Preview 5 steps to a 5 ap calculus ab

Page 1

5

TM

5 STEPS TO A

AP Calculus AB 2014--2015 William Ma

Preview - tusachduhoc.com

CONTENTS About the Author xi Preface xii Acknowledgments xiii Introduction: The Five-Step Program xiv STEP 1

Set Up Your Study Plan 1

2

STEP 2

Determine Your Test Readiness 3

STEP 3

What You Need to Know About the AP Calculus AB Exam 3 1.1 What Is Covered on the AP Calculus Exam? 4 1.2 What Is the Format of the AP Calculus AB Exam? 4 1.3 What Are the Advanced Placement Exam Grades? 5 How Is the AP Calculus AB Exam Grade Calculated? 5 1.4 Which Graphing Calculators Are Allowed for the Exam? 6 Calculators and Other Devices Not Allowed for the AP Calculus AB Exam 7 Other Restrictions on Calculators 7 How to Plan Your Time 8 2.1 Three Approaches to Preparing for the AP Calculus AB Exam 8 Overview of the Three Plans 8 2.2 Calendar for Each Plan 10 Summary of the Three Study Plans 13

Take a Diagnostic Exam 17 3.1 Getting Started! 20 3.2 Diagnostic Test 20 3.3 Answers to Diagnostic Test 26 3.4 Solutions to Diagnostic Test 27 3.5 Calculate Your Score 35 Short-Answer Questions 35 AP Calculus AB Diagnostic Exam 35

Develop Strategies for Success 4

How to Approach Each Question Type 39 4.1 The Multiple-Choice Questions 40 4.2 The Free-Response Questions 40 4.3 Using a Graphing Calculator 41 4.4 Taking the Exam 42 What Do I Need to Bring to the Exam? 42 Tips for Taking the Exam 43

v

Preview - tusachduhoc.com

vi

Contents

STEP 4

Review the Knowledge You Need to Score High 5

6

Review of Precalculus 47 5.1 Lines 48 Slope of a Line 48 Equations of a Line 48 Parallel and Perpendicular Lines 49 5.2 Absolute Values and Inequalities 52 Absolute Values 52 Inequalities and the Real Number Line 53 Solving Absolute Value Inequalities 54 Solving Polynomial Inequalities 55 Solving Rational Inequalities 57 5.3 Functions 59 Definition of a Function 59 Operations on Functions 60 Inverse Functions 62 Trigonometric and Inverse Trigonometric Functions 65 Exponential and Logarithmic Functions 68 5.4 Graphs of Functions 72 Increasing and Decreasing Functions 72 Intercepts and Zeros 74 Odd and Even Functions 75 Shifting, Reflecting, and Stretching Graphs 77 5.5 Rapid Review 80 5.6 Practice Problems 81 5.7 Cumulative Review Problems 82 5.8 Solutions to Practice Problems 82 5.9 Solutions to Cumulative Review Problems 85 Limits and Continuity 86 6.1 The Limit of a Function 87 Definition and Properties of Limits 87 Evaluating Limits 87 One-Sided Limits 89 Squeeze Theorem 92 6.2 Limits Involving Infinities 94 Infinite Limits (as x → a) 94 Limits at Infinity (as x → ±∞) 96 Horizontal and Vertical Asymptotes 98 6.3 Continuity of a Function 101 Continuity of a Function at a Number 101 Continuity of a Function over an Interval 101 Theorems on Continuity 101

Preview - tusachduhoc.com

Contents

7

8

6.4 Rapid Review 104 6.5 Practice Problems 105 6.6 Cumulative Review Problems 106 6.7 Solutions to Practice Problems 107 6.8 Solutions to Cumulative Review Problems 109 Differentiation 111 7.1 Derivatives of Algebraic Functions 112 Definition of the Derivative of a Function 112 Power Rule 115 The Sum, Difference, Product, and Quotient Rules 116 The Chain Rule 117 7.2 Derivatives of Trigonometric, Inverse Trigonometric, Exponential, and Logarithmic Functions 118 Derivatives of Trigonometric Functions 118 Derivatives of Inverse Trigonometric Functions 120 Derivatives of Exponential and Logarithmic Functions 121 7.3 Implicit Differentiation 123 Procedure for Implicit Differentiation 123 7.4 Approximating a Derivative 126 7.5 Derivatives of Inverse Functions 128 7.6 Higher Order Derivatives 130 7.7 Rapid Review 131 7.8 Practice Problems 132 7.9 Cumulative Review Problems 132 7.10 Solutions to Practice Problems 133 7.11 Solutions to Cumulative Review Problems 136 Graphs of Functions and Derivatives 138 8.1 Rolle’s Theorem, Mean Value Theorem, and Extreme Value Theorem 138 Rolle’s Theorem 139 Mean Value Theorem 139 Extreme Value Theorem 142 8.2 Determining the Behavior of Functions 143 Test for Increasing and Decreasing Functions 143 First Derivative Test and Second Derivative Test for Relative Extrema 146 Test for Concavity and Points of Inflection 149 8.3 Sketching the Graphs of Functions 155 Graphing without Calculators 155 Graphing with Calculators 156 8.4 Graphs of Derivatives 158 8.5 Rapid Review 163 8.6 Practice Problems 165 8.7 Cumulative Review Problems 168

vii

Preview - tusachduhoc.com

viii

Contents

8.8 Solutions to Practice Problems 168 8.9 Solutions to Cumulative Review Problems 175 9 Applications of Derivatives 177 9.1 Related Rate 177 General Procedure for Solving Related Rate Problems 177 Common Related Rate Problems 178 Inverted Cone (Water Tank) Problem 179 Shadow Problem 180 Angle of Elevation Problem 181 9.2 Applied Maximum and Minimum Problems 183 General Procedure for Solving Applied Maximum and Minimum Problems 183 Distance Problem 183 Area and Volume Problems 184 Business Problems 187 9.3 Rapid Review 188 9.4 Practice Problems 189 9.5 Cumulative Review Problems 191 9.6 Solutions to Practice Problems 192 9.7 Solutions to Cumulative Review Problems 199 10 More Applications of Derivatives 202 10.1 Tangent and Normal Lines 202 Tangent Lines 202 Normal Lines 208 10.2 Linear Approximations 211 Tangent Line Approximation (or Linear Approximation) 211 Estimating the nth Root of a Number 213 Estimating the Value of a Trigonometric Function of an Angle 213 10.3 Motion Along a Line 214 Instantaneous Velocity and Acceleration 214 Vertical Motion 216 Horizontal Motion 216 10.4 Rapid Review 218 10.5 Practice Problems 219 10.6 Cumulative Review Problems 220 10.7 Solutions to Practice Problems 221 10.8 Solutions to Cumulative Review Problems 225 11 Integration 227 11.1 Evaluating Basic Integrals 228 Antiderivatives and Integration Formulas 228 Evaluating Integrals 230

Preview - tusachduhoc.com

Contents

11.2

12

13

Integration by U-Substitution 233 The U-Substitution Method 233 U-Substitution and Algebraic Functions 233 U-Substitution and Trigonometric Functions 235 U-Substitution and Inverse Trigonometric Functions 236 U-Substitution and Logarithmic and Exponential Functions 238 11.3 Rapid Review 241 11.4 Practice Problems 242 11.5 Cumulative Review Problems 243 11.6 Solutions to Practice Problems 244 11.7 Solutions to Cumulative Review Problems 246 Definite Integrals 247 12.1 Riemann Sums and Definite Integrals 248 Sigma Notation or Summation Notation 248 Definition of a Riemann Sum 249 Definition of a Definite Integral 250 Properties of Definite Integrals 251 12.2 Fundamental Theorems of Calculus 253 First Fundamental Theorem of Calculus 253 Second Fundamental Theorem of Calculus 254 12.3 Evaluating Definite Integrals 257 Definite Integrals Involving Algebraic Functions 257 Definite Integrals Involving Absolute Value 258 Definite Integrals Involving Trigonometric, Logarithmic, and Exponential Functions 259 Definite Integrals Involving Odd and Even Functions 261 12.4 Rapid Review 262 12.5 Practice Problems 263 12.6 Cumulative Review Problems 264 12.7 Solutions to Practice Problems 265 12.8 Solutions to Cumulative Review Problems 268 Areas and Volumes 270 x 13.1 The Function F (x) = a f (t )dt 271 13.2 Approximating the Area Under a Curve 275 Rectangular Approximations 275 Trapezoidal Approximations 279 13.3 Area and Definite Integrals 280 Area Under a Curve 280 Area Between Two Curves 285

ix

Preview - tusachduhoc.com

x

Contents

13.4

14

STEP 5

Volumes and Definite Integrals 289 Solids with Known Cross Sections 289 The Disc Method 293 The Washer Method 298 13.5 Rapid Review 301 13.6 Practice Problems 303 13.7 Cumulative Review Problems 305 13.8 Solutions to Practice Problems 305 13.9 Solutions to Cumulative Review Problems 312 More Applications of Definite Integrals 315 14.1 Average Value of a Function 316 Mean Value Theorem for Integrals 316 Average Value of a Function on [a, b] 317 14.2 Distance Traveled Problems 319 14.3 Definite Integral as Accumulated Change 322 Business Problems 322 Temperature Problem 323 Leakage Problems 324 Growth Problem 324 14.4 Differential Equations 325 Exponential Growth/Decay Problems 325 Separable Differential Equations 327 14.5 Slope Fields 330 14.6 Rapid Review 334 14.7 Practice Problems 335 14.8 Cumulative Review Problems 337 14.9 Solutions to Practice Problems 338 14.10 Solutions to Cumulative Review Problems 342

Build Your Test-Taking Confidence AP Calculus AB Practice Exam 1 347 AP Calculus AB Practice Exam 2 373 AP Calculus AB Practice Exam 3 401 Appendix 427 Bibliography and Websites 431

Preview - tusachduhoc.com

ABOUT THE AUTHOR WILLIAM MA has taught calculus for many years. He received his B.A. and M.A. from Columbia University. He was the chairman of the Math Department at the Herricks School District on Long Island, New York, for many years before retiring. He also taught as adjunct instructor at Baruch

College, Fordham University, and Columbia University. He is the author of several books, including test preparation books for the SAT and ACT, and an online review course for New York State’s Math A Regents Exam. He is currently a math consultant.

xi

PREFACE

Preview - tusachduhoc.com

Congratulations! You are an AP Calculus student. Not too shabby! As you know, AP Calculus is one of the most challenging subjects in high school. You are studying mathematical ideas that helped change the world. Not that long ago, calculus was taught at the graduate level. Today, smart young people like yourself study calculus in high school. Most colleges will give you credit if you score a 3 or more on the AP Calculus Exam. So how do you do well on the AP Calculus Exam? How do you get a 5? Well, you’ve already taken the first step. You’re reading this book. The next thing you need to do is to make sure that you understand the materials and do the practice problems. In recent years, the AP Calculus exam has gone through many changes. For example, today the questions no longer stress long and tedious algebraic manipulations. Instead, you are expected to be able to solve a broad range of problems, including problems presented to you in the form of a graph, a chart, or a word problem. For many of the questions, you are also expected to use your calculator to find the solutions. After having taught AP Calculus for many years and having spoken to students and other calculus teachers, we understand some of the difficulties that students might encounter with the AP Calculus exam. For example, some students have complained about not being able to visualize what the question was asking and other students said that even when the solution was given, they could not follow the steps. Under these circumstances, who wouldn’t be frustrated? In this book, we have addressed these issues. Whenever possible, problems are accompanied by diagrams and solutions are presented in a step-by-step manner. The graphing calculator is used extensively whenever it is permitted. To make things even easier, this book begins with a chapter that reviews precalculus. So, if you need to look up a formula, definition, or concept in precalculus, it is right here in the book. If you’re familiar with these concepts, you might skip this chapter and begin with Chapter 6. So how do you get a 5 on the AP Calculus Exam? Step 1: Set up your study program by selecting one of the three study plans in Chapter 2 of this book. Step 2: Determine your test readiness by taking the Diagnostic Exam in Chapter 3. Step 3: Develop strategies for success by learning the test-taking techniques offered in Chapter 4. Step 4: Review the knowledge you need to score high by studying the subject materials in Chapter 5 through Chapter 14. Step 5: Build your test-taking confidence by taking the Practice Exams provided in this book. As an old martial artist once said, “First you must understand. Then you must practice.” Have fun and good luck!

xii

Preview - tusachduhoc.com

ACKNOWLEDGMENTS I could not have written this book without the help of the following people: My high school calculus teacher, Michael Cantor, who taught me calculus. Professor Leslie Beebe, who taught me how to write. David Pickman, who fixed my computer and taught me Equation Editor. Jennifer Tobin, who tirelessly edited many parts of the manuscript and with whom I look forward to coauthoring a math book in the future. Robert Teseo and his calculus students who field-tested many of the problems. Allison Litvack, Rich Peck, and Liz Spiegel, who proofread sections of the Practice Tests. And a special thanks to Trisha Ho, who edited some of the materials. Mark Reynolds, who proofread part of the manuscript. Maxine Lifsfitz, who offered many helpful comments and suggestions. Grace Freedson, the series editor, Vastavikta Sharma, project manager, and Bev Weiler, the tech editor, for all their assistance. Sam Lee and Derek Ma, who were on 24-hour call for technical support. My older daughter, Janet, for not killing me for missing one of her concerts. My younger daughter, Karen, who helped me with many of the computer graphics. My wife, Mary, who gave me many ideas for the book and who often has more confidence in me than I have in myself.

xiii

INTRODUCTION: THE FIVE-STEP PROGRAM How Is This Book Organized?

Preview - tusachduhoc.com

This book begins with an introduction to the Five-Step Program followed by 14 chapters reflecting the five steps. •

• •

•

•

Step 1 provides an overview of the AP Calculus AB Exam and offers three study plans for preparing for this exam. Step 2 contains a diagnostic test with answers and explanations. Step 3 offers test-taking strategies for answering both multiple-choice and free-response questions, and for using a graphing calculator. Step 4 consists of 10 chapters providing a comprehensive review of all topics covered on the AP Calculus AB Exam. At the end of each chapter (beginning with Chapter 5), you will find a set of practice problems with solutions, a set of cumulative review problems with solutions, and a Rapid Review section giving you the highlights of the chapter. Step 5 provides three full practice AP Calculus Exams with answers, explanations, and worksheets to compute your score.

The book concludes with a summary of math formulas and theorems related to the AP Calculus Exams. (Please note that the exercises in this book are done with the TI-89 Graphing Calculator.)

Introducing the Five-Step Preparation Program This book is organized as a five-step program to prepare you to succeed in the AP Calculus AB Exam. These steps are designed to provide you with vital skills, strategies, and the practice that can lead you to that perfect 5. Here are the five steps.

Step 1: Set Up Your Study Program In this step you will read an overview of the AP Calculus AB Exam, including a summary of topics covered in the exam and a description of the format of the exam. You will also follow a process to help determine which of the following preparation programs is right for you: • • •

Full school year: September through May. One semester: January through May. Six weeks: Basic training for the exam.

Step 2: Determine Your Test Readiness In this step you will take a diagnostic multiple-choice exam in calculus. This pre-test should give you an idea of how prepared you are to take the real exam before beginning to study for the actual AP Calculus AB Exam.

xiv

Preview - tusachduhoc.com

Introduction: The 5-Step Program

xv

Step 3: Develop Strategies for Success In this step you will learn strategies that will help you do your best on the exam. These strategies cover both the multiple-choice and free-response sections of the exam. • • •

Learn to read multiple-choice questions. Lean how to answer multiple-choice questions. Learn how to plan and write answers to the free-response questions.

Step 4: Review the Knowledge You Need to Score High In this step you will learn or review the material you need to know for the test. This review section takes up the bulk of this book. It contains: • • • •

A comprehensive review of AP Calculus AB. A set of practice problems. A set of cumulative review problems beginning with Chapter 5. A rapid review summarizing the highlights of the chapter.

Step 5: Build Your Test-Taking Confidence In this step you will complete your preparation by testing yourself on practice exams. We have provided you with three complete AP Calculus AB exams, along with scoring guides for all of them. Although these practice exams are not reproduced questions from the actual AP calculus exams, they mirror both the material tested by AP and the way in which it is tested. Finally, at the back of this book you will find additional resources to aid your preparation. These include: • • •

A brief bibliography. A list of websites related to the AP Calculus exam. A summary of formulas and theorems related to the AP Calculus exam.

Introduction to the Graphics Used in this Book To emphasize particular skills and strategies, we use several icons throughout this book. An icon in the margin will alert you that you should pay particular attention to the accompanying text. We use these icons: This icon points out a very important concept or fact that you should not pass over.

This icon calls your attention to a strategy that you may want to try.

This icon indicates a tip that you might find useful.

Preview - tusachduhoc.com

This page intentionally left blank

Preview - tusachduhoc.com

STEP

1

Set Up Your Study Plan CHAPTER

CHAPTER

What You Need to Know About the AP Calculus AB Exam 2 How to Plan Your Time 1

Preview - tusachduhoc.com

20

STEP 2. Determine Your Test Readiness

3.1 Getting Started! Taking the Diagnostic Test helps you assess your strengths and weaknesses as you begin preparing for the AP Calculus AB exam. The questions in the Diagnostic Test contain both multiple-choice and open-ended questions. They are arranged by topic and designed to review concepts tested on the AP Calculus AB exam. All questions in the diagnostic test should be done without the use of a graphing calculator, except in a few cases where you need to find the numerical value of a logarithmic or exponential function.

3.2 Diagnostic Test Chapter 5

8. If

1. Write an equation of a line passing through the origin and perpendicular to the line 5x − 2y = 10. 2. Solve the inequality 3|2x − 4| > 6 graphically. Write the solution in interval notation. 3. Given f (x ) = x 2 + 1, evaluate

f (3 + h) − f (3) . h

4. Solve for x : 4 ln x − 2 = 6.

√

h(x ) =

if x > 4 if x ≤ 4

x x − 12 2

find lim h(x ). x →4

9. If f (x ) = |2x e x |, what is the value of lim+ f (x )? x →0

Chapter 7

π 10. If f (x ) = −2 csc (5x ), find f . 6

5. Given f (x ) = x 3 − 8 andg (x ) = 2x − 2, find f (g (2)).

Chapter 6 6. A function f is continuous on [−2, 0] and some of the values of f are shown below.

11. Given the equation y = (x + 1)(x − 3)2 , what is the instantaneous rate of change of y at x = −1? π π + Δx − tan tan 4 4 ? 12. What is lim Δx Δx →0

Chapter 8

x

−2

−1

0

f

4

b

4

13. The graph of f is shown in Figure D-1. Draw a possible graph of f on (a , b). y

If f (x ) = 2 has no solution on [−2, 0], then b could be

f

(A) 3 (B) 2 a

(C) 1

c

0

(D) 0 (E) −2

√

7. Evaluate lim

x→ − ∞

x2 − 4 . 2x

Figure D-1

d

e

f

b

x

Preview - tusachduhoc.com

21

Take a Diagnostic Exam

14. The graph of the function g is shown in Figure D-2. Which of the following is true for g on (a , b)?

16. The graph of f , the derivative of f , is shown in Figure D-4. At what value(s) of x is the graph of f concave up?

I. g is monotonic on (a , b).

y

II. g is continuous on (a , b).

f′

III. g >0 on (a , b).

y

x

x1 0

x2

x3

x4

g

x 0

a

b

Figure D-4

Figure D-2 15. The graph of f is shown in Figure D-3 and f is twice differentiable, which of the following statements is true?

17. How many points of inflection does the graph of y = sin(x 2 ) have on the interval [−π, π ]? x 18. If g (x ) = f (t)d t and the graph of f is a

shown in Figure D-5, which of the graphs in Figure D-6 on the next page is a possible graph of g ?

y y

f

f (t) t x 0

a

0

b

10

Figure D-3 (A) f (10) < f (10) < f (10) (B) f (10) < f (10) < f (10) (C) f (10) < f (10) < f (10) (D) f (10) < f (10) < f (10) (E) f (10) < f (10) < f (10)

Figure D-5

19. The graphs of f , g , p , and q are shown in Figure D-7 on the next page. Which of the functions f, g, p, or q have a point of inflection on (a , b)?

Preview - tusachduhoc.com

27

Take a Diagnostic Exam

3.4 Solutions to Diagnostic Test Chapter 5

Chapter 6

1. Write 5x − 2y = 10 in slope-intercept form 5 y = mx + b and obtain y = x − 5. The slope 2 5 5 of the line y = x − 5 is . Therefore the 2 2 5 slope of the line perpendicular to y = x − 5 is 2 2 − . Since the line perpendicular to 5 5 y = x − 5 also passes through the origin, its 2 2 y -intercept is 0 and its equation is y = − x . 5

6. See Figure DS-2. If b = 2, then x = −1 would be a solution for f (x ) = 2. If b = 1, 0, or −2, f (x ) = 2 would have two solutions. Thus, b = 3, choice (A).

y (−2, 4)

4

(0, 4)

3

2. Using your calculator, set y 1 = 3 |2x − 4| and y 2 = 6. Examine Figure DS-1 and note that 3 |2x − 4| > 6 when x < 1 or x > 3. Thus the solution in interval notation is (− ∞, 1) ∪ (3, ∞).

y=2

2 1

x −2

−1

0 −1 −2

Figure DS-2

√ x2 − 4 − x2 √ 7. lim x→ − ∞ 2x − x 2 √ (Note: as x → − ∞, x = − x 2 .)

− (x 2 − 4) x 2 = lim x→ − ∞ 2 √ − 1 − (4/x 2 ) = lim x→ − ∞ 2 √ 1 1 =− =− 2 2 √ x if x > 4 8. h (x ) = 2 x − 12 if x ≤ 4 √ √ lim+ h(x ) = lim+ x = 4 = 2 √

[−3, 7] by [−3, 10]

Figure DS-1 3. Since f (x ) = x 2 + 1, f (3) = 10 and f (3 + h) = (3 + h)2 + 1 = 9 + 6h + h 2 + 1 = 10 + 6h + h 2 . f (3 + h) − f (3) The quotient h 10 + 6h + h 2 − 10 6h + h 2 = = = 6 + h. h h 4. Simplify the equation 4 ln x − 2 = 6 and obtain ln x = 2. (Note that e x and ln x are inverse functions.) Thus, e ln x = e 2 or x = e 2 . 5. Since g (x ) = 2x − 2, g (2) = 2(2) − 2 = 2. Therefore f (g (2)) = f (2). Also f (x ) = x 3 − 8. Thus f (2) = 23 − 8 = 0.

x2 − 4 = lim x→ − ∞ 2x

x →4

√

x →4

lim− h(x ) = lim− (x 2 − 12) = (42 − 12) = 4

x →4

x →4

Since lim+ h(x ) = lim− h(x ) , thus lim h(x ) x →4

does not exist.

x →4

x →4

(3) f (10) > 0, since f is concave upward. Thus, f (10) < f (10) < f (10), choice (C).

2x e x if x ≼ 0 −2x e x if x < 0 If x ≼ 0, f (x ) = 2e x + e x (2x ) = 2e x + 2x e x lim+ f (x ) = lim+ (2e x + 2x e x ) =

9. f (x ) = |2x e | = x

x →0

2e 0 + 0 = 2

Based on the graph of f: f

x →0

decr.

incr.

[ a

Chapter 7 10.

incr.

f′

f (x ) = −2 csc (5x ) f (x ) = −2(−csc 5x ) [cot (5x )] (5) = 10 csc (5x ) cot (5x )

Ď€ 5Ď€ 5Ď€ = 10 csc cot f 6 6 6 √ √ = 10(2)(− 3) = −20 3

0

f

e

−

0

+

b

0

+

Concave upward

Concave downward

[

Preview - tusachduhoc.com

STEP 2. Determine Your Test Readiness

[

28

[ a

11. y = (x + 1)(x − 3)2 ; dy = (1)(x − 3)2 + 2(x − 3)(x + 1) dx

d

b

f″

−

+

f′

decr.

incr.

A possible graph of f ′

= (x − 3)2 + 2(x − 3)(x + 1) d y = (−1 − 3)2 + 2(−1 − 3)(−1 + 1) d x x = −1

y

= (−4)2 + 0 = 16 f (x 1 + Δx ) − f (x 1 ) Δx Δx →0 Ď€ Ď€ tan + Δx − tan 4 4 Thus, lim Δ x Δx →0 Ď€ d (tan x ) at x = = dx 4 Ď€ √ = ( 2)2 = 2 = sec2 4

12. f (x 1 ) = lim

a

f′

I. Since the graph of g is decreasing and then increasing, it is not monotonic.

b

incr.

decr.

x2

III. Since the graph of g is concave upward, g > 0. Thus, only statements II and III are true. 15. The graph indicates that (1) f (10) = 0, (2) f (10) < 0, since f is decreasing; and

e

16. See Figure DS-4. The graph of f is concave upward for x < x2.

13. See Figure DS-3.

II. Since the graph of g is a smooth curve, g is continuous.

d

Figure DS-3

Chapter 8 14.

0

f″

+

−

f

Concave upward

Concave downward

Figure DS-4 17. See Figure DS-5. Enter y 1 = sin(x 2 ). Using the [Inflection]

x

Preview - tusachduhoc.com

CHAPTER

4

How to Approach Each Question Type IN THIS CHAPTER Summary: Knowing and applying question-answering strategies helps you succeed on tests. This chapter provides you with many test-taking tips to help you earn a 5 on the AP Calculus Exams. Key Ideas ✪ Read each question carefully. ✪ Do not linger on a question. Time yourself accordingly. ✪ For multiple-choice questions, sometimes it is easier to work backward by trying ✪ ✪ ✪ ✪ ✪ ✪

each of the given choices. You will able to eliminate some of the choices quickly. For free-response questions, always show sufficient work so that your line of reasoning is clear. Write legibly. Always use calculus notations instead of calculator syntax. If the question involves decimals, round your final answer to 3 decimal places unless the question indicates otherwise. Trust your instincts. Your first approach to solving a problem is usually the correct one. Get a good night’s sleep the night before.

39

Preview - tusachduhoc.com

40

STEP 3. Develop Strategies for Success

4.1 The Multiple-Choice Questions •

•

•

•

• •

•

•

•

There are 45 multiple-choice questions for the AP Calculus AB exam. These questions are divided into Section 1–Part A, which consists of 28 questions for which the use of a calculator is not permitted; and Section 1–Part B with 17 questions, for which the use of a graphing calculator is allowed. The multiple-choice questions account for 50% of the grade for the whole test. Do the easy questions first because all multiple-choice questions are worth the same amount of credit. You have 55 minutes for the 28 questions in Section I–Part A and 50 minutes for the 17 questions in Section I–Part B. Do not linger on any one question. Time yourself accordingly. There is no partial credit for multiple-choice questions, and you do not need to show work to receive credit for the correct answer. Read the question carefully. If there is a graph or a chart, look at it carefully. For example, be sure to know if the given graph is that of f (x ) or f (x ). Pay attention to the scale of the x and y axes, and the unit of measurement. Never leave a question blank since there is no penalty for incorrect answers. If a question involves finding the derivative of a function, you must first find the derivative, and then see if you need to do additional work to get the final answer to the question. For example, if a question asks for an equation of the tangent line to a curve at a given point, you must first find the derivative, evaluate it at the given point (which gives you the slope of the line), and then proceed to find an equation of the tangent line. For some questions, finding the derivative of a given function (or sometimes, the antiderivative), is only the first step to solving the problem. It is not the final answer to the question. You might need to do more work to get the final answer. Sometimes, it is easier to work backward by trying each of the given choices as the final answer. Often, you will able to eliminate some of the given choices quickly. If a question involves decimal numbers, do not round until the final answer, and at that point, the final answer is usually rounded to 3 decimal places. Look at the number of decimal places of the answers in the given choices. Trust your instincts. Usually your first approach to solving a problem is the correct one.

4.2 The Free-Response Questions •

•

•

There are 6 free-response questions in Section 2: Part A consisting of 2 questions which allow the use of a calculator, and Part B with 4 questions which do not permit the use of a calculator. The 6 free-response questions account for 50% of the grade for the whole test. Read, Read, Read. Read the question carefully. Know what information is given, what quantity is being sought, and what additional information you need to find in order to answer the question. Always show a sufficient amount of work so that your line of reasoning is clear. This is particularly important in determining partial credit. In general, use complete sentences to explain your reasoning. Include all graphs, charts, relevant procedures, and theorems. Clearly indicate all the important steps that you have taken in solving the problem. A correct answer with insufficient work will receive minimal credit.

Preview - tusachduhoc.com

143

Graphs of Functions and Derivatives

8.2 Determining the Behavior of Functions Main Concepts: Test for Increasing and Decreasing Functions, First Derivative Test and Second Derivative Test for Relative Extrema, Test for Concavity and Points of Inflection

Test for Increasing and Decreasing Functions Let f be a continuous function on the closed interval [a , b] and differentiable on the open interval (a , b). 1. If f (x ) > 0 on (a , b), then f is increasing on [a , b]. 2. If f (x ) < 0 on (a , b), then f is decreasing on [a , b]. 3. If f (x ) = 0 on (a , b), then f is constant on [a , b]. Definition: Let f be a function defined at a number c . Then c is a critical number of f if either f (c ) = 0 or f (c ) does not exist. (See Figure 8.2-1.) y f(x)

f′ < 0 f decreasing

f′

0

f′ < 0 f decreasing

f′ > 0 f increasing 0

f′ = 0 constant x

f′ = 0

Figure 8.2-1

Example 1 Find the critical numbers of f (x ) = 4x 3 + 2x 2 . To find the critical numbers of f (x ), you have to determine where f (x ) = 0 and where f (x ) does not exist. Note f (x ) = 12x 2 + 4x , and f (x ) is defined for all real numbers. Let f (x ) = 0 and thus 12x 2 + 4x = 0, which implies 4x (3x + 1) = 0 ⇒ x = −1/3 or x = 0. Therefore, the critical numbers of f are 0 and −1/3. (See Figure 8.2-2.)

[–1, 1] by [–1, 1]

Figure 8.2-2

Preview - tusachduhoc.com

144

STEP 4. Review the Knowledge You Need to Score High

Example 2 Find the critical numbers of f (x ) = (x − 3)2/5 . 2 2 f (x ) = (x − 3)− 3/5 = . Note that f (x ) is undefined at x = 3 and that 5 5(x − 3)3/5 f (x ) = 0. Therefore, 3 is the only critical number of f . (See Figure 8.2-3.)

[–3, 8] by [–4, 4]

Figure 8.2-3

Example 3 The graph of f on (1, 6) is shown in Figure 8.2-4. Find the intervals on which f is increasing or decreasing. y

3

f′

x 0

1

2

3

4

5

6

Figure 8.2-4 Solution: (See Figure 8.2-5.) f′

–

–

+

x 2

1 f

decr.

5 incr.

6 decr.

Figure 8.2-5 Thus, f is decreasing on [1, 2] and [5, 6] and increasing on [2, 5].

Preview - tusachduhoc.com

228

STEP 4. Review the Knowledge You Need to Score High

11.1 Evaluating Basic Integrals Main Concepts: Antiderivatives and Integration Formulas, Evaluating Integrals •

Answer all parts of a question from Section II even if you think your answer to an earlier part of the question might not be correct. Also, if you do not know the answer to part one of a question, and you need it to answer part two, just make it up and continue.

Antiderivatives and Integration Formulas Definition: A function F is an antiderivative of another function f if F (x ) = f (x ) for all x in some open interval. Any two antiderivatives of f differ by an additive constant C . We denote the set of antiderivatives of f by f (x )d x , called the indefinite integral of f .

Integration Rules: 1. f (x )d x = F (x ) + C ⇔ F (x ) = f (x ) a f (x )d x = a f (x )d x 2. − f (x )d x = − f (x )d x 3. f (x )d x ± g (x )d x 4. [ f (x ) ± g (x )] d x = Differentiation Formulas: 1. 2. 3. 4. 5. 6. 7. 8. 9.

d (x ) = 1 dx d (a x ) = a dx d n (x ) = nx n − 1 dx d (cos x ) = − sin x dx d (sin x ) = cos x dx d (tan x ) = sec2 x dx d (cot x ) = − csc2 x dx d (sec x ) = sec x tan x dx d (csc x ) = − csc x (cot x ) dx

Integration Formulas: 1. 1d x = x + C 2. a d x = a x + C xn+1 3. x n d x = + C, n = − 1 n+1 4. sin x d x = − cos x + C 5. cos x d x = sin x + C 6. sec2 x d x = tan x + C 7. csc2 x d x = − cot x + C 8. sec x (tan x ) d x = sec x + C 9. csc x (cot x ) d x = − csc x + C

Preview - tusachduhoc.com

Integration

Differentiation Formulas (cont.): 10. 11. 12. 13. 14. 15.

1 d (ln x ) = dx x d x (e ) = e x dx d x (a ) = (ln a )a x dx 1 d (sin− 1 x ) = √ dx 1 − x2 d 1 (tan− 1 x ) = dx 1 + x2 1 d √ (sec− 1 x ) = dx |x | x 2 − 1

229

Integration Formulas (cont.): 1 10. d x = ln |x | + C x 11. e x d x = e x + C ax 12. a x d x = + C a > 0, a = 1 ln a 1 √ 13. d x = sin− 1 x + C 2 1−x 1 14. d x = tan− 1 x + C 1 + x2 1 √ 15. d x = sec− 1 x + C |x | x 2 − 1

More Integration Formulas: 16. tan x d x = ln |sec x | + C or − ln |cos x | + C 17. cot x d x = ln |sin x | + C or − ln |csc x | + C 18. sec x d x = ln |sec x + tan x | + C 19. csc x d x = ln |csc x − cot x | + C 20. ln x d x = x ln |x | − x + C x 1 √ d x = sin− 1 +C 21. a a2 − x2 x 1 1 −1 22. tan d x = +C a2 + x2 a a 1 − 1 x 1 1 −1 a √ 23. d x = sec + C or cos + C a a a x x x2 − a2 x sin(2x ) 1 − cos 2x 24. sin2 x d x = − + C. Note: sin2 x = 2 4 2 Note: After evaluating an integral, always check the result by taking the derivative of the answer (i.e., taking the derivative of the antiderivative). •

1 Remember that the volume of a right-circular cone is v = πr 2 h where r is the radius 3 of the base and h is the height of the cone.

Preview - tusachduhoc.com

238

STEP 4. Review the Knowledge You Need to Score High

U-Substitution and Logarithmic and Exponential Functions Example 1 Evaluate

x3 dx. x4 − 1

Step 1: Let u = x 4 − 1. du Step 2: Differentiate: d u = 4x 3 d x ⇒ = x 3d x . 4 1 du 1 1 Step 3: Rewrite: = d u. u 4 4 u 1 Step 4: Integrate: ln |u| + C . 4 1 Step 5: Replace u: ln x 4 − 1 + C . 4

x3 1 1 3 4x = . Step 6: Differentiate and Check: 4 x4 − 1 x4 − 1

Example 2 sin x Evaluate dx. cos x + 1 Step 1: Let u = cos x + 1. Step 2: Differentiate: d u = − sin x d x ⇒ − d u = sin x d x . − du du Step 3: Rewrite: =− . u u Step 4: Integrate: − ln |u| + C . Step 5: Replace u: − ln |cos x + 1| + C .

sin x 1 . Step 6: Differentiate and Check: − (− sin x ) = cos x + 1 cos x + 1

Example 32 x +3 Evaluate dx. x −1 x2 + 3 4 Step 1: Rewrite =x +1+ ; by dividing (x 2 + 3) by (x − 1).

2 x −1 x −1 4 x +3 4 x +1+ dx = dx = dx (x + 1) d x + x −1 x −1 x −1 1 x2 dx = +x +4 2 x −1 Let u = x − 1. Step 2: Differentiate: d u = d x .

Preview - tusachduhoc.com

294

STEP 4. Review the Knowledge You Need to Score High y f(x)

x a

0

b

y=k k

y

x=h

d

g(y)

x h

0 c

Figure 13.4-6

Example 1 Find the volume of the √ solid generated by revolving about the x -axis the region bounded by the graph of f (x ) = x − 1, the x-axis, and the line x = 5. Step 1: Draw a sketch. (See Figure 13.4-7.) y y=

0

1

Figure 13.4-7

5

x–1

x

Preview - tusachduhoc.com

Areas and Volumes

295

Step 2: Determine the radius of a disc from a cross section. r = f (x ) =

√

x −1

Step 3: Set up an integral.

5

V =π

( f (x )) d x = π 2

5 √

1

2 x − 1 dx

1

Step 4: Evaluate the integral.

2 5 2 x 5 V =π −x x − 1 d x = π [(x − 1)]1 = π 2 1 1 2 2

5 1 =π = 8π −5 − −1 2 2 5 √

Verify your result with a calculator.

Example 2 Find the volume of the solid generated by revolving about the x -axis the region bounded by √ π the graph of y = cos x where 0 ≤ x ≤ , the x -axis, and the y -axis. 2 Step 1: Draw a sketch. (See Figure 13.4-8.) y y=

1

cos x

π 2

0

Figure 13.4-8 Step 2: Determine the radius from a cross section. r = f (x ) =

√ cos x

Step 3: Set up an integral.

V =π 0

π/2

√

cos x

2

dx = π

π/2

cos x d x 0

x

Preview - tusachduhoc.com

STEP

5

Build Your Test-Taking Confidence AP Calculus AB Practice Exam 1 AP Calculus AB Practice Exam 2 AP Calculus AB Practice Exam 3

Preview - tusachduhoc.com

349

AP Calculus AB Practice Exam 1

Section I—Part A Number of Questions

Time

Use of Calculator

28

55 Minutes

No

Directions: Use the answer sheet provided in the previous page. All questions are given equal weight. There is no penalty for unanswered questions. Unless otherwise indicated, the domain of a function f is the set of all real numbers. The use of a calculator is not permitted in this part of the exam.

1. The lim

x→ − ∞

(A) − 1 (D) 2

2x − 1 is 1 + 2x (B) 0 (E) nonexistent

6. The graph of f is shown in Figure 1T-1. (C) 1

y

f¢

x

2.

cos t d t π/2

(A) cos x (D) sin x + 1

(B) − sin x (E) − sin x + 1

(C) sin x − 1

a

0

x

b

3. The radius of a sphere is increasing at a constant of 2 cm/sec. At the instant when the volume of the sphere is increasing at 32π cm3 /sec, the surface area of the sphere is 32π (C) 16π 3 256π (D) 64π (E) 3 √ 3 (5s − 1)2 , what is 4. Given the equation A = 4 the instantaneous rate of change of A with respect to s at s = 1? √ √ 5√ (B) 2 3 (C) 3 (A) 2 3 + 5 2 √ √ (D) 4 3 (E) 10 3 (A) 8π

5. What is the lim g (x ), if x →ln2 e x if x > ln 2 g (x ) = ? 4 − e x if x ≤ ln 2 (A) − 2 (D) 2

Figure 1T-1

(B)

(B) ln 2 (E) nonexistent

(C) e 2

A possible graph of f is (see Figure 1T-2 on the next page): 7. If g (x ) = − 2|x + 3|, what is the lim − g (x )? x→ − 3

(A) − 6 (D) 6

(B) − 2 (C) 2 (E) nonexistent π π sin + Δx − sin 3 3 ? 8. What is lim Δx Δx →0 1 1 (B) 0 (C) (A) − 2 2 √ 3 (D) (E) nonexistent 2 GO ON TO THE NEXT PAGE

Preview - tusachduhoc.com

350

STEP 5. Build Your Test-Taking Confidence y

(A)

a

x

b

a

y

(D)

a

0

y

(B)

x

x

b

a

0

x

b

y

(E)

b

0

y

(C)

a

0

b

x

Figure 1T-2 9. If f (x ) is an antiderivative of x e − x and f (0) = 1, then f (1) = 1 1 3 1 1 (A) (B) − (C) − e 2e 2 2e 2 1 3 1 1 (D) − + (E) − + 2e 2 2e 2 π 10. If g (x ) = 3 tan2 (2x ), then g is 8 √ (A) 6 √ (B) 6 2 (C) 12 (D) 12 2 (E) 24 2

y

f

x a

0

b

Figure 1T-3 11. The graph of a continuous and twice differentiable function f is shown in Figure 1T-3. Which of the following statements is/are true? I. f (0) = 0 II. f has an absolute maximum value on [a , b] III. f < 0 on (0, b) (A) (B) (C) (D) (E)

III only I and II only II and III only I and III only I, II, and III

12.

1+x √ dx = x

√ x2 (A) 2 x + + C 2 √ x 3 3/2 + x +C (B) 2 2 √ 2 (C) 2 x + x 3/2 + C 3 2 3/2 (D) x + x + C 3 (E) 0 GO ON TO THE NEXT PAGE

Preview - tusachduhoc.com

422

STEP 5. Build Your Test-Taking ConďŹ dence

Solutions to AB Practice Exam 3—Section II

Section II Part A 1. (A)

d y 4x + 1 = ; (2, 4) dx 2y d y 4(2) + 1 9 = = dx 2(4) 8 (2,4)

9 Equation of tangent line: y − 4 = (x − 2) 8 9 or y = (x − 2) + 4 8 9 0.9 (B) f (2.1) ≈ (2.1 − 2) + 4 ≈ +4 ≈ 8 8 4.1125 ≈ 4.113 (C) 2y d y = (4x + 1) d x 2y d y = (4x + 1) d x y 2 = 2x 2 + x + c ; f (2) = 4 42 = 2(2)2 + 2 + c ⇒ c = 6 2 = 2x 2 + x + 6 or Thus y√ y = Âą 2x 2 + x + 6 Since √the point (2, 4) is on the graph of f , y = 2x 2 + x + 6. √ 2+x +6 (D) y = 2x √ √ f (2.1) = 2(2.1)2 + 2.1 + 6 = 16.92 ≈ 4.11339 ≈ 4.113 2. (A) The amount of oil leaked out after 6 minutes 6 Ď€t 10 sin dt = 12 ⎥0 ⎤ Ď€t 6 − 10 cos ⎢ 12 ⎼ =⎣ ⎌ Ď€ 12 0 Ď€t 6 120 cos = − 12 0 Ď€ 120 = ≈ 38.1972 gallons Ď€ ≈ 38 gallons (B) Average amount of oil leaked out per minute from t = 0 to t = 6:

6 Ď€t 1 120 1 10 sin dt = = 6−0 0 12 6 Ď€ = 6.3662 ≈ 6 gallons

(C) The amount of oil in the drum at t = t1 : t1 Ď€t 10 sin dt f (t) = 100 − 12 0 (D) Let a be the value of t: a Ď€t 100 − 10 sin d t = 40 12 0

a Ď€t 120 − cos = 40 100 − Ď€ 12 0

120 aĎ€ 100 − − cos Ď€ 12

120 − − cos(0) = 40 Ď€

120 aπ 100 + cos π 12

120 − = 40 Ď€

120 aĎ€ 120 cos = + 40 − 100 Ď€ 12 Ď€

a Ď€ 120 Ď€ = cos − 60 12 Ď€ 120 aĎ€ − Ď€ + 2 cos = ≈ − 0.570796 12 2 aĎ€ ≈ cos− 1 (− 0.570796) ≈ 2.17827 12

12 ≈ 8.32038 a = (2.17827) Ď€ a ≈ 8 minutes

Section II Part B 3. (A) f (x ) = x e 2x f (x ) = e 2x + x (e 2x )(2) = e 2x + 2x e 2x = e 2x (1 + 2x ) Set f (x ) = 0 ⇒ e 2x (1 + 2x ) = 0. Since e 2x > 0, thus 1 + 2x = 0 or x = − 0.5.

Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.