 Integration by Parts—A Way to Integrate Products 434

9-3 Rapid Repeated Integration by Parts 438

9-4 Reduction Formulas and Computer Algebra Systems 444

9-5 Integrating Special Powers of Trigonometric Functions 449

9-6 Integration by Trigonometric Substitution 454

9-7 Integration of Rational Functions by Partial Fractions 460

9-8 Integrals of the Inverse Trigonometric Functions 466

9-9 Calculus of the Hyperbolic and Inverse Hyperbolic Functions 469

9-10 Improper Integrals 481

9-11 Miscellaneous Integrals and Derivatives 488

9-12 Integrals in Journal 493

9-13 Chapter Review and Test 494

10

The Calculus of Motion—Averages,

Extremes, and Vectors

10-1 Introduction to Distance and Displacement for

Motion Along a Line 501

10-2 Distance, Displacement, and Acceleration for Linear Motion 502

10-3 Average Value Problems in Motion and Elsewhere 508

10-4 Minimal Path Problems 514

10-5 Maximum and Minimum Problems in Motion and Elsewhere 520

10-6 Vector Functions for Motion in a Plane 522

10-7 Chapter Review and Test 538

C H A P T E

R

11

The Calculus of Variable-Factor Products

11-1 Review of Work—Force Times Displacement

547

11-2 Work Done by a Variable Force 548

11-3 Mass of a Variable-Density Object 553

11-4 Moments, Centroids, Center of Mass, and the Theorem of Pappus

558 11-5 Force Exerted by a Variable Pressure—Center of

Pressure

567 C H A P T E R

11-6 Other Variable-Factor Products 573

11-7 Chapter Review and Test 580

C H A P T E

R

12

The Calculus of Functions Defined by Power Series

12-2 Geometric Sequences and Series as Mathematical Models 590

12-3 Power Series for an Exponential Function 597

12-4 Power Series for Other Elementary Functions 598

12-5 Taylor and Maclaurin Series, and Operations on These Series 605 12-6 Interval of Convergence for a Series—The Ratio Technique 613 12-7 Convergence of Series at the Ends of the Convergence Interval 621

12-8 Error Analysis for Series—The Lagrange Error Bound 635

12-9 Chapter Review and Test 643

Cumulative Reviews 648

Final Examination: A Guided Tour Through Calculus

Appendix: Summary of Properties of

Trigonometric Functions

Answers to Selected Problems

Glossary

Index of Problem Titles

General Index

Photograph Credits

12-9

xii © 2005 Key Curriculum Press 12-10

A Note to the Student

from the Author

In earlier courses you have learned about functions. Functions express the way one variable quantity, such as distance you travel, is related to another quantity, such as time. Calculus was invented over 300 years ago to deal with the rate at which a quantity varies, particularly if that rate does not stay constant.

In your calculus course you will learn the algebraic formulas for variable rates that will tie together the mathematics you have learned in earlier courses. Fortunately, computers and graphing calculators (“graphers”) will give you graphical and numerical methods to understand the concepts even before you develop the formulas. In this way you will be able to work calculus problems from the real world starting on day one. Later, once you understand the concepts, the formulas will give you time-efficient ways to work these problems.

The time you save by using technology for solving problems and learning concepts can be used to develop your ability to write about mathematics. You will be asked to keep a written journal recording the concepts and techniques you have been learning, and verbalizing things you may not yet have mastered. Thus, you will learn calculus in four ways—algebraically, graphically, numerically, and verbally. In whichever of these areas your talents lie, you will have the opportunity to excel.

As in any mathematics course, you must learn calculus by doing it. Mathematics is not a “spectator sport.” As you work on the

Explorations that introduce you to new concepts and techniques, you will have a chance to participate in cooperative groups, learning from your classmates and improving your skills.

The Quick Review problems at the beginning of each problem set ask you to recall quickly things that you may have forgotten from earlier in the text or from previous courses. Other problems, marked by a shaded star, will prepare you for a topic in a later section. Prior to the Chapter Test at the end of each chapter, you will find review problems keyed to each section. Additionally, the Concept Problems give you a

chance to apply your knowledge to new and challenging situations. So, keeping up with your homework will help to ensure your success. At times you may feel you are becoming submerged in details. When that happens, just remember that calculus involves only four concepts:

• Limits • Derivatives • Integrals (one kind) • Integrals (another kind)

Ask yourself, “Which of these concepts does my present work apply to?” That way, you will better see the big picture. Best wishes as you venture into the world of higher mathematics!

Paul A. Foerster

Alamo Heights High School San Antonio, Texas

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CHAPTER

1

1

Limits, Derivatives,

Integrals, and Integrals

Integrals, and Integrals

Automakers have recently begun to produce electric cars, which

utilize electricity instead of gasoline to run their engines. Engineers

are constantly looking for ways to design an electric car that can

match the performance of a conventional gasoline-powered car.

Engineers can predict a car’s performance characteristics even

before the first prototype is built. From information about the

acceleration, they can calculate the car’s velocity as a function of

time. From the velocity, they can predict the distance it will travel

while it is accelerating. Calculus provides the mathematical tools

to analyze quantities that change at variable rates.

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Mathematical Overview

Calculus deals with calculating things that change at variable rates.

The four concepts invented to do this are

•

Limits

•

Derivatives

•

Integrals (one kind)

•

Integrals (another kind)

In Chapter 1, you will study three of these concepts in four ways.

Graphically

Graphically

The icon at the top of each

even-numbered page of this chapter

illustrates a limit, a derivative, and one

type of integral.

Numerically

Numerically

x

x

− d

Slope

2.1

0.1

1.071666

...

2.01

0.01

1.007466

...

2.001

0.001

1.000749

...

..

.

..

.

..

.

Algebraically

Algebraically

Average rate of change

=

f

(

x

)

− f

(2)

x

−

2

Verbally

Verbally

_{I have learned that a definite integral is used to measure the product}

of x and f(x). For instance, velocity multiplied by time gives the

distance traveled by an object. The definite integral is used to find

this distance if the velocity varies.

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1-1

The Concept of Instantaneous Rate

If you push open a door that has an automatic closer, it opens fast at first, slows down, stops, starts closing, then slams shut. As the door moves, the number of degrees,d, it is from its closed position depends on how many seconds it has been since you pushed it. Figure 1-1a shows such a door from above.

Door

d

Figure 1-1a

The questions to be answered here are, “At any particular instant in time, is the door opening or closing?” and “How fast is it moving?” As you progress through this course, you will learn to write equations expressing the rate of change of one variable quantity in terms of another. For the time being, you will answer such questions graphically and numerically.

OBJECTIVE

100 1 d t 7 Figure 1-1b

Suppose that a door is pushed open at timet₌0 s and slams shut again at time

t₌7 s. While the door is in motion, assume that the number of degrees,d, from its closed position is modeled by this equation.

d=200t·2−t for 0≤ t ≤7

How fast is the door moving at the instant whent=1 s? Figure 1-1b shows this equation on a grapher (graphing calculator or computer). Whentis 1, the graph is going up astincreases from left to right. So the angle is increasing and the door is opening. You can estimate the rate numerically by calculating values ofd

for values oft close to 1.

t₌1: d₌200(1)_·2−1 ₌₁₀₀◦

t=1.1: d=200(1.1)·2−1.1 =102.633629...◦

The door’s angle increased by 2.633...◦in 0.1 s, meaning that it moved at a rate of about (2.633...)/0.1, or 26.33... deg/s. However, this rate is an average rate, and the question was about an instantaneous rate. In an “instant” that is 0 s long, the door moves 0◦. Thus, the rate would be 0/0, which is awkward because division by zero is undefined.

To get closer to the instantaneous rate att₌1 s, finddatt₌1.01 s and at

t₌1.001 s.

t₌1.01: d₌200(1.01)_·2−1.01 _{=100.30234..., a change of 0.30234...}◦

t=1.001: d=200(1.001)·2−1.001 =100.03064..., a change of 0.03064...◦

Here are the average rates for the time intervals 1 s to 1.01 s and 1 s to 1.001 s. 1 s to 1.01 s: average rate ₌0.30234...

0.01 =30.234...deg/s 1 s to 1.001 s: average rate ₌0.03064...

0.001 =30.64...deg/s

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The important thing for you to notice is that as the time interval gets smaller and smaller, the number of degrees per second doesn’t change much.

Figure 1-1c shows why. As you zoom in on the point (1, 100), the graph appears to be straighter, so the change inddivided by the change int becomes closer to the slope of a straight line.

If you list the average rates in a table, another interesting feature appears. The values stay the same for more and more decimal places.

100 1 d t 7 0.01 t = 1 t = 1.01 d = 100 0.30… d = 100.3023… Figure 1-1c t (s) Average Rate 1 to 1.01 30.23420... 1 to 1.001 30.64000... 1 to 1.0001 30.68075... 1 to 1.00001 30.68482... 1 to 1.000001 30.68524...

There seems to be a limiting number that the values are approaching.

To estimate the instantaneous rate att₌3 s, follow the same steps as fort₌1 s.

t₌3: d₌200(3)_·2−3 ₌₇₅◦

t₌3.1: d₌200(3.1)_·2−3.1 _{=72.310056...}◦

t₌3.01: d₌200(3.01)_·2−3.01 _{=74.730210...}◦

t₌3.001: d₌200(3.001)_·2−3.001 _{=74.973014...}◦ Here are the corresponding average rates.

3 s to 3.1 s: average rate ₌72.310056...−75 3.1₋3 = −26.899...deg/s 3 s to 3.01 s: average rate ₌74.730210...−75 3.01−3 = −26.978...deg/s 3 s to 3.001 s: average rate =74.973014...−75 3.001₋3 = −26.985...deg/s Again, the rates seem to be approaching some limiting number, this time, around₋27. So the instantaneous rate att₌3 s should be somewhere close to₋27 deg/s. The negative sign tells you that the number of degrees,d,is decreasing as time goes on. Thus, the door is closing whent₌3 s. It is opening whent₌1 because the rate of change is positive.

For the door example shown above, the angle is said to be a function of time. Time is the independent variable and angle is the dependent variable. These names make sense, because the number of degrees the door is open depends on the number of seconds since it was pushed. The instantaneous rate of change of the dependent variable is said to be the limit of the average rates as the time interval gets closer to zero. This limiting value is called the derivative of the dependent variable with respect to the independent variable.

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Problem Set 1-1

1. Pendulum Problem: A pendulum hangs from the ceiling (Figure 1-1d). As the pendulum swings, its distance,d,in centimeters from one wall of the room depends on the number of seconds,t, since it was set in motion. Assume that the equation fordas a function oftis

d=80+30 cosπ₃t, t≥0

You want to find out how fast the pendulum is moving at a given instant,t, and whether it is approaching or going away from the wall.

d

Figure 1-1d

a. Finddwhent₌5. If you don’t get 95 for the answer, make sure your calculator is in radian mode.

b. Estimate the instantaneous rate of change of

datt=5 by finding the average rates from

t₌5 to 5.1,t₌5 to 5.01, andt₌5 to 5.001. c. Why can’t the actual instantaneous rate of

change ofdwith respect totbe calculated using the method in part b?

d. Estimate the instantaneous rate of change of

dwith respect totatt₌1.5. At that time, is the pendulum approaching the wall or moving away from it? Explain.

e. How is the instantaneous rate of change related to the average rates? What name is given to the instantaneous rate?

f. What is the reason for the domain restrictiont_≥0? Can you think of any reason that there would be an upper bound to the domain?

2. Board Price Problem: If you check the prices of various lengths of lumber, you will find that a board twice as long as another of the same type does not necessarily cost twice as much. Letxbe the length, in feet, of a 2 ×6 board (Figure 1-1e) and letybe the price, in cents, that you pay for the board. Assume thatyis given by y₌0.2x3₋4.8x2₊80x x 6″ 2″ 2-by-6 Figure 1-1e

a. Find the price of 2 _×6 boards that are 5 ft long, 10 ft long, and 20 ft long.

b. Find the average rate of change of the price in cents per foot forx₌5 to 5.1,x₌5 to 5.01, andx₌5 to 5.001.

c. The average number of cents per foot in part b is approaching an integer as the change inxgets smaller and smaller. What integer? What is the name given to this rate of change?

d. Estimate the instantaneous rate of change in price ifxis 10 ft and ifxis 20 ft. You should find that each of these rates is an integer. e. One of the principles of marketing is that

when you buy in larger quantities, you usually pay less per unit. Explain how the numbers in Problem 2 show that this principle does not apply to buying longer boards. Think of a reason why it does not apply.

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1-2

Rate of Change by Equation, Graph,

or Table

In Section 1-1, you explored functions for which an equation related two variable quantities. You found average rates of change off(x) over an interval ofx-values, and used these to estimate the instantaneous rate of change at a particular value ofx. The instantaneous rate is called the derivative of the function at that value ofx. In this section you will estimate instantaneous rates for functions specified graphically or numerically, as well as algebraically (by equations).

OBJECTIVE

increases through a particular value, and estimate the instantaneous rate of change ofyat that value ofx.

Background: Function Terminology and

Types of Functions

The price you pay for a certain type of board depends on how long it is. In mathematics the symbolf(x) (pronounced “f ofx” or “f atx”) is often used for the dependent variable. The letterf is the name of the function, and the number in parentheses is either a value of the independent variable or the variable itself. Iff(x)₌3x₊7, thenf(5) is 3(5)₊7, or 22.

The equationf(x)₌3x₊7 is the particular equation for a linear function. The general equation for a linear function is writteny_{= mx + b}, orf(x)_{= mx + b}, wheremandbrepresent the constants. The following box shows the names of some types of functions and their general equations.

DEFINITIONS: Types of Functions

DEFINITIONS: Types of Functions

Linear:f(x)_{= mx + b};mandbstand for constants,m_≠0

Quadratic:f(x)_{= ax}2_{+ bx + c;a,b, andcstand for constants,a≠0}

Polynomial:f(x)_{= a}0+ a1x+ a2x2+ a3x3+ a4x4+ · · · + anxn;a0, a1, . . . stand for constants,nis a positive integer,an≠0 (nth degree polynomial function)

Power:f(x)_{= ax}n_{;aandnstand for constants}

Exponential:f(x)_{= ab}x_{;aandbstand for constants,a≠0,b >0,b≠1}

Rational Algebraic:f(x)=(polynomial)/(polynomial) Absolute value:f(x) contains|(variable expression)|

Trigonometric or Circular:f(x) contains cosx, sinx, tanx, cotx, secx, or cscx

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EXAMPLE 1

x_{= b}, andx_{= c}, state whethery is increasing, decreasing, or neither asxincreases. Then state whether the rate of change is fast or slow.

Solution Atx= a, yis increasing quickly as you go from left to right.

Atx_{= b, y}is decreasing slowly becausey is dropping asxgoes from left to right, but it’s not dropping very quickly.

Atx_{= c, y}is neither increasing nor decreasing, as shown by the fact that the graph has leveled off

atx_{= c}.

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EXAMPLE 2

function of the time,t, in seconds since it was kicked into the air.

2 4 6 8 10 20 40 60 t (s) h(t) (ft) Figure 1-2b

a. Estimate the instantaneous rate of change ofh(t) at timet₌5.

b. Give the mathematical name of this instantaneous rate, and state why the rate is negative.

Solution a. Draw a line tangent to the graph atx₌5 by laying a ruler against it, as shown in Figure 1-2c. You will be able to estimate the tangent line more accurately if you put the ruler on the concave side of the graph.

The instantaneous rate is the slope of this tangent line. From the point wheret₌5, run over a convenient distance in thet-direction, say 4 seconds. Then draw a vertical line to the tangent line. As shown in the figure, this rise is about 56 feet in the negative direction.

Instantaneous rate₌slope of tangent_≈ −56

4 = −14 ft/s 2 4 6 8 10 20 40 60 t (s) h (t) (ft) Ruler Tangent line Run 4 Rise –56 Figure 1-2c

b. The mathematical name is derivative. The rate is negative becauseh(t) is

decreasing att=5.

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EXAMPLE 3

P(x)=40(0.6x), the probability that it rains a number of inches,

x, at a particular place during a particular thunderstorm.

a. The probability that it rains 1 inch isP(1)=24%. By how much, and in which direction, does the probability change fromx₌1 tox₌1.1? What is the average rate of change from 1 inch to 1.1 inches? Make sure to include units in

your answer. Why is the rate negative?

1 2 3 4 10 20 30 40 50 P (x) x (in.) Figure 1-2d

b. Write an equation forr(x), the average rate of change ofP(x) from 1 tox. Make a table of values ofr(x) for each 0.01 unit ofxfrom 0.97 to 1.03. Explain whyr(x) is undefined atx₌1.

c. The instantaneous rate atx=1 is the limit that the average rate approaches asxapproaches 1. Estimate the instantaneous rate using information from part b. Name the concept of calculus that is given to this instantaneous rate. Solution a. To find the average rate, first you must findP(1) andP(1.1).

P(1)₌40(0.61)₌24

P(1.1)₌40(0.61.1)₌22.8048...

Change=22.8048...−24= −1.1951... Change is always final minus initial.

Average rate₌−1.1951...

0.1 = −11.1951 %/in.

The rate is negative because the probability decreases as the number of inches increases.

b. The average rate of change ofP(x) from 1 toxis equal to the change inP(x) divided by the change inx.

r(x)=P(x)−24 x₋1 = 40(0.6x₎₋₂₄ x₋1 change inP(x) change inx

StoreP(x) asy1andr(x) asy2in your grapher. Make a table of values ofx andr(x). x r (x) 0.97 ₋12.3542... 0.98 ₋12.3226... 0.99 ₋12.2911... 1.00 Error 1.01 −12.2285... 1.02 −12.1974... 1.03 −12.1663...

Note thatr(1) is undefined because you would be dividing by zero. When

x=1,x−1=0.

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c. Averager(0.99) andr(1.01), the values in the table closest tox=1.

Instantaneous rate_≈ 1

2[r(0.99)+ r(1.01)]= −12.2598...

The percentage is decreasing at about 12.26% per inch. (The percentage decreases because it is less likely to rain greater quantities.) The name is

derivative.

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EXAMPLE 4

(Figure 1-2e). Its distance,y, in feet, from the ceiling is measured by a calculator distance probe each 1/10 s, giving this table of values, in whicht is time in seconds. t (s) y (ft) 0.2 3.99 0.3 5.84 0.4 7.37 0.5 8.00 0.6 7.48 0.7 6.01 0.8 4.16 0.9 2.63 1.0 2.00 1.1 2.52 y y Mass Figure 1-2e

a. How fast isy changing at each time? i. t=0.3

ii. t₌0.6 iii. t₌1.0

b. At timet₌0.3, is the mass going up or down? Justify your answer. Solution a. If data are given in numerical form, you cannot get better estimates of the

rate by taking values oftcloser and closer to 0.3. However, youcan get a better estimate by using the closestt-values on both sides of the given value. A time-efficient way to do the computations is shown in the following table. If youlike, do the computations mentally and write only the final answer.

t y Difference Rate Average Rate 0.2 3.99 0.3 5.84 0.4 7.37 0.5 8.00 0.6 7.48 0.7 6.01 0.8 4.16 0.9 2.63 1.0 2.00 1.1 2.52 1.85 1.85/0.1₌ 18.5 1.53 1.53/0.1₌ 15.3 −0.52 ₋0.52/0.1_{= −}5.2 −1.47 −1.47/0.1= −14.7 −0.63 −0.63/0.1= −6.3 0.52 0.52/0.1= 5.2 16.9 −9.95 −0.55

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All you need to write on your paper are the results, as shown here. i. t₌0.3: increasing at about 16.9 ft/s

ii. t₌0.6: decreasing at about 9.95 ft/s

iii. t=1.0: decreasing at about 0.55 ft/s Write real-world answers with units.

b. Att₌0.3, the rate is about 16.9 ft/s, a positive number. This fact implies thaty is increasing. Asyincreases, the mass goes downward.

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1 5

y

t

Figure 1-2f

The technique in Example 4 for estimating instantaneous rates by going forward and backward from the given value ofxcan also be applied to functions

specified by an equation. The result is usually more accurate than the rate estimated by only going forward as you did in the last section.

As you learned in Section 1-1, the instantaneous rate of change off(x) atx= c

is the limit of the average rate of change over the interval fromctoxasx

approachesc. The value of the instantaneous rate is called the derivative off(x) with respect toxatx= c. The meaning of the word derivative is shown here. You will learn the precise definition when it is time to calculate derivatives exactly.

Meaning of Derivative

Meaning of Derivative

The derivative of functionf(x) atx_{= c}is the instantaneous rate of change of

f(x) with respect toxatx_{= c}. It is found

•

toxasxapproachesc

•

Note that “with respect tox” implies that you are finding how fasty changes as x changes.

Preview: Definition of Limit

In Section 1-1, you saw that the average rate of change of they-value of a function got closer and closer to some fixed number as the change in thex-value got closer and closer to zero. That fixed number is called the limit of the average rate as the change inxapproaches zero. The following is a verbal definition of limit. The full meaning will become clearer to you as the course progresses.

Verbal Definition of Limit

Verbal Definition of Limit

Lis the limit off(x) asxapproachesc

if and only if

Lis the one number you can keepf(x) arbitrarily close to just by keepingxclose enough toc, but not equal toc.

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Problem Set 1-2

Quick Review 5min

From now on, there will be ten short problems at the beginning of most problem sets. Some of the problems will help you review skills from previous sections or chapters. Others will test your general knowledge. Speed is the key here, not detailed work. You should be able to do all ten problems in less than five minutes.

Q1. Name the type of function:f(x)= x3_.

Q2. Findf(2) for the function in Problem Q1.

Q3. Name the type of function:g(x)=3x.

Q4. Findg(2) for the function in Problem Q3.

Q5. Sketch the graph:h(x)_{= x}2_.

Q6. Findh(5) for the function in Problem Q5.

Q7. Write the general equation for a quadratic function.

Q8. Write the particular equation for the function in Figure 1-2g. x y 1 1 Figure 1-2g x y 1 1 Figure 1-2h

Q9. Write the particular equation for the function in Figure 1-2h.

Q10. What name is given to the instantaneous rate of change of a function?

Problems 1–10 show graphs of functions with values ofxmarked a, b, and so on. At each marked value, state whether the function is increasing, decreasing, or neither asxincreases from left to right, and also whether the rate of increase or decrease is fast or slow.

1. 2. a b x f (x) a b x f (x) 3. 4. a b x f (x) a x b f (x) 5. 6. a b c d x f (x) a b c d x f (x) 7. 8. a b c x f (x) a b c x f (x) 9. 10. a b c d x f (x) a b c d x f (x)

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11. Boiling Water Problem: Figure 1-2i shows the temperature,T(x), in degrees Celsius, of a kettle of water at timex, in seconds, since the burner was turned on.

100 200 50 100 x (s) T(x) (_°C) Figure 1-2i

a. On a copy of the figure, draw tangent lines at the points wherex₌40, 100, and 140. Use the tangent lines to estimate the

instantaneous rate of change of temperature at these times.

b. What do yousuppose is happening to the water for 0< x <80? For 80< x <120? For

x >120?

12. Roller Coaster Velocity Problem: Figure 1-2j shows the velocity,v(x), in ft/s, of a roller coaster car at timex, in seconds, after it starts down the first hill.

1 2 3 4 5 6 7 8 10 20 30 40 50 60 70 x (s) v (x) (ft/s) Figure 1-2j

a. On a copy of the figure, draw tangent lines at the points wherex=2,5, and 6. Use the tangent lines to estimate the instantaneous rate of change of velocity at these times. b. The instantaneous rates in part a are

derivatives ofv(x) with respect tox. What units must you include in your answers? What physical quantity is this?

13. Rock in the Air Problem: A small rock is tied to an inflated balloon, then the rock and balloon are thrown into the air. While the rock and

balloon are moving, the height of the rock is given by

h(x)_{= −x}2₊8x₊2

whereh(x) is in feet above the ground at timex, in seconds, after the rock was thrown.

a. Plot the graph of functionh. Sketch the result. Based on the graph, ish(x) increasing or decreasing atx=3? Atx=7?

b. How high is the rock atx₌3? Atx₌3.1? What is the average rate of change of its height from 3 to 3.1 seconds?

c. Find the average rate of change from 3 to 3.01 seconds, and from 3 to 3.001 seconds. Based on the answers, what limit does the average rate seem to be approaching as the time interval gets shorter and shorter? d. The limit of the average rates in part c is

called the instantaneous rate atx₌3. It is also called the derivative ofh(x) atx₌3. Estimate the derivative ofh(x) atx₌7. Make sure to include units in your answer. Why is the derivative negative atx₌7?

14. Fox Population Problem: The population of foxes in a particular region varies periodically due to fluctuating food supplies. Assume that the number of foxes,f(t), is given by

f(t)=300+200 sint

wheretis time in years after a certain date. a. Store the equation forf(t) asy1in your

grapher, and plot the graph using a window with [0, 10] fort. Sketch the graph. On the sketch, show a point wheref(t) is

increasing, a point where it is decreasing, and a point where it is not changing much. b. The change inf(t) from 1 year totis

(f(t)_{− f}(1)). So for the time interval [1, t], f(t) changes at the average rater(t) given by

r(t)=f(t)− f(1)

t₋1

Enterr(t) asy2in your grapher. Then make a table of values ofr(t) for each 0.01 year from 0.97 through 1.03.

c. The instantaneous rate of change off(t) at

t=1 is the limitf(t) approaches ast

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approaches 1. Explain why your grapher gives an error message if you try to calculate

r(1). Find an estimate for the instantaneous rate by taking values oftcloser and closer to 1. What special name is given to this instantaneous rate?

d. At approximately what instantaneous rate is the fox population changing att=4? Explain why the answer is negative.

15. Bacteria Culture Problem: Bacteria in a laboratory culture are multiplying in such a way that the surface area of the culture,a(t), in mm2_{, is given by}

a(t)₌200(1.2t)

wheret is the number of hours since the culture was started.

a. Find the average rate of increase of bacteria fromt₌2 tot₌2.1.

b. Write an equation forr(t), the average rate of change ofa(t), from 2 hours tot. Plot the graph ofr using a friendly window that includest₌2 as a grid point. What do you notice when you trace the graph ofr to

t₌2?

c. The instantaneous rate of change (the derivative) ofa(t) att₌2 is

52.508608...mm2_{/h. How close to this value} isr(2.01)? How close musttbe kept to 2 on the positive side so that the average rate is within 0.01 unit of this derivative?

16. Sphere Volume Problem: Recall from geometry that the volume of a sphere is

V(x)₌4 3π x

3

whereV(x) is volume in cubic centimeters and

xis the radius in centimeters.

a. FindV(6). Write the answer as a multiple ofπ. b. Find the average rate of change ofV(x) from

x=6 tox=6.1. Find the average rate from

x=5.9 tox=6. Use the answers to find an estimate of the instantaneous rate atx=6. c. Write an equation forr(x), the average rate of change ofV(x) from 6 tox. Plot the graph ofr using a friendly window that hasx=6 as a grid point. What do you notice when you trace the graph tox=6?

d. The derivative ofV(x) atx₌6 equals 4π62_, the surface area of a sphere of radius 6 cm. How close isr(6.1) to this derivative? How close to 6 on the positive side must the radius be kept forr(x) to be within 0.01 unit of this derivative?

17. Rolling Tire Problem: Apebble is stuck in the tread of a car tire (Figure 1-2k). As the wheel turns, the distance,y, in inches, between the pebble and the road at various times,t, in seconds, is given by the table below.

t (s) y (in.) 1.2 0.63 1.3 0.54 1.4 0.45 1.5 0.34 1.6 0.22 1.7 0.00 1.8 0.22 1.9 0.34 2.0 0.45 y Figure 1-2k

a. About how fast isychanging at each time? i.t₌1.4

ii.t=1.7 iii.t₌1.9

b. At what time does the stone strike the pavement? Justify your answer.

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18. Flat Tire Problem: Atire is punctured by a nail. As the air leaks out, the distance,y, in inches, between the rim and the pavement (Figure 1-2l) depends on the time,t, in minutes, since the tire was punctured. Values oftandy are given in the table below.

t (min) y (in.) 0 6.00 2 4.88 4 4.42 6 4.06 8 3.76 10 3.50 12 3.26 14 3.04 16 2.84 y Figure 1-2l

a. About how fast isychanging at each time? i.t=2 ii.t=8 iii.t=14

b. How do you interpret the sign of the rate at whichyis changing?

For Problems 19–28,

a. Give the type of function (linear, quadratic, and so on).

b. State whetherf(x) is increasing or

decreasing atx_{= c}, and how you know this. 19.f(x)_{= x}2_{+5x+6, c=3} 20.f(x)_{= −x}2_{+8x+5, c=1} 21.f(x)=3x, c=2 22.f(x)=2x, c= −3 23.f(x)₌ 1 x₋5, c=4 24.f(x)_{= −}1 x, c= −2 25.f(x)_{= −}3x₊7, c₌5 26.f(x)₌0.2x₋5, c₌8

27.f(x)₌sinx, c₌2 (Radian mode!)

28.f(x)=cosx, c=1 (Radian mode!)

29. Derivative Meaning Problem: What is the physical meaning of the derivative of a function? How can you estimate the derivative graphically? Numerically? How does the numerical computation of a derivative illustrate the meaning of limit?

30. Limit Meaning Problem: From memory, write the verbal meaning of limit. Compare it with the statement in the text. If you did not state all parts correctly, try writing it again until you get it completely correct. How do the results of Problems 13 and 14 of this problem set

illustrate the meaning of limit?

1-3

One Type of Integral of a Function

The title of this chapter is Limits, Derivatives, Integrals, and Integrals. In Section 1-2, you

30 60 70 100 v (t) (ft/s) t (s) Area = distance traveled Figure 1-3a estimated the derivative of a function, which is

the instantaneous rate of change ofy with respect tox. In this section you will learn about one type of integral, the definite integral.

Suppose you start driving your car. The velocity increases for a while, then levels off. Figure 1-3a shows the velocity,v(t), increasing from zero, then approaching and leveling off at 60 ft/s.

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In the 30 seconds between timet=70 andt=100, the velocity is a constant 60 ft/s. Because distance=rate×time, the distance you go in this time interval is

60 ft/s_×30 s₌1800 ft

Geometrically, 1800 is the area of the rectangle shown in Figure 1-3a. The width is 30 and the length is 60. Between 0 s and 30 s, where the velocity is changing, the area of the region under the graph also equals the distance traveled. Because the length varies, you cannot find the area simply by multiplying two numbers. The process of evaluating a product in which one factor varies is called finding a definite integral. You can evaluate definite integrals by finding the

corresponding area. In this section you will find the approximate area by counting squares on graph paper (by “brute force”!). Later, you will apply the concept of limit to calculate definite integrals exactly.

OBJECTIVE

definite integral of the function betweenx= aandx= bby counting squares.

If you are given only the equation, you can plot it with your grapher’s grid-on feature, estimating the number of squares in this way. However, it is more accurate to use a plot on graph paper to count squares. You can get plotting data by using your grapher’sTRACEorTABLEfeature.

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EXAMPLE 1

x₌1 tox₌7.

Solution You can get reasonable accuracy by plottingf(x) at each integer value ofx

(Figure 1-3b). x f (x) 0 f(0)=8 1 f(1)=5.6 2 f(2)≈3.9 3 f(3)≈2.7 4 f(4)≈1.9 5 f(5)_≈1.3 6 f(6)_≈0.9 7 f(7)_≈0.7 x 7 8 1 f (x) Figure 1-3b

The integral equals the area under the graph fromx₌1 tox₌7. “Under” the graph means “between the graph and thex-axis.” To find the area, first count the whole squares. Put a dot in each square as you count it to keep track, then estimate the area of each partial square to the nearest 0.1 unit. For instance, less than half a square is 0.1, 0.2, 0.3, or 0.4. You be the judge. You should get about 13.9 square units for the area, so the definite integral is approximately 13.9. Answers anywhere from 13.5 to 14.3 are reasonable.

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If the graph is already given, you need only count the squares. Be sure you know how much area each square represents! Example 2 shows you how to do this.

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EXAMPLE 2

v(t)= −100t2_{+90t+14, wheretis in seconds and}

v(t) is in feet per second. Estimate the definite integral ofv(t) with respect totfromt₌0.1 tot₌1. Solution Notice that each space in thet-direction is 0.1 s and

each space in the direction ofv(t) is 2 ft/s. Thus, each square represents (0.1)(2), or 0.2 ft. You should count about 119.2 squares for the area. So, the definite integral will be about

(119.2)(0.2)_≈23.8 ft

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The following box gives the meaning of definite integral. The precise definition is given in Chapter 5, where you will learn an algebraic technique for calculating exact values of definite integrals.

Meaning of Definite Integral

Meaning of Definite Integral

The definite integral of the functionf fromx= atox= bgives a way to find the product of (b− a) andf(x), even iff(x) is not a constant. See Figure 1-3d. (b – a) f (x) varies x b a Integral = area, representing f(x) . (b – a) f (x) Figure 1-3d

Problem Set 1-3

Quick Review 5min

6 ft 10 ft 14 ft 1 1 _x y 1 1 x y 1 1 _x y 1 1 x y

Figure 1-3e Figure 1-3f Figure 1-3g Figure 1-3h Figure 1-3i

Q1. Find the area of the trapezoid in Figure 1-3e.

Q2. Write the particular equation for the function graphed in Figure 1-3f.

Q3. Write the particular equation for the function graphed in Figure 1-3g.

Q4. Write the particular equation for the function graphed in Figure 1-3h.

Q5. Write the particular equation for the function graphed in Figure 1-3i.

Q6. Findf(5) iff(x)_{= x −}1.

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Q7. Sketch the graph of a linear function with positivey-intercept and negative slope.

Q8. Sketch the graph of a quadratic function opening downward.

Q9. Sketch the graph of a decreasing exponential function.

Q10. At what value(s) ofxisf(x)₌(x₋4)/(x₋3) undefined?

For Problems 1–4, estimate the definite integral by counting squares on a graph.

1. f(x)_{= −}0.1x2₊₇ a.x₌0 tox₌5 b.x= −1 tox=6 2.f(x)= −0.2x2+8 a.x₌0 tox₌3 b.x= −2 tox=5 3.h(x)₌sinx a.x=0 tox= π b.x₌0 tox_{= π/}2 4.g(x)₌2x₊₅ a.x=1 tox=2 b.x_{= −}1 tox₌1

5. In Figure 1-3j, a car is slowing down from velocityv=60 ft/s. Estimate the distance it travels from timet=5 s tot=25 s by finding the definite integral.

60

t (s) v (ft/s)

5 25

Figure 1-3j

6. In Figure 1-3k, a car slowly speeds up from

v₌55 mi/h during a long trip. Estimate the distance it travels from timet=0 h tot=4 h by finding the definite integral.

t (h) v (mi/h)

70

1 2 3 4

Figure 1-3k

For Problems 7 and 8, estimate the derivative of the function at the given value ofx.

7.f(x)=tanx, x=1 8.h(x)= −7x+100, x=5

9. Electric Car Problem: You have been hired by an automobile manufacturer to analyze the predicted motion of a new electric car they are building. When accelerated hard from a standing start, the velocity of the car,v(t), in ft/s, is expected to vary exponentially with time,t, in seconds, according to the equation

v(t)₌50(1₋0.9t)

a. Plot the graph of functionv in the domain [0, 10]. What is the corresponding range of the function?

b. Approximately how many seconds will it take the car to reach a velocity of 30 ft/s? c. Approximately how far will the car have

traveled when it reaches 30 ft/s? Which of the four concepts of calculus is used to find this distance?

d. At approximately what rate is the velocity changing whent=5? Which of the four concepts of calculus is used to find this rate? What is the physical meaning of the rate of change of velocity?

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10. Slide Problem: Phoebe sits atop the swimming pool slide (Figure 1-3l). At timet=0 s she pushes off. Calvin finds that her velocity,v(t), in ft/s, is given by

v(t)₌10 sin 0.3t

Figure 1-3l

Phoebe splashes into the water at timet=4 s. a. Plot the graph of functionv. Use radian mode. b. How fast was Phoebe going when she hit the water? What, then, are the domain and range of the velocity function?

c. Find, approximately, the definite integral of the velocity function fromt=0 tot=4. What are the units of the integral? What real-world quantity does this integral give you?

d. What, approximately, was the derivative of the velocity function whent₌3? What are

the units of the derivative? What is the physical meaning of the derivative in this case?

11. Negative Velocity Problem: Velocity differs from speed in that it can be negative. If the velocity of a moving object is negative, then its distance from its starting point is decreasing as time increases. The graph in Figure 1-3m shows

v(t), in cm/s, as a function oft, in seconds, after its motion started. How far is the object from its starting point whent₌9?

1 9 5 –5 t (s) v (t) (cm/s) Figure 1-3m

12. Write the meaning of derivative. 13. Write the meaning of definite integral. 14. Write the verbal definition of limit.

1-4

Definite Integrals by Trapezoids,

fromEquations and Data

In Section 1-3, you learned that the definite integral of a function is the product ofx- andy-values, where they-values may be different for various values ofx. Because the integral is represented by the area of a region under the graph, you were able to estimate it by counting squares. In this section you will learn a more efficient way of estimating definite integrals.

0.1 0.4 0.7 1 10 20 30 22 34 28 4 x f (x) 0.3 Figure 1-4a

Figure 1-4a shows the graph of

f(x)= −100x2+90x+14

which is the function in Example 2 of Section 1-3 usingf(x) instead ofv(t). Instead of counting squares to find the area of the region under the graph, the region is divided into vertical strips. Line segments connect the points where the strip boundaries meet the graph. The result is a set of trapezoids whose areas add up to a number approximately equal to the area of the region.