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AP Calculus Premium, 2022-2023 (2021)

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    THE TRUSTED NAME IN TEST PREP 4 P 12 full-length practice tests with detailed answer explanations Online practice with a timed test option and scoring Comprehensive review and practice for all topics on the exam Expert tips plus Barron's “Essential 5” things you need to know Dennis Donovan, M.S., W. ONLINE PRACTICE David Bock, M.S., and Shirley O. Hockett, M AP® isa Tetered ple’ of the College Board, which was not involved

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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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    BARRON'S THE TRUSTED NAME IN TEST PREP Premium 2022-2023 Dennis Donovan, M.S. David Bock, M.S. Shirley O. Hockett, M.A.

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    Continue Your Practice Online To access Barron's Online Learning Hub, go to: online.barronsbhooks.com BARRON'S

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    © Copyright 2022, 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. No part of this publication may be reproduced or distributed in any form or by any means without the written permission of the copyright owner. Published by Kaplan, Inc., d/b/a Barron’s Educational Series 1515 W Cypress Creek Road Fort Lauderdale, FL. 33309 www.barronseduc.com ISBN: 978-1-5062-7959-6 10987654321 Kaplan, Inc., d/b/a Barron’s Educational Series print books are available at special quantity discounts to use for sales promotions, employee premiums, or educational purposes. For more information or to purchase books, please call the Simon & Schuster special sales department at 866-506-1949. + StudyXY

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    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 T® 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. 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. + StudyXxy

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    Review material on topics covered only in Calculus BC is indicated by an asterisk (*). How To Use This Book 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 DIAGNOSTIC TESTS Diagnostic Test Calculus AB Diagnostic Test Calculus BC TOPICAL REVIEW AND PRACTICE 1 Functions A. Definitions B. Special Functions C. Polynomial and Other Rational Functions D. Trigonometric Functions E. Exponential and Logarithmic Functions + StudyXxy

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    *F. Parametrically Defined Functions *G. Polar Functions 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 E1. Numerically E2. Graphically *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’Hospital’s Rule 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

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    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 L. Tangent-Line Approximations M. Related Rates *N. Slope of a Polar Curve Practice Exercises 5 Antidifferentiation A. Antiderivatives B. Basic Formulas *C. Integration by Partial Fractions *D. Integration by Parts E. Applications of Antiderivatives; Differential Equations Practice Exercises 6 Definite Integrals A. Fundamental Theorem of Calculus (FTC); Evaluation of Definite Integrals 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

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    F. Graphing a Function from Its Derivative; Another Look G. Interpreting In x as an Area H. Average Value Practice Exercises 7 Applications of Integration to Geometry A. Area Al. Area Between Curves A2. Using Symmetry *A3. Region Bounded by Polar Curve B. Volume B1. Solids with Known Cross Sections B2. Solids of Revolution *C. Length of Curve (Arc Length) *D. Improper Integrals Practice Exercises 8 Further Applications of Integration A. Motion Along a Straight Line *B. Motion Along a Plane Curve 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 D. Solving First-Order Differential Equations Analytically E. Exponential Growth and Decay Case I: Exponential Growth Case II: Restricted Growth *Case III: Logistic Growth Practice Exercises

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    10 *Sequences and Series 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 11 Miscellaneous Multiple-Choice Practice Questions 12 Miscellaneous Free-Response Practice Exercises AB PRACTICE TESTS AB Practice Test 1 AB Practice Test 2 AB Practice Test 3 BC PRACTICE TESTS BC Practice Test 1 BC Practice Test 2 BC Practice Test 3 Appendix: Formulas and Theorems for Reference + StudyXxy

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    Index

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    How To Use This Book Diagnostic Tests The diagnostic tests for both AB and BC Calculus are practice AP tests. They are followed by solutions keyed to the corresponding topical review chapter. Review and Practice “Topical Review and Practice” includes 10 chapters with notes on the main topics of the Calculus AB and BC syllabi and numerous carefully worked-out examples. Each chapter concludes with a set of multiple-choice questions, usually divided into calculator and no-calculator sections, followed immediately by answers and solutions. This review is followed by further practice: (1) Chapter 11, which includes a set of multiple-choice questions on miscellaneous topics and an answer key and answer explanations; (2) Chapter 12, a set of miscellaneous free-response problems that are intended to be similar to those in Section II of the AP examinations. They are followed by detailed solutions. In this ebook, review material on topics covered only in Calculus BC is indicated by an asterisk (*), as are both multiple-choice questions and free- response-type problems that are likely to occur only on a BC examination. Practice Tests The next part of this book, titled “AB Practice Tests” and “BC Practice Tests,” has three AB and three BC practice tests that simulate the actual AP examinations. Each is followed by answers and explanations.

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    Online Practice In addition to the two diagnostic tests and six practice tests within this book, there are also four full-length online practice tests. You may take these tests in practice (or untimed) mode or in timed mode. Two of these online tests mirror the AB test, and the other two mirror the BC test. Detailed answers and explanations are provided for every question. Refer to the card at the beginning of this book, which provides instructions for accessing these online tests. For Teachers The teacher who uses this book with a class may profitably do so in any of several ways. If the book is used throughout a year’s course, the teacher can assign all or part of each set of multiple-choice questions and some miscellaneous exercises after the topic has been covered. These sets can also be used for review purposes shortly before examination time. The practice tests will also be very helpful when reviewing toward the end of the year. Teachers may also assemble examinations by choosing appropriate problems from the sample miscellaneous practice questions in Chapters 11 and 12. For Students Students who use this book independently will improve their performance by studying the illustrative examples carefully and trying to complete practice problems before referring to the solutions.

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    bd 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: 1 Learn the basic facts: + derivatives (p. 115) and antiderivatives (p. 213) of common functions; + the Product (p. 115), Quotient (p. 115), and Chain Rules (p. 116) for finding derivatives; + the midpoint, left and right rectangle, and trapezoid approximations for estimating definite integrals (p. 248); + finding antiderivatives by substitution (p. 224); + the important theorems: Rolle’s Theorem (p. 132), the Mean Value Theorem (p. 131), and especially the Fundamental Theorem of Calculus (p. 241). 2 Understand that a derivative is an instantaneous rate of change, and be able to apply that concept to: + use L’Hospital’s Rule to find limits of indeterminate forms (only 2 and 2) (p. 133); + find equations of tangent lines (p. 161); + determine where a function is increasing/decreasing (p. 163), is concave up/down (p. 164), or has maxima, minima, or points of inflection (pp. 165, 170); + analyze the speed, velocity, and acceleration of an object in motion (p. 180); + solve related rates problems (p. 188), using implicit differentiation (p. 128) when necessary.

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    3 Understand that integrals represent accumulation functions based on antiderivatives, and be able to apply those concepts to: + the average value of a function (p. 261); + area (p. 283) and volume (p. 290); + the position of objects in motion and distance traveled (p. 337); + total amount when given the rate of accumulation (p. 344); + differential equations, including solutions and slope fields (p. 357). 4 Be able to apply any of the above calculus concepts to functions defined algebraically, graphically, or in tables. 5 Be able to maximize your score on the exam by: + answering all the multiple-choice questions; + knowing how and when to use your calculator, and when not to; + 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.)

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