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AP Calculus: With 8 Practice Tests, 15th Edition (2020)

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    So RRNA _ PET UDI \ Vr ok ACK: Rs, K OICE A “AN i \ | L HB For 2020 & 2021 tests Comprehensive content review > Dennis Donovan, M.S., David Bock, M.S., and Shirley 0. Hockett, M.A. { AP" Is a registered trademark of the College Board, which was not 1 involved in the production of, and does not endorse, this product. ————T E

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    Downloaded from StudyXY.com ® + StudyXY Sd Ye. o> \ | iF ’ pr E \ 3 S Stu dy Anything This ContentHas been Posted On StudyXY.com as supplementary learning material. StudyXY does not endrose any university, college or publisher. Allmaterials posted are under the liability of the contributors. wv 8) www.studyxy.com

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    Lb 9. 9 9 9 99 © © 9% N) A i With 8 Practice Tests FIFTEENTH EDITION Dennis Donovan, M.S. Xaverian Brothers High School Westwood, Massachusetts David Bock, M.S. Ithaca High School (retired) Shirley 0. Hockett, M.A. Formerly, Professor of Mathematics Ithaca College BARRON'S AP is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this product. © © © © © © © © © © WO) + StudyXY

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    About the Authors Dennis Donovan has been a math teacher at Xaverian Brothers High School for more than 25 years, teaching AP Calculus, both AB and BC, for more than 20 years. 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 13 Regional Instructor for Texas Instruments. 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.

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    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.

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    © Copyright 2019, 2017, 2015, 2013, 2012, 2010, 2008 by Kaplan, Inc., d/b/a Barron’s Educational Series Prior editions © Copyright 2005, 2002, 1998 by Kaplan, Inc., d/b/a Barron’s Educational Series 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. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this eBook on screen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of the publisher. Published by Kaplan, Inc., d/b/a Barron’s Educational Series 750 Third Avenue New York, NY 10017 www.barronseduc.com ISBN: 978-1-5062-7192-7 International Standard Serial No.: 1940-3089 10987654321

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    Contents Barron’s Essential 5 Introduction The Courses Topic Outline for the AB and BC Calculus Exams The Examinations The Graphing Calculator: Using Your Graphing Calculator on the AP Exam Grading the Examinations This Review Book Flashcards DIAGNOSTIC TESTS Calculus AB Calculus BC TOPICAL REVIEW AND PRACTICE 1 Functions A. Definitions B. Special Functions C. Polynomial and Other Rational Functions D. Trigonometric Functions

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    E. Exponential and Logarithmic Functions F. Parametrically Defined Functions (BC Only) G. Polar Functions (BC Only) Practice Exercises 2 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 3 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 El. Numerically E2. Graphically F. Derivatives of Parametrically Defined Functions (BC Only) G. Implicit Differentiation H. Derivative of the Inverse of a Function I. The Mean Value Theorem J. Indeterminate Forms and L’Hospital’s Rule

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    K. Recognizing a Given Limit as a Derivative Practice Exercises 4 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 K. Motion Along a Curve: Velocity and Acceleration Vectors (BC Only) L. Tangent-Line Approximations

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    M. Related Rates N. Slope of a Polar Curve (BC Only) Practice Exercises 5 Antidifferentiation A. Antiderivatives B. Basic Formulas C. Integration by Partial Fractions (BC Only) D. Integration by Parts (BC Only) E. Applications of Antiderivatives; Differential Equations Practice Exercises 6 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 El. Using Rectangles E2. Using Trapezoids E3. Comparing Approximating Sums F. Graphing a Function from Its Derivative; Another Look G. Interpreting In x as an Area H. Average Value Practice Exercises

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    7 Applications of Integration to Geometry A. Area Al. Area Between Curves A2. Using Symmetry B. Volume B1. Solids with Known Cross Sections B2. Solids of Revolution C. Length of Curve (Arc Length) (BC Only) D. Improper Integrals (BC Only) Practice Exercises 8 Further Applications of Integration A. Motion Along a Straight Line B. Motion Along a Plane Curve (BC Only) C. Other Applications of Riemann Sums D. FTC: Definite Integral of a Rate Is Net Change Practice Exercises 9 Differential Equations A. Basic Definitions B. Slope Fields C. Euler’s Method (BC Only) D. Solving First-Order Differential Equations Analytically E. Exponential Growth and Decay Case I: Exponential Growth Case II: Restricted Growth Case III: Logistic Growth (BC Only)

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    Practice Exercises l0 Sequences and Series (BC Only) A. Sequences of Real Numbers 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 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 C7. Power Series over Complex Numbers Practice Exercises [1 Miscellaneous Multiple-Choice Practice Questions 2 Miscellaneous Free-Response Practice Exercises AB PRACTICE EXAMINATIONS

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    AB1 AB?2 AB 3 BC PRACTICE EXAMINATIONS BC1 BC 2 BC3 Appendix: Formulas and Theorems for Reference Index

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    5 . Barron’s Essential 5 As you review the content in this book to work toward earning that 5 on your AP CALCULUS AB exam, here are five things that you MUST know above everything else: Learn the basic facts: ® derivatives (p. 119) and antiderivatives (p. 217) of common functions; eo the Product (p. 119), Quotient (p. 119), and Chain Rules (p. 120) for finding derivatives; o the midpoint, left and right rectangle, and trapezoid approximations for estimating definite integrals (p. 254); o finding antiderivatives by substitution (p. 219); o the important theorems: Rolle’s Theorem (p. 135), the Mean Value Theorem (p. 134), and especially the Fundamental Theorem of Calculus (p. 247). (Barron’s AP Calculus Flashcards are another great way to study these!) Understand that a derivative is an instantaneous rate of change, and be able to apply that concept to: e using L’Hospital’s Rule to find limits of indeterminate forms (onlygand 2) (p. 136); eo find equations of tangent lines (p. 165);

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    ® determine where a function is increasing/decreasing (p. 166), is concave up/down (p. 167), or has maxima, minima, or points of inflection (pp. 168, 173); ® analyze the speed, velocity, and acceleration of an object in motion (p. 183); ® solve related rates problems (p. 191), using implicit differentiation (p. 131) when necessary. Understand that integrals represent accumulation functions based on antiderivatives, and be able to apply those concepts to: eo the average value of a function (p. 267); ® area (p. 287) and volume (p. 294); ® position of object in motion and distance traveled (p. 341); eo total amount when given the rate of accumulation (p. 346); eo differential equations, including solutions and slope fields (p. 359). Be able to apply any of the above calculus concepts to functions defined algebraically, graphically, or in tables. Be able to maximize your score on the exam by: e answering all the multiple-choice questions;

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    ® knowing how and when to use your calculator, and when not to; o understanding what work you need to show; ® knowing how to explain, interpret, and justify answers when a question requires that. (The free-response solutions in this book model such answers.) As you review the content in this book to work toward earning that 5 on your AP CALCULUS BC exam, here are five things that you MUST know above everything else: Master the Essential 5 listed for the AB Calculus Exam. These form the core for questions that determine your AB subscore, and provide the essential knowledge base you’ll need for questions related to the additional BC topics. Understand how to extend AB Calculus concepts to more advanced situations, including: ® using limits to analyze improper integrals (p. 303); ® solving logistic differential equations (p. 379) and estimating solutions using Euler’s method (p. 365); o finding antiderivatives using integration by parts (p. 225) or partial fractions (p. 224); o finding length of curve (arc length) (p. 301).

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