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AMSCO’S AP Calculus AB/BC Preparing for the Advanced Placement Examinations Maxine Lifshitz Mathematics Department Chairperson Friends Academy Locust Valley, New York with Martha Green Mathematics Teacher Baldwin High School Baldwin, New York AMSCO Amsco School Publications, Inc 315 Hudson Street, New York, N.Y 10013 Author Maxine Lifshitz is Chair of the Math Department at Friends Academy in Locust Valley, New York She received her A.B degree from Barnard College with Honors in Mathematics and her Ph.D from New York University in Mathematics Education She has been a mathematics consultant for the College Board and a reader of Advanced Placement Calculus Examinations Dr Lifshitz has conducted workshops in applications of the graphing calculator both locally at Calculators Help All Teachers (CHAT) and Long Island Mathematics (Limaỗon) conferences, and nationally at National Council of Teachers of Mathematics (NCTM) and Teachers Teaching with Technology (T3) conferences Dr Lifshitz has published articles in Mathematics Teacher and The New York State Mathematics Teachers Journal Collaborator Martha Green has taught mathematics at Baldwin High School for the past 17 years and is currently a reader for the AP Calculus Examinations She received a Bachelor’s degree in Engineering from Hofstra University and a Masters degree in Secondary Education from Adelphi University She instructs a graduate-level class on Teaching AP Calculus through The Effective Teachers Program of the New York State United Teachers (NYSUT) She has conducted numerous workshops on using calculators to enhance the teaching of mathematics and has presented at CHAT, Limaỗon, and regional NCTM conferences She has previously collaborated with Dr Lifshitz to conduct workshops at T3 International Conferences In 2001, the Nassau County Mathematics Teachers Association named Martha Green Teacher of the Year Reviewers Steven J Balasiano Assistant Principal Supervising Mathematics Canarsie High School Brooklyn, NY Brad Huff Headmaster University High School Fresno, CA Terrence Kent Mathematics Teacher Downers Grove High School Downers Grove, IL Text design by One Dot Inc Composition and Line Art by Nesbitt Graphics, Inc Please visit our Web site at: www.amscopub.com When ordering this book, please specify: either R 781 W or AP CALCULUS AB/BC: PREPARING FOR THE ADVANCED PLACEMENT EXAMINATIONS ISBN 1-56765-562-9 NYC Item 56765-562-8 Copyright © 2004 by Amsco School Publications, Inc No part of this book may be reproduced in any form without written permission from the publisher Printed in the United States of America 10 09 08 07 06 05 04 03 To all the AP Calculus teachers, from the ones who accept the challenge a few weeks before the course begins to those who continuously refresh and renew themselves after years of teaching And to Seymour, Alissa, and Mariel, who provide the base and the encouragement for all my efforts Maxine Lifshitz To my parents, Robert and Martha Sweeney, who raised me to believe I could accomplish anything and who helped me realize my dreams Martha Green CONTENTS AB/BC Topics Chapter Introduction About the Book How to Use This Book Prerequisites to AP Calculus The AP Calculus AB and BC Courses The AP Calculus Examinations Factors Leading to Success in AP Calculus 4 Chapter Functions and Their Properties 2.1 A Review of Basic Functions 2.2 Lines 2.3 Properties of Functions 2.4 Inverses 2.5 Translations and Reflections 2.6 Parametric Equations Chapter Assessment 20 22 28 31 32 35 Contents v Chapter Limits and Continuity 39 3.1 Functions and Asymptotes 3.2 Evaluating Limits as x Approaches a Finite Number c 3.3 Evaluating Limits as x Approaches ; q 3.4 Special Limits: lim sin and lim Ϫ cos x x xS0 x xS0 3.5 Evaluating Limits of a Piecewise-Defined Function 3.6 Continuity of a Function Chapter Assessment 39 43 46 48 50 52 54 Chapter The Derivative 58 4.1 The Derivative of a Function 4.2 The Average Rate of Change of a Function on an Interval 4.3 The Definition of the Derivative 4.4 Rules for Derivatives 4.5 Recognizing the Form of the Derivative 4.6 The Equation of a Tangent Line 4.7 Differentiability vs Continuity 4.8 Particle Motion 4.9 Motion of a Freely Falling Object 4.10 Implicit Differentiation 4.11 Related Rates 4.12 Derivatives of Parametric Equations Chapter Assessment Chapter Applications of the Derivative 98 5.1 Three Theorems: The Extreme Value Theorem, 5.1 Rolle’s Theorem, and the Mean Value Theorem 5.2 Critical Values 5.3 Concavity and the Second Derivative 5.4 Curve Sketching and the Graphing Calculator 5.5 Optimization Chapter Assessment Chapter Techniques and Applications of Antidifferentiation 98 102 110 113 118 121 125 6.1 Antiderivatives 6.2 Area Under a Curve: Approximation by Riemann Sums 6.3 The Fundamental Theorem of Calculus 6.4 The Accumulation Function: An Application of 6.4 Part Two of the Fundamental Theorem 6.5 Integration by the Change of Variable or 6.4 u-Substitution Method 6.6 Applications of the Integral: Average Value of a Function 6.7 Volumes 6.8 The Trapezoidal Rule 6.9 Arc Length and Area of a Surface of Revolution Chapter Assessment vi Contents 58 63 66 67 73 78 81 82 86 88 91 93 94 125 131 138 146 150 155 157 168 172 176 Chapter Separable Differential Equations and Slope Fields 180 7.1 Separable Differential Equations 7.2 Slope Fields 7.3 The Connection Between a Slope Field 7.3 and Its Differential Equation Chapter Assessment 180 185 188 192 BC Topics Chapter Methods of Integration 199 8.1 Integration by Parts 8.2 Integration by the Method of Partial Fractions 8.3 Improper Integrals Chapter Assessment 199 202 206 209 Chapter Polynomial Approximations and Infinite Series 213 9.1 Introduction 9.2 Sigma Notation 9.3 Derivation of the Taylor Polynomial Formula 9.4 Finding New Polynomials from Old 9.5 Error Formula for Taylor Polynomial Approximation 9.6 Sequences and Series 9.7 Power Series Chapter Assessment Chapter 10 Logistic Growth, Euler’s Method, and Other BC Topics 213 215 217 221 225 227 242 248 253 10.1 The Logistic Growth Model 10.2 Euler’s Approximation Method 10.3 Logarithmic Differentiation 10.4 L’Hôpital’s Rule 10.5 Polar Curves Chapter Assessment Model Examinations 253 257 261 263 266 277 281 AB Model Examination AB Model Examination BC Model Examination BC Model Examination 283 293 302 311 Contents vii Answer Key Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter 10 Contents 323 331 335 346 350 356 360 361 367 Model Examinations 371 AB Model Examination AB Model Examination BC Model Examination BC Model Examination 371 378 383 389 Index viii 321 395 AB/BC Topics CHAPTER Introduction CHAPTER Functions and Their Properties CHAPTER Limits and Continuity CHAPTER The Derivative CHAPTER Applications of the Derivative CHAPTER Techniques and Applications of Antidifferentiation CHAPTER Separable Differential Equations and Slope Fields (a) (a) points dv ϭ Ϫ2v ϩ dt dv ϭ 2: defining both integrals Ϫ2v ϩ dt v ϭ (Ϫ2v ϩ 8)t ϩ c 2: solving the integrals 2: finding v(t) ϭ Solve for v 8t ϩ c 2t ϩ 8(0) ϩ c v(0) ϭ ϭ 20 2(0) ϩ 8t ϩ c 2t ϩ 1: answer v(t) ϭ c ϭ 20 8t ϩ 20 2t ϩ 8t ϩ 20 (b) lim ϭ cm/sec t S q 2t ϩ v(t) ϭ (b) points 1: solving the limit 1: answer BC Model Examination Section I Part A (pages 302–305) (C) ln(6) ϩ ln a b 10 35 (A) 10 sin 10x 10 ϭ2 (E) x ϭ Ϫ e 10 ϭ ln(6) ϩ ln a b (E) I, II, and III (B) f(x) is continuous but not differentiable at x ϭ e2 ϩ (A) u ϭ ln x dv ϭ x dx du ϭ x dx vϭ x2 e x ln x dx e x2 x ln x Ϫ dx 21 e ϭ ϩ dx xϪ4 xϩ5 (A) ϭ 35 ϭ ln(x Ϫ 4) ϩ ln(x ϩ 5) (A) 2(et Ϫ e ϩ 1) 21 14x ϩ 16 dx x2 ϩ x Ϫ 20 x2 x2 ln x Ϫ 2 e2 e2 ln e Ϫ R Ϫ Q ln Ϫ R 4 e ϭ ϩ1 4 ϭQ 20 "1 ϩ cos2 x dx 10 (B) a ϭ , c ϭ Ϫ1 For f(x) to be continuous and differentiable, two conditions must be satisfied: the two functions must have the same f value at 2, and the two functions must have the same derivative at f¿(x) ϭ for x Ͼ f¿(2) ϭ f¿(x) ϭ 2ax for x Յ f¿(2) ϭ 4a ϭ aϭ1 f(x) ϭ x2 (first condition satisfied) Answer Key 383 f(2) ϭ # 22 ϭ 23 (E) does not exist ϩ c ϭ (second condition satisfied) 24 (A) Ϫ2 tan x 11 (B) y ϭ eϪx dy ϭ ϪeϪx>2 is the only graph whose dx slope is always positive and decreasing 25 (B) y ϭ Ϫ 1e x ϩ e 26 (A) divergent b b xex2 dx ϭ lim eu du 1 bSq2 b S q2 lim 12 (C) "34 ϭ lim c dx ϭ 2t ϩ dt When t ϭ 1, bSq eb2 e Ϫ bSq ϭ lim dx ϭ dt 27 (B) dy ϭ3 dt By the Pythagorean Theorem, 52 ϩ 32 ϭ s2 13 (B) 28 (D) F¿(x) ϭ f(x) Ϫ g(x) (g(x))3 22 u ϭ f(x), u¿ ϭ g(x), v ϭ (g(x))2, v¿ ϭ 2g(x) f(x) 22 ϩ F(2) ϭ F(2) ϭ (g(x))3g(x) 2(g(x)) Ϫ ϭ Ϫ ϭ (g(x))4 (g(x))4 g(x) (g(x))3 15 (B) Section I Part B (pages 306–308) (C) 5.278 Q"xR dx ϭ 16 (D) 4y ϩ 10 20 3x5>3 96 ` ϭ 5 3k5>3 48 ϭ 5 k ϭ "2 L 5.278 18 (A) 6x Ϫ y ϭ 11 19 (C) F¿(x)dx ϭ F(x) ϭ f(x) (g(x))2g(x) f(x) Ϫ 2 ((g(x)) ) ((g(x))2)2 14 (E) I, II, and III f¿(x)dx ϭ 2(g(x)) 17 (E) b ex2 d f(4) Ϫ f(0) (D) II and III (A) 20 (A) Ϫ cos 2t t (C) 21 (C) 16 3Ϫ3 a2 ( "9 Ϫ x2) b dx ϭ 16 22 (E) 36 384 Answer Key x ln y ϩ ex ϭ y dy x2 ln y x ϭ ϩ y ϩ ex dx 02 ln ϩ ϩ e0 ϭ (D) 10 (B) one relative minimum and one relative maximum f Ј(x) 11 (C) x Ϫ3 Ϫ2 Ϫ1 Ϫ1 f–(x) ϭ 2x(x2 Ϫ 3) (x2 ϩ 1)3 x ϭ 0, x ϭ "3, or x ϭ Ϫ "3 f¿(0) ϭ 0, f¿Q "3R ϭ Ϫ , and f¿QϪ "3R ϭ Ϫ (E) does not exist f(x) ϭ f¿(x) ϭ (x2 Ϫ x)Ϫ1>2(2x Ϫ 1) ϭ 2x Ϫ 2Q "x(x Ϫ 1)R 12 (E) ϪeϪt ϩ c 13 (E) 4 2 20 (C) lim f(x) ϭ Q(2 Ϫ cos u) Ϫ cos uR du xS2 14 (E) diverges by the Comparison Test (D) D Q xR (E) e ϭ a , which simplifies to (E) n! nϭ0 n 15 (E) 14.697 q x 16 (A) 10.0 mph 17 (D) Section II Part A (page 309) (a) 30 (cos2x Ϫ sin2x)dx ϩ 3 34 (a) points (sin2x Ϫ cos2x)dx 1: limits ϩ 334 2: definite integral 1: integrand (cos2x Ϫ sin2x)dx 1: answer ϭ 1ϩ1ϩ1ϭ2 2 3 4 2 (b) a cos2x Ϫ b dx ϩ a sin2 Ϫ b dx 2 34 30 ϩ 334 a cos2x Ϫ b dx 2 2 2 2 ϭ ϩ ϩ ϭ 32 16 32 (b) points 2: definite integral 1: limits 1: integrand 1: answer Answer Key 385 (c) 30 (cos2x Ϫ sin2x)2 dx ϩ ϩ 334 3 34 (c) points (sin2x Ϫ cos2x)2 dx 2: definite integral 1: limits (cos2x Ϫ sin2x)2 dx 1: integrand 1: answer 5 ϩ ϩ ϭ 4 (a) dy ϭ 0.03y(100 Ϫ y) dt (a) points 1: finding y ϭ and y ϭ 100 0.03y(100 Ϫ y) ϭ 1: answer y ϭ and y ϭ 100 y(Ϫ1)¿ ϭ Ϫ101.03, so the interval before y ϭ is decreasing y(1) ϭ 2.97, so the interval [0, 100] is increasing y(101) ϭ Ϫ303, so the interval after y ϭ 100 is decreasing Thus, the interval where y is increasing is Ͻ y Ͻ 100 (b) y– ϭ Ϫ 0.06y (b) points Ϫ 0.06y ϭ yϭ 1: the second derivative 1: answer ϭ 50 0.06 Thus, y ϭ 50 is increasing the fastest (c) y(t) ϭ 10 ϩ (0.03(y)(100 Ϫ y)t) (c) points y(0) ϭ 10 3: definite integral y(0.1) ϭ 10 ϩ (0.03(10)(100 Ϫ 10))0.1 ϭ 12.7 2: integrand y(0.2) ϭ 12.7 ϩ (0.03(12.7)(100 Ϫ 12.7))0.1 ϭ 16.026 1: finding the value of c y(0.3) ϭ 16.026 ϩ (0.03(100 Ϫ 16.026))0.1 ϭ 20.063 2: answer (a) absolute minimum ϭ (2, Ϫ1.18) (a) points absolute maximum ϭ (1.55, 2.55) To find the absolute maximum, set 1: derivative dy ϭ dt dy ϭ Ϫ et>2 ϭ dt t ϭ ln 16 x L 1.55, y L 2.55 Since there is only one solution found when we set the derivative equal to zero, the absolute minimum must occur at one of the endpoints It occurs at t ϭ 386 Answer Key 1: absolute minimum 1: absolute maximum (b) dy dx ϭ Ϫ et>2 ϭ Ϫ e when t ϭ and ϭ 2 dt dt dy (b) points 2: derivative 1: answer dy dt ϭ 10 Ϫ e L 3.20 ϭ dx dx dt The slope of the tangent line at t ϭ y ϭ 3.20x Ϫ 2.2 3.20x Ϫ 2.2 ϭ 0.69 is the x-intercept (c) t ϭ corresponds to the point (1.4, 2.28) Use this with the slope to get the x-intercept METHOD The arc length is METHOD a b ϩ a Ϫ e b dt L 6.43 30 A t 2 (c) points 2: definite integral 1: limit 1: integrand 1: answer t xϭ ϩ1 t ϭ 5(x Ϫ 1) y ϭ 2(5x Ϫ 5) Ϫ e y ϭ 10x Ϫ e 30 A 5xϪ5 5xϪ5 ϩ1 ϩ9 5xϪ5 a ϩ a 10 Ϫ e b b dx L 6.43 Section II Part B (page 310) x2 "(x3 Ϫ 1)2 x ϭ for the horizontal tangent line (a) y¿ ϭ 3 Ϫ 1)2 ϭ for the vertical line "(x The tangent line is vertical at (1, 0), and horizontal at (0, Ϫ1) (b) lim xSq 3 3 "x Ϫ1 "x Ϫ1 and lim ϭ ϭ1 x x x S Ϫq With these limits, we can use the Direct Comparison Test to find if the graph converges on a point or diverges The graph does not converge (c) y¿ ϭ x2 "(x3 Ϫ 1)2 x2 ϭ0 "(x3 Ϫ 1)2 3 Ϫ 1) is "(0 Ϫ 1) ϭ Thus, the point is When x ϭ 0, "(x (0, 1) (a) points 1: solving derivative 1: the point where the tangent line is vertical 1: the point where the tangent line is horizontal (b) points 2: solving both limits 1: justifying the answer 1: answer (c) points 2: answer Answer Key 387 (a) q ϭ Ϫ 2x ϩ 4x2 Ϫ 8x3 ϩ % ϭ a (Ϫ1)n(2x)n ϩ 2x nϭ0 The interval of convergence is ƒ x ƒ (b) d ϭ Ϫ1 Q R dx ϩ 2x (1 ϩ 2x)2 (a) points 1: first four terms of Maclaurin series 1: sigma notation 1: interval of convergence (b) points ϭ Ϫ (Ϫ2 ϩ 8x Ϫ 24x2 ϩ % ) ϭ Ϫ 4x ϩ 12x2 Ϫ % 1: first four terms of Maclaurin series 1: sigma notation 1: interval of convergence q ϭ a (Ϫ1)nϩ1 n(2x)nϪ1 nϭ1 The interval of convergence is ƒ x ƒ (c) ln(1 ϩ 2x) ϭ dx ϩ 2x ϭ c ϩ 2x Ϫ 2x2 (c) points ϩ x3 Ϫ 4x4 ϩ % Let x ϭ Then c ϭ 0, and 1: first four terms of Maclaurin series 1: sigma notation 1: interval of convergence ln(1 ϩ 2x) ϭ 2x Ϫ 2x2 ϩ x3 Ϫ 4x4 ϩ % q (2x)nϩ1 ϭ a (Ϫ1)n nϩ1 nϭ0 The interval of convergence is Ϫ x Յ 2 (a) g(0) ϭ g(2) ϭ 20 (a) points f(t) dt ϭ 1: g(0) 20 f(t) dt ϭ ϩ 1: g(2) (b) Set g¿(x) ϭ Since g¿ ϭ f by the Fundamental Theorem, solve f(x) ϭ x ϭ ; gЈ – –2 – + – 2 There is a relative minimum at x ϭ Ϫ , and a relative maxi2 mum at x ϭ 388 Answer Key (b) points 1: critical points of g 1: relative minimum at xϭϪ , relative maximum at xϭ 1: justification The First Derivative Test 3 Since g¿ changes from Ϫ to ϩ at x ϭ Ϫ , gQϪ R is a relative 2 minimum 3 Since g¿ changes from ϩ to Ϫ at x ϭ , gQ R is a relative max2 imum (c) Solve g–(x) ϭ or is undefined Since g– ϭ f¿ , find values where f ¿(x) ϭ or f¿ is undefined (c) points 1: points of inflection If f ¿(x) ϭ 0, then x ϭ If f¿ is undefined, then x ϭ ;1 _ + + gЈЈ –2 –1 1: justification _ Since g– changes sign at x ϭ 0, it is the only point of inflection (d) Graph of g(x) (d) points 1: one relative maximum and one relative minimum g (x ) 1: one point of inflection x –2 – –1 –2 BC Model Examination Section I Part A (pages 311–314) (E) 16 (A) 3x ϩ y ϭ (B) a ϭ c ϭ Ϫ1 For f(x) to be continuous and differentiable, two conditions must be satisfied: the two functions must have the same f value at 2, and the two functions must have the same derivative at f¿(x) ϭ for x Ͼ f¿(2) ϭ 3 (D) f¿(x) ϭ 2ax for x Յ f¿(2) ϭ 4a ϭ (D) Ϫ5 aϭ1 (E) divergent q 30 ln(x) (ln(x))2 q ` x dx ϭ f(x) ϭ x2 (first condition satisfied) f(2) ϭ # 22 ϭ ϩ c ϭ (second condition satisfied) Answer Key 389 (E) 13 (C) [Ϫ1, 1] If you try to substitute directly, you get an indeterminate form; therefore, use L’Hôpital’s Rule 1 ln(ln(x)) ln(x) x ϭ lim lim ln(x) xSq xSq x 14 (C) ln(x Ϫ 1) 15 (C) # ϭ lim xSq ϭ0 ln(x) There is no x in dy d2y , so is for all values dx dx of x 16 (E) $50.50 (C) Let x be the number of widgets not sold ϭqϫ0ϭ0 lim x5e Ϫx>5 ϭ lim x5 x>5 e xSq xSq Then R(x) ϭ (1,000 Ϫ x)(0.10x ϩ 1.00) ϭ 1,000 ϩ 99x Ϫ (E) Ϫ 24 x2 10 dR ϭ 0, you get x ϭ 495, so the dx selling price is (0.10)495 ϩ 1.00 If you set dx ϭ dt 5x2 ϩcϭt 5x cϭ 25 17 (A) (x ϩ 1)7>2 Ϫ (x ϩ 1)5>2 3>2 + (x ϩ 1) ϩ C ϩ ϭ1 5x 25 18 (C) 2h f(h2) xϭ Ϫ 24 19 (E) II and III q 10 (B) 20 (A) a ( Ϫ 1)n(5x)n Ϫ1 tcos(t) dt ϭ t sin(t) ϩ cos(t) ` ϭ Ϫ 32 11 (C) x3 ϩx e3 dy ϭ (x2 ϩ 1) dx (y ϩ 1) ϩx 20 ϭ Ϫ1 12 (A) ln ƒ x2 Ϫ 2x ƒ ϭ ln(x2 Ϫ 2x), when x or x ln ƒ x2 Ϫ 2x ƒ ϭ ln(2x Ϫ x2), when x In either case, the derivative is 390 Answer Key v(1) ϭ Because the arc length formula requires the derivative of x(t), we not need to integrate again in order to find that derivative dy ϭ y(x2 ϩ 1) ϩ (x2 ϩ 1) dx x3 If a(t) ϭ 2t ϩ 2, then v(t) ϭ t2 ϩ 2t ϩ C Cϭ1 Ϫ1 y ϭ e3 21 (C) 21 2x Ϫ x2 Ϫ 2x "1 ϩ (x¿(t))2 dt 20 "1 ϩ (t2 ϩ 2t ϩ 1)2 dt 22 (C) y ϭ ( Ϫ 4) xϩ 16 First, find the slope of the tangent line by using the derivative Section I Part B (pages 315–317) (A) 0.0024 g(x) ϭ 1, g¿(x) ϭ , g–(x) ϭ , g–(x) ϭ arctan(x) at x ϭ is x2 ϩ 1Ϫ1ϩ1Ϫ 48 2 b with the slope to find 16 the equation of the line Use the point a 1, 23 (E) Ϫ "3 Ϫ Ϫ ϩ Ϫ ϭ 0.0024 48 "e (E) I, II, and III (A) 0.443 24 (C) 1 g¿(x) ϭ ln(1 ϩ x) (g¿(x))2 ϭ (ln(1 ϩ 30 x))2 (2x) dx ϩ 312 a1 x Ϫ b dx Substitute for x (E) 1.0986 25 (D) 2x Ϫ x2 Ϫ 2x b Ϫ cos (2 ϫ 3 2) ϭ (sin2(3 2)) 2a 27 (D) 19 (Ϫ2x2 ϩ 4x)dx ϭ Ϫ2 x3 ϩ 2x2 ϩ C F(0) ϭ Cϭ5 28 (A) 4(e Ϫ 2) Consider two points on a parabola that have the same y value (E) does not exist (D) F(x) ϭ Q (2 # cos(t))2 ϩ a Ϫ # sin(t) b dx sin(t) A (B) f has a point of inflection in 3a, b4 ϭ (sin2(3 2)) 26 (B) dy dx R ϩ Q R dt 2a A dt dt b The length is The integral is arcsin(x) (A) 2 20 (2 ϩ cosu)du (A) (1.333, 5.333) The intercepts are at x ϭ and x ϭ dy ϭ Ϫ 3x2 dx The tangent line at x ϭ is y ϭ 4x The tangent line at x ϭ is y ϭ Ϫ8x ϩ 16 10 (A) 11 (B) 0.267 12 (B) 1.771 Answer Key 391 13 (E) 0.73024 h¿(x) ϭ 16 (A) f¿(x) # cos(f(x)) # "sin(f(x)) e # cosQ R g(x) ϭ (ln(1 ϩ t))2 dt 20 ϭ (x ϩ 1) ln(x ϩ 1)2 Ϫ 2(x ϩ 1) ln(x ϩ 1) # "sin Q R Ϫy 14 (B) x ϩ 2x g¿(x) ϭ ln(x ϩ 1)2 L 0.73024 g¿(e Ϫ 1) ϭ Ϫ y tan(ln(xy)) x sec2(ln(xy)) ϩ 2y2 sec2(ln(xy)) 17 (A) none 18 (B) 1.185 15 (B) p ϭ Ϫ4 Section II Part A (page 318) b (a) 2a h(x)dx ϭ b 2a (a) points g(x)dx f–(g(0)) f ¿–(g(0)) x ϩ x (b) f(g(0)) ϩ f¿(g(0))x ϩ 2! 3! 2: answer (b) points 2: correct expansion (c) We conclude that there can be distinct values on 3a, b4 where f(x) ϭ 1: answer (c) points 2: understanding that to have distinct values of x, the polynomial of f (3)(x) must be a fourth-degree polynomial 1: derivatives 1: answer (a) (e ϩ 1) Ϫ 22 eϩ1 (a) points ln(x Ϫ 1)dx 1: identifying the integral 1: solving the integral L 0.3387 eϪ1 1: answer (b) points (b) The area under the curve from to e ϩ is (Integrate with that bound.) k ln(x ϩ 1)dx ϭ 22 (k ϩ 1) ln(k ϩ 1) Ϫ k ϩ ϭ ; therefore, k L 2.43 (c) Use the integral 2 (4 Ϫ e) L 4.03 392 Answer Key 20 (1 Ϫ y)(e ϩ Ϫ (ey ϩ 1))dy 1: identifying the integral 1: solving the integral 1: answer (c) points 1: identifying the integral 1: solving the integral 1: answer (a) Equation of tangent line at x ϭ a: yϪ (a2 (a) points ϩ 2) ϭ 2a(x Ϫ a) 1: f¿(x) 1: equation of tangent line at xϭa a2 Ϫ 4a (b) Base of triangle ϭ Ϫ Area of triangle ϭ Q4 Ϫ (c) A¿ ϭ (b) points a2 Ϫ R (a2 ϩ 2) 2a 16a3 Ϫ 3a4 Ϫ 4a2 1: finding A¿ 1: setting A¿ ϭ + AЈ 1: area of triangle (c) points If A¿ ϭ 0, then x ϭ 0.658 _ 1: base of triangle 1: solving A¿ ϭ 0.658 Since A¿ changes from Ϫ to ϩ at x ϭ 0.658, A(0.658) is a relative minimum 1: justification that solution is a minimum 1: only one solution shown between and Section II Part B (page 319) (a) (a) points y 5: graph Ϫ3 Ϫ2 Ϫ1 Ϫ1 (b) y ϭ ϩ 4e5t t (b) points 2: answer (c) y ϭ (c) points 2: answer Answer Key 393 (a) (a) points dr ϭ r dt 3: finding the formula 1: answer (b) (c) dr ϭ ϭ L 0.0159154 (102) 20 dt (b) point dE ϭ ev>3 dt (c) points dr dE dV ϭ 4r2 Ϫ dt dt dt e3 1: answer 2: finding the formula 1: answer V 20 Ϫ dr ϭ 4r2 dt (d) (0) e3 20 Ϫ ϭ 4r2 dr dt dr ϭ 0.0151197 dt (d) points 1: using v ϭ 1: answer (a) y¿ ϭ e2Ϫ2x2(Ϫ4x2 Ϫ 1) ; odd x2 (a) points 1: the derivative 2: answer (b) (Ϫq, 0) ´ (0, q) (b) point 1: the answer (c) The graph has no points of inflection since the part that is concave down (where x is negative) is separated by a vertical asymptote at x ϭ from the part that is concave up Substitute positive and negative values into the second derivative to verify this (c) points (d) (0, q) (d) point 2: answer 1: answer (e) (Ϫq, 0) ´ (0, q) (e) points 2: answer 394 Answer Key Index A Absolute convergence, 238–239 Acceleration of a particle, 83 Accumulation function, 146–148 Addition-Subtraction Rule, 127 Algebraic functions, 10 Algebraic methods of evaluating limits, 43–45 Alternating series, 237–238 error formula for, 238 Alternative form of derivative, 67 Antiderivative(s), 125–129 formulas for, 125–127 of a power series, 244–246 Antidifferentiation, 125 AP Calculus Examinations, 5–6 factors leading to success in, 6–7 use of graphing calculator on, 5–6 Approximation by Riemann sums, 131–136 Archimedes, 131 Arc length, 172–173 of polar curve, 274 Area antiderivative and, 138–141 under a curve, 131–136 in polar coordinates, 273–274 of a surface of revolution, 174–175 Argument, 70 Asymptotes, 11, 23, 39–41 Average rate of change of a function on an interval, 63–65 Average value of a function, 155–156 Axis of revolution, 157–159, 174 Calculator Notes (continued) Functions and Asymptotes, 40–41 Infinite Series, 226 Inverses, 29 Parametric Equations, 33 Polar Curves, 271, 274–276 Properties of Functions, 25 A Review of Basic Functions, 9, 17–18 Slope Fields, 185 Techniques and Applications of Antidifferentiation, 134–135 Trapezoid Rule, 169–171 Calculus, Fundamental Theorem of, 138–143 Cardioid, 270 Chain Rule Extension, 68 Circle with its center on the line perpendicular to the polar axis, 270 Circle with its center on the polar axis, 269 Concave down, 110, 113 Concave up, 110, 113 Concavity, 110 Conditional convergence, 238–239 Constant c, 126 Constant-Multiple Rule, 68 Constant Rule, 67, 127 Continuity versus differentiability, 81 of a function, 52–53 Convergence, 227 absolute, 238–239 conditional, 238–239 interval of, 242 radius of, 242 C Cosine, double-angle formula Calculator Notes for, 14 Average Rate of Change of a Critical points, 59 Function, 64–65 Critical values, 59, 102–109 Curve Sketching and the Cross sections, 162 Graphing Calculator, 115–117 solids with known, 162–166 Curve(s) area under, 131–136 graphing calculator and sketching of, 113–117 petal, 270–271 polar, 266–276 D Definite integral, 139 Degree of the polynomial, 9, 10 Derivation of the Taylor polynomial formula, 217–220 Derivative(s) alternative form of, 67 definition of, 66–67 of a function, 58–61 of inverse functions, 74–75 of parametric equations, 93 of a power series, 244–246 recognizing the form of, 73–76 rules for, 67–72 second, 75–76, 83 Differentiability versus continuity, 81 Differential equations, 180–188 Differentiation implicit, 88–90 logarithmic, 261–262 Direct Comparison Test, 233–236 Disc method, 157 Discontinuity, removable, 53 Divergence, 227 Divergence Test, 231 Domain, 22–23 Double-angle formulas for sine and cosine, 14 E End behavior of a function, 40 of a polynomial, 10 Index 395 Equation(s) differential, 180–183, 188 of horizontal line, 20 of a line, 20 parametric, 32–34, 93 separable differential, 180–183 of a tangent line, 78–80 trigonometric, 13 of a vertical line, 20 Eudoxus, 131 Euler’s approximation method, 257–259 Even functions, 23, 25 Exponential functions, 15–16 Exponential rules, 69, 128 Exponents, properties of, 16 Extreme Value Theorem, 98–100 F First Derivative Test, 103–109 Fractions, integration by the method of partial, 202–205 Functional notation, 23 Function(s), accumulation, 146–148 algebraic, 10 average value of, 155–156 continuity of, 52–53 derivative of, 58–61 end behavior of, 40 even, 23, 25 exponential, 15–16 inverse of, 15 inverse trigonometric, 15 multiple representations of a, 23–24 odd, 23, 25 piecewise, 16–17, 50–51 properties of, 22–27 rational, 11–12, 39 roots of, 26 symmetry of, 23 trigonometric, 12–14 Fundamental Theorem of Calculus, 138–143 Part One, 138–141 Part Two, 141–143 G Geometric series, 228–229, 242 Graphing calculator curve sketching and, 113–117 use of, on AP Calculus Examination, 5–6 See also Calculator Notes 396 Index H Harmonic series, 230–231 Horizontal asymptotes, 11, 23, 39–41 Horizontal line, equation of, 20 Horizontal line test, 28 I Identities Pythagorean, 14 quotient, 14 reciprocal, 14 trigonometric, 14 Implicit differentiation, 88–90 Improper integrals, 206–208 Indefinite integral, 125 Independent variable, 22 Infinite series, 213–252 alternating, 237–238 geometric, 228–229, 242 harmonic, 230–231 p-, 229 power, 242–246 strategy for testing, 231–238 Taylor, 214, 242, 246 telescoping, 230 Infinity, evaluating limits as x approaches, 46–47 Instantaneous rate of change, 66 Integral(s) definite, 139 improper, 206–208 properties of, 129 Integral Test, 236–237 Integrand, 125, 126 Integration, 125 by the change of variable, 150–153 by the method of partial fractions, 202–205 by parts, 199–201 Interval average rate of change of a function on an, 63–65 of convergence, 242 Inverse functions, 15 derivatives of, 74–75 Inverses, 28–29 Inverse trigonometric functions, 15 Inverse trigonometric rules, 70 L Leading coefficient, sign of, 10 Left-hand sum, 134 L’Hôpital’s Rule, 45, 263–265 tips for using, 264 Limaỗon, 270 Limit Comparison Test, 233 Limits, 39 evaluating algebraic methods of, 45 of piecewise-defined function, 50–51 as x approaches a finite number c, 43–45 as x approaches infinity, 46–47 special, 48–49 Limit symbol, 39–40 Line(s), 20–21 equation of, 20 horizontal, 20 parallel, 20 perpendicular, 20 slope of, 20 vertical, 20 Local maximum, Local minimum, Logarithmic differentiation, 261–262 Logarithmic function, properties of, 16 Logistic growth, 254 Logistic growth model, 253–256 Logarithmic rules, 69 Lower sum, 134 M Maclaurin polynomial, 218, 221–223 Mean Value Theorem, 98, 100–101, 102 Method of partial fractions, integration by, 202–205 Midpoint sum, 134 Motion of a freely falling object, 86–87 particle, 82–84 O Odd function, 23, 25 One-to-one function, 28 Optimization, 118–120 P Parallel lines, 20 Parameter, 32, 93 Parametric equations, 32–34 derivatives of, 93 Partial fractions, integration by the method of, 202–205 Particle motion, 82–84 Parts, integration by, 199–201 Perpendicular lines, 20 Petal curves, 270 Piecewise functions, 16–17 evaluating limits of, 50–51 Point-slope form, 78 Points of inflection, 111–113 Points of intersection of polar curves, 271–273 Polar coordinates, converting, to rectangular coordinates, 267 Polar coordinate system, 266 Polar curves, 266–276 arc length of, 274 points of intersection of, 271–273 slope of the line tangent to, 274 Polynomials, 9–10 end behavior of, 10 finding new, from old, 218, 221–223 Maclaurin, 218, 221–223 Taylor, 213–214 Population growth, 183 Position of a particle, 82 Power Rule, 68, 126 Power series, 242–246 antiderivative of, 244–246 derivative of, 244–246 Product Rule, 71, 199 p-series, 229 Pythagorean identities, 14 Pythagorean Theorem, 173 Q Quotient identities, 14 Quotient Rule, 71 R Radius of convergence, 242 Range of a function, 23 Rate of change average, of a function on an interval, 63–65 instantaneous, 66 Rates, related, 91–92 Rational functions, 11–12, 39 Ratio Test, 231–233, 242 Reciprocal identities, 14 Rectangular coordinates, converting polar coordinates to, 267, 268 Reflections, 31 Related rates, 91–92 Relative extrema, 103 Relative maxima, 103 Relative minima, 103 Removal discontinuity, 53 Riemann sums, 136 approximation by, 131–136 Right-hand sum, 134 Rolle’s Theorem, 98–100 Roots of a function, 26 Rule of Four, 67 S Second derivative, 75–76, 83 Second Derivative Test, 111 Separable differential equations, 180–183 Sequences and series, 227–239 Sigma notation, 215–216 Sign of the leading coefficient, 10 Sine, double-angle formula for, 14 Slope field(s), 185–187, 257 connection between, and its differential equation, 188 Slope of a line, 20 Slope of the line tangent to a polar curve, 274 Solids of revolution, 157–159 Solids with known cross sections, 162–166 Special limits, 48–49 Speed of a particle, 82 Split functions, 16–17 See also Piecewise functions Sum and Difference Rule, 71 Symmetry of a function, 23 T Tangent line, 58 equation of, 78–80 shape of, in a polar curve, 274 Taylor polynomial(s), 213–214 Taylor polynomial approximation, error formula for, 225–226 Taylor polynomial formula, derivation of, 217–220 Telescoping series, 230 Translations, 31 Trapezoidal Rule, 168–171 error in, 169 Trigonometric equations, solving, 13 Trigonometric functions, 12–14 graphing, 13 inverse, 15 Trigonometric identities, 14 Trigonometric rules, 70, 128–129 Turning points, 9, 59, 98 U Upper sum, 134 u-substitution method, 150–153 V Variable(s) independent, 22 integration by the change of, 150–153 Velocity of a particle, 82 Vertical asymptotes, 11, 23, 39–41 Vertical line, equation of, 20 Vertical line test, Volumes, 157–166 W Washer method, 157, 159 X x-coordinate, 22 Y y-coordinate, 23 Z Zeros of a function, 26 Index 397 ... be proficient in the use of a graphing calculator The AP Calculus AB and BC Courses There are two levels of AP Calculus, Calculus AB and Calculus BC The Calculus AB course is intended to be a yearlong... dreams Martha Green CONTENTS AB/ BC Topics Chapter Introduction About the Book How to Use This Book Prerequisites to AP Calculus The AP Calculus AB and BC Courses The AP Calculus Examinations Factors... AB Model Examination BC Model Examination BC Model Examination 283 293 302 311 Contents vii Answer Key Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter 10 Contents 323
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