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AP Calculus Premium, 2025: Prep Book with 12 Practice Tests + Comprehensive Review (2025)

Ensure exam success with AP Calculus Premium, 2025: Prep Book with 12 Practice Tests + Comprehensive Review (2025) , a trusted certification guide packed with high-yield content and practice exercises.

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    AP ® is a registered trademark of the College Board, which is not affiliated with Barron’s and was not involved in the production of, and does not endorse, this product. © Copyright 2024, 2023, 2022, 2019, 2017, 2015, 2013, 2012, 2010, 2008 by Kaplan North America, LLC, d/b/a Barron’s Educational Series Prior editions © Copyright 2005, 2002, 1998 by Kaplan North America, LLC, 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, including but not limited to generative artificial intelligence (gen AI) systems and machine learning systems, 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 North America, LLC, d/b/a Barron’s Educational Series 1515 W. Cypress Creek Road Fort Lauderdale, FL 33309 www.barronseduc.com ISBN: 978-1-5062-9169-7 10 9 8 7 6 5 4 3 2 1

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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 T 3 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.

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    Table of Contents Review material on topics covered only in Calculus BC is indicated by an asterisk (*). How to Use This Book Barron’s Essential 5 Introduction Content Areas Exam Format Scoring of the Exams Using Your Graphing Calculator on the AP Exam DIAGNOSTIC TESTS Diagnostic Test Calculus AB Answer Explanations Diagnostic Test Calculus BC Answer Explanations 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 *F. Parametrically Defined Functions

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    *G. Polar Functions Practice Exercises Answer Explanations 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 Answer Explanations 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 Answer Explanations 4 Applications of Differential Calculus A. Slope; Critical Points B. Tangents to a Curve

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    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 L. Tangent-Line Approximations M. Related Rates *N. Slope of a Polar Curve Practice Exercises Answer Explanations 5 Antidifferentiation A. Antiderivatives B. Basic Formulas *C. Integration by Partial Fractions *D. Integration by Parts E. Applications of Antiderivatives; Differential Equations Practice Exercises Answer Explanations 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

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    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 Answer Explanations 7 Applications of Integration to Geometry A. Area A1. 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 Answer Explanations 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 Answer Explanations 9 Differential Equations A. Basic Definitions B. Slope Fields

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    *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 Answer Explanations 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 Answer Explanations 11 Miscellaneous Multiple-Choice Practice Questions Answer Explanations 12 Miscellaneous Free-Response Practice Exercises Answer Explanations

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    AB PRACTICE TESTS AB Practice Test 1 Answer Explanations AB Practice Test 2 Answer Explanations BC PRACTICE TESTS BC Practice Test 1 Answer Explanations BC Practice Test 2 Answer Explanations Appendix: Formulas and Theorems for Reference Visit Barron’s Online Learning Hub for more full-length practice tests.

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    How to Use This Book This book provides comprehensive review and extensive practice for both of the AP exams in Calculus. It is based on the latest Course and Exam Description published by the College Board and covers the topics listed there for both AP Calculus AB and AP Calculus BC. Introduction and Diagnostic Tests Start with the Introduction, which provides an overview of the AP Calculus AB and BC courses and exams. Then, proceed to the diagnostic tests, which allow you to assess your strengths and areas for improvement. There is one AB diagnostic and one BC diagnostic, and both are followed by solutions keyed to the corresponding 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 answer explanations. This review is followed by further practice. Chapter 11 includes a set of multiple-choice questions on miscellaneous topics and clear answer explanations. Chapter 12 features a set of miscellaneous free-response problems that are intended to be similar to those in Section II of the AP exams. They are followed by detailed solutions. In this book, review material on topics covered only in Calculus BC is indicated by a BC icon and a vertical lines, as are both multiple-choice questions and free-response-type problems that are likely to only appear on a BC exam.

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    Practice Tests This book concludes with two full-length AB practice tests and two full- length BC practice tests, all of which mirror the actual exams in format, content, and level of difficulty. Each test is followed by detailed answer explanations. Online Practice There are also six additional full-length practice tests online. You may take these tests in practice (untimed) mode or in timed mode. Three of these online tests mirror the AB test, and the other three mirror the BC test. All questions are answered and explained. 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. For Teachers The teacher who uses this book with a class may profitably do so in 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 exam time. The practice tests will also be very helpful when reviewing toward the end of the year.

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    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: Images Learn the basic facts: derivatives ( p. 99) and antiderivatives ( p. 191) of common functions; the Product ( p. 99), Quotient ( p. 99), and Chain Rules ( p. 100) for finding derivatives; the midpoint, left and right rectangle, and trapezoid approximations for estimating definite integrals ( p. 224); finding antiderivatives by substitution ( p. 202); the important theorems: Rolle’s Theorem ( p. 115), the Mean Value Theorem ( p. 114), and especially the Fundamental Theorem of Calculus ( p. 217). Imag es 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 Images and Images) ( p. 116); find equations of tangent lines ( p. 141); determine where a function is increasing/decreasing ( p. 143), is concave up/down ( p. 144), or has maxima, minima, or points of inflection ( pp. 145, 150); analyze the speed, velocity, and acceleration of an object in motion ( p. 160); solve related rates problems ( p. 168), using implicit differentiation ( p. 111) when necessary.

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    Imag es Understand that integrals represent accumulation functions based on antiderivatives, and be able to apply those concepts to: the average value of a function ( p. 237); area ( p. 255) and volume ( p. 262); the position of objects in motion and distance traveled ( p. 307); total amount when given the rate of accumulation ( p. 313); differential equations, including solutions and slope fields ( p. 325). Imag es Be able to apply any of the aforementioned calculus concepts to functions defined algebraically, graphically, or in tables. Images 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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    BARRON’S ESSENTIAL 5 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: Imag es Review and practice 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. Imag es Understand how to extend AB Calculus concepts to more advanced situations, including: •using limits to analyze improper integrals ( p. 271); •solving logistic differential equations ( p. 345) and estimating solutions using Euler’s method ( p. 331); •finding antiderivatives using integration by parts ( p. 199) or partial fractions ( p. 198); •finding length of curve (arc length) ( p. 269). Imag es Be able to apply calculus concepts to parametrically defined functions ( pp. 62, 109, 309) and polar functions ( pp. 66, 170, 260) . Imag es Know how to analyze the position, velocity, speed, acceleration, and distance traveled for an object in motion in two dimensions by applying calculus concepts to vectors ( pp. 162, 309) .

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