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Go to barronsbooks.com/AP/calcab/ or barronsbooks.com/AP/calcbc/ to take a free sample AP Calculus AB or BC test, complete with answer explanations and automated scoring *These online tests can be accessed on most mobile devices, including tablets and smartphones About the Authors David Bock taught AP Calculus during his 35 years at Ithaca High School, and served for several years as an Exam Reader for the College Board He also taught mathematics at Tompkins-Cortland Community College, Ithaca College, and Cornell University A recipient of several local, state, and national teaching awards, Dave has coauthored five textbooks, and now leads workshops for AP teachers Dennis Donovan has been a math teacher for 22 years, teaching AP Calculus for the past 19 years (8 years AB, 11 years BC) He has served as an AP Calculus Reader and as one of nine national Question Leaders for the AP Calculus exam Dennis leads professional development workshops for math teachers as a consultant for the College Board and a T3 Regional Instructor for Texas Instruments IN MEMORIAM Shirley O Hockett (1920-2013) Shirley Hockett taught mathematics for 45 years, first at Cornell University and later at Ithaca College, where she was named Professor Emerita in 1991 An outstanding teacher, she won numerous awards and authored six mathematics textbooks Shirley’s experiences as an Exam Reader and Table Leader for AP Calculus led her to write the first ever AP Calculus review book, published in 1971 Her knowledge of calculus, attention to detail, pedagogical creativity, and dedication to students continue to shine throughout this book On behalf of the thousands of AP Calculus students who have benefited from her tireless efforts, we gratefully dedicate this review book to Shirley © Copyright 2017, 2015, 2013, 2012, 2010, 2008 by Barron’s Educational Series, Inc © Prior editions copyright 2005, 2002, 1998 under the title How to Prepare for the AP Advanced Placement Examination in Calculus and 1995, 1992, 1987, 1983, 1971 under the title How to Prepare for the Advanced Placement Examination: Mathematics by Barron’s Educational Series, Inc All rights reserved No part of this work may be reproduced or distributed in any form or by any means without the written permission of the copyright owner All inquiries should be addressed to: Barron’s Educational Series, Inc 250 Wireless Boulevard Hauppauge, New York 11788 www.barronseduc.com Replace deleted text with the following: eISBN: 978-1-4380-6892-3 Revised eBook publication: April 2017 Contents Barron’s Essential Introduction The Courses Topics That May Be Tested on the Calculus AB Exam Topics That May Be Tested on the Calculus BC Exam The Examinations The Graphing Calculator: Using Your Graphing Calculator on the AP Exam Grading the Examinations The CLEP Calculus Examination This Review Book Flash Cards DIAGNOSTIC TESTS Calculus AB Calculus BC TOPICAL REVIEW AND PRACTICE Functions A Definitions B Special Functions C Polynomial and Other Rational Functions D Trigonometric Functions E Exponential and Logarithmic Functions BC ONLY F Parametrically Defined Functions BC ONLY G Polar Functions Practice Exercises Limits and Continuity A Definitions and Examples B Asymptotes C Theorems on Limits D Limit of a Quotient of Polynomials E Other Basic Limits F Continuity Practice Exercises Differentiation A Definition of Derivative B Formulas C The Chain Rule; the Derivative of a Composite Function D Differentiability and Continuity E Estimating a Derivative E1 Numerically E2 Graphically BC ONLY F Derivatives of Parametrically Defined Functions G Implicit Differentiation H Derivative of the Inverse of a Function I The Mean Value Theorem J Indeterminate Forms and L’Hôpital’s Rule K Recognizing a Given Limit as a Derivative Practice Exercises Applications of Differential Calculus A Slope; Critical Points B Tangents to a Curve C Increasing and Decreasing Functions Case I Functions with Continuous Derivatives Case II Functions Whose Derivatives Have Discontinuities D Maximum, Minimum, Concavity, and Inflection Points: Definitions E Maximum, Minimum, and Inflection Points: Curve Sketching Case I Functions That Are Everywhere Differentiable Case II Functions Whose Derivatives May Not Exist Everywhere F Global Maximum or Minimum Case I Differentiable Functions Case II Functions That Are Not Everywhere Differentiable G Further Aids in Sketching H Optimization: Problems Involving Maxima and Minima I Relating a Function and Its Derivatives Graphically J Motion Along a Line BC ONLY K Motion Along a Curve: Velocity and Acceleration Vectors L Tangent-Line Approximations M Related Rates BC ONLY N Slope of a Polar Curve Practice Exercises Antidifferentiation A Antiderivatives B Basic Formulas BC ONLY C Integration by Partial Fractions BC ONLY D Integration by Parts E Applications of Antiderivatives; Differential Equations Practice Exercises Definite Integrals A Fundamental Theorem of Calculus (FTC); Evaluation of Definite Integral B Properties of Definite Integrals C Definition of Definite Integral as the Limit of a Riemann Sum D The Fundamental Theorem Again E Approximations of the Definite Integral; Riemann Sums E1 Using Rectangles E2 Using Trapezoids E3 Comparing Approximating Sums F Graphing a Function from Its Derivative; Another Look G Interpreting ln x as an Area H Average Value Practice Exercises Applications of Integration to Geometry A Area A1 Area Between Curves A2 Using Symmetry B Volume B1 Solids with Known Cross Sections B2 Solids of Revolution BC ONLY C Arc Length BC ONLY D Improper Integrals Practice Exercises Further Applications of Integration A Motion Along a Straight Line BC ONLY B Motion Along a Plane Curve C Other Applications of Riemann Sums D FTC: Definite Integral of a Rate Is Net Change Practice Exercises Differential Equations A Basic Definitions B Slope Fields BC ONLY C Euler’s Method D Solving First-Order Differential Equations Analytically E Exponential Growth and Decay Case I: Exponential Growth Case II: Restricted Growth BC ONLY Case III: Logistic Growth Practice Exercises 10 Sequences and Series BC ONLY A Sequences of Real Numbers BC ONLY B Infinite Series B1 Definitions B2 Theorems About Convergence or Divergence of Infinite Series B3 Tests for Convergence of Infinite Series B4 Tests for Convergence of Nonnegative Series B5 Alternating Series and Absolute Convergence BC ONLY C Power Series C1 Definitions; Convergence C2 Functions Defined by Power Series C3 Finding a Power Series for a Function: Taylor and Maclaurin Series C4 Approximating Functions with Taylor and Maclaurin Polynomials C5 Taylor’s Formula with Remainder; Lagrange Error Bound C6 Computations with Power Series BC ONLY C7 Power Series over Complex Numbers Practice Exercises 11 Miscellaneous Multiple-Choice Practice Questions 12 Miscellaneous Free-Response Practice Exercises AB Practice Examinations −1, b = 3, so Replace x with xk and replace dx with Δx in the integrand to get the general term in the summation 30 (A) From the integral we get a = Part B 31 (B) Expressed parametrically, x = sin 3θ cos θ, y = sin 3θ sin θ undefined where is sin θ + cos 3θ cos θ = Use your calculator to solve for θ CALCULATOR TIP: Graph in function mode: y = −sin(3x) · sin(x) + 3cos(3x) · cos(x), and find the roots (zeros) of the function on the interval 34 (C) See the figure below The roots of f (x) = x2 − 4x − = (x − 5)(x + 1) are x = −1 and Since areas A and B are equal, therefore, Thus, A calculator yields k = 37 (C) It is given that a(t) = sin t, e−t Use a calculator to find that the object’s maximum speed is 2.217 39 (C) Use the Ratio Test: which is less than if −3 < x < When x = −3, the convergent alternating harmonic series is obtained When x = 3, the divergent harmonic series is obtained 40 (A) Total distance is 41 (A) Arc length is given by dx Here the integrand implies that hence, y = ln x + C Since the curve contains (1,2), = ln + C, which yields C = 42 (C) The velocity functions are v1 = −2t sin (t2 + 1) and When these functions are graphed on a calculator, it is clear that they intersect four times during the first sec, as shown below Section II Free-Response Part A (a) The following table shows x- and y-components of acceleration, velocity, and position: The last line in the table is the answer to part (a) (b) To determine how far above the ground the ball is when it hits the wall, find out when x = 315, and evaluate y at that time (c) The ball’s speed at the moment of impact in part (b) is | v(t) | evaluated at See solution for AB/BC Part B See solution for AB/BC See solution for AB/BC (a) The table below is constructed from the information given in Question (b) f (5.1) ≈ − 2(5.1 − 5) − (5.1 − 5)2 + (5.1 − 5)3 ≈ − 0.2 − 0.005 + 0.001 = 1796 (c) Use Taylor’s Theorem around x = g (x) ≈ − 4x − 2x2 + 8x3 (a) At (−1,8), so the tangent line is y − = 5(x − (−1)) Therefore f (x) ≈ + 5(x + 1) (b) f (3) ≈ + 5(0 + 1) = 13 (c) At (−1,8), For Δx = 0.5, Δy = 0.5(5) = 2.5, so move to (−1 + 0.5, + 2.5) = (−0.5,10.5) At (−0.5,10.5), For Δx = 0.5, Δy = 0.5(8) = 4, so move to (−0.5 + 0.5, 10.5 + 4) Thus f (0) ≈ 14.5 (d) ... textbooks, and now leads workshops for AP teachers Dennis Donovan has been a math teacher for 22 years, teaching AP Calculus for the past 19 years (8 years AB, 11 years BC) He has served as an AP Calculus. .. a power series; • find bounds on the error for estimates based on series Welcome to Barron? ? ?s AP Calculus AB and BC eBook! All equations, tables, graphs, and other illustrations may appear differently... COURSES Calculus AB and BC are both full-year courses in the calculus of functions of a single variable Both courses emphasize: (1) student understanding of concepts and applications of calculus