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

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AP®is a registered trademark of the College Board, which is not affiliatedwith Barron’s and was not involved in the production of, and does notendorse, this product.© Copyright 2024, 2023, 2022, 2019, 2017, 2015, 2013, 2012, 2010, 2008by Kaplan North America, LLC, d/b/a Barron’s Educational SeriesPrior editions © Copyright 2005, 2002, 1998 by Kaplan North America,LLC, d/b/a Barron’s Educational Series under the titleHow to Prepare forthe AP Advanced Placement Examination in Calculusand 1995, 1992,1987, 1983, 1971 under the titleHow to Prepare for the AdvancedPlacement Examination: Mathematics.All rights reserved under International and Pan-American CopyrightConventions. By payment of the required fees, you have been granted thenon-exclusive, non-transferable right to access and read the text of thiseBook on screen. No part of this text may be reproduced, transmitted,downloaded, decompiled, reverse engineered, or stored in or introducedinto any information storage and retrieval system, including but not limitedto generative artificial intelligence (gen AI) systems and machine learningsystems, in any form or by any means, whether electronic or mechanical,now known or hereinafter invented, without the express written permissionof the publisher.Published by Kaplan North America, LLC, d/b/a Barron’s EducationalSeries1515 W. Cypress Creek RoadFort Lauderdale, FL 33309www.barronseduc.comISBN: 978-1-5062-9169-710 9 8 7 6 5 4 3 2 1

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About the AuthorsDennis Donovanhas been a math teacher at Xaverian Brothers HighSchool for more than 25 years, teaching AP Calculus, both AB and BC, formore than 20 years. He has served as an AP Calculus Reader and as one ofnine national Question Leaders for the AP Calculus exam. Dennis leadsprofessional development workshops for math teachers as a consultant forthe College Board and a T3Regional Instructor for Texas Instruments.David Bocktaught AP Calculus during his 35 years at Ithaca High School,and served for several years as an Exam Reader for the College Board. Healso taught mathematics at Tompkins Cortland Community College, IthacaCollege, and Cornell University. A recipient of several local, state, andnational teaching awards, Dave has coauthored five textbooks, and nowleads workshops for AP teachers.IN MEMORIAMShirley O. Hockett(1920–2013)Shirley Hocketttaught mathematics for 45 years, first at CornellUniversity and later at Ithaca College, where she was named ProfessorEmerita in 1991. An outstanding teacher, she won numerous awards andauthored six mathematics textbooks. Shirley’s experiences as an ExamReader and Table Leader for AP Calculus led her to write the first ever APCalculus review book, published in 1971. Her knowledge of calculus,attention to detail, pedagogical creativity, and dedication to studentscontinue to shine throughout this book. On behalf of the thousands of APCalculus students who have benefited from her tireless efforts, wegratefully dedicate this review book to Shirley.

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Table of ContentsReview material on topics covered only in Calculus BC is indicatedby an asterisk (*).How to Use This BookBarron’s Essential 5IntroductionContent AreasExam FormatScoring of the ExamsUsing Your Graphing Calculator on the AP ExamDIAGNOSTIC TESTSDiagnostic Test Calculus ABAnswer ExplanationsDiagnostic Test Calculus BCAnswer ExplanationsTOPICAL REVIEW AND PRACTICE1FunctionsA.DefinitionsB.Special FunctionsC.Polynomial and Other Rational FunctionsD.Trigonometric FunctionsE.Exponential and Logarithmic Functions*F.Parametrically Defined Functions

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*G.Polar FunctionsPractice ExercisesAnswer Explanations2Limits and ContinuityA.Definitions and ExamplesB.AsymptotesC.Theorems on LimitsD.Limit of a Quotient of PolynomialsE.Other Basic LimitsF.ContinuityPractice ExercisesAnswer Explanations3DifferentiationA.Definition of DerivativeB.FormulasC.The Chain Rule: The Derivative of a Composite FunctionD.Differentiability and ContinuityE.Estimating a DerivativeE1. NumericallyE2. Graphically*F.Derivatives of Parametrically Defined FunctionsG.Implicit DifferentiationH.Derivative of the Inverse of a FunctionI.The Mean Value TheoremJ.Indeterminate Forms and L’Hospital’s RuleK.Recognizing a Given Limit as a DerivativePractice ExercisesAnswer Explanations4Applications of Differential CalculusA.Slope; Critical PointsB.Tangents to a Curve

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C.Increasing and Decreasing FunctionsCase I. Functions with Continuous DerivativesCase II. Functions Whose Derivatives Have DiscontinuitiesD.Maximum, Minimum, Concavity, and Inflection Points: DefinitionsE.Maximum, Minimum, and Inflection Points: Curve SketchingCase I. Functions That Are Everywhere DifferentiableCase II. Functions Whose Derivatives May Not Exist EverywhereF.Global Maximum or MinimumCase I. Differentiable FunctionsCase II. Functions That Are Not Everywhere DifferentiableG.Further Aids in SketchingH.Optimization: Problems Involving Maxima and MinimaI.Relating a Function and Its Derivatives GraphicallyJ.Motion Along a Line*K.Motion Along a Curve: Velocity and Acceleration VectorsL.Tangent-Line ApproximationsM.Related Rates*N.Slope of a Polar CurvePractice ExercisesAnswer Explanations5AntidifferentiationA.AntiderivativesB.Basic Formulas*C.Integration by Partial Fractions*D.Integration by PartsE.Applications of Antiderivatives; Differential EquationsPractice ExercisesAnswer Explanations6Definite IntegralsA.Fundamental Theorem of Calculus (FTC); Evaluation of Definite IntegralsB.Properties of Definite IntegralsC.Definition of Definite Integral as the Limit of a Riemann SumD.The Fundamental Theorem Again

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E.Approximations of the Definite Integral; Riemann SumsE1. Using RectanglesE2. Using TrapezoidsE3. Comparing Approximating SumsF.Graphing a Function from Its Derivative; Another LookG.Interpreting lnxas an AreaH.Average ValuePractice ExercisesAnswer Explanations7Applications of Integration to GeometryA.AreaA1. Area Between CurvesA2. Using Symmetry*A3. Region Bounded by Polar CurveB.VolumeB1. Solids with Known Cross SectionsB2. Solids of Revolution*C.Length of Curve (Arc Length)*D.Improper IntegralsPractice ExercisesAnswer Explanations8Further Applications of IntegrationA.Motion Along a Straight Line*B.Motion Along a Plane CurveC.Other Applications of Riemann SumsD.FTC: Definite Integral of a Rate Is Net ChangePractice ExercisesAnswer Explanations9Differential EquationsA.Basic DefinitionsB.Slope Fields

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*C.Euler’s MethodD.Solving First-Order Differential Equations AnalyticallyE.Exponential Growth and DecayCase I: Exponential GrowthCase II: Restricted Growth*Case III: Logistic GrowthPractice ExercisesAnswer Explanations10*Sequences and SeriesA.Sequences of Real NumbersB.Infinite SeriesB1. DefinitionsB2. Theorems About Convergence or Divergence of Infinite SeriesB3. Tests for Convergence of Infinite SeriesB4. Tests for Convergence of Nonnegative SeriesB5. Alternating Series and Absolute ConvergenceC.Power SeriesC1. Definitions; ConvergenceC2. Functions Defined by Power SeriesC3. Finding a Power Series for a Function: Taylor and Maclaurin SeriesC4. Approximating Functions with Taylor and Maclaurin PolynomialsC5. Taylor’s Formula with Remainder; Lagrange Error BoundC6. Computations with Power SeriesC7. Power Series over Complex NumbersPractice ExercisesAnswer Explanations11Miscellaneous Multiple-Choice Practice QuestionsAnswer Explanations12Miscellaneous Free-Response Practice ExercisesAnswer Explanations

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AB PRACTICE TESTSAB Practice Test 1Answer ExplanationsAB Practice Test 2Answer ExplanationsBC PRACTICE TESTSBC Practice Test 1Answer ExplanationsBC Practice Test 2Answer ExplanationsAppendix: Formulas and Theorems for ReferenceVisit Barron’s Online Learning Hub for more full-length practicetests.

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How to Use This BookThis book provides comprehensive review and extensive practice for bothof the AP exams in Calculus. It is based on the latest Course and ExamDescription published by the College Board and covers the topics listedthere for both AP Calculus AB and AP Calculus BC.Introduction and Diagnostic TestsStart with the Introduction, which provides an overview of the AP CalculusAB and BC courses and exams. Then, proceed to the diagnostic tests,which allow you to assess your strengths and areas for improvement. Thereis one AB diagnostic and one BC diagnostic, and both are followed bysolutions keyed to the corresponding review chapter.Review and Practice“Topical Review and Practice” includes 10 chapters with notes on the maintopics of the Calculus AB and BC syllabi and numerous carefully worked-out examples. Each chapter concludes with a set of multiple-choicequestions, 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 ofmultiple-choice questions on miscellaneous topics and clear answerexplanations.Chapter 12 features a set of miscellaneous free-responseproblems that are intended to be similar to those in Section II of the APexams. They are followed by detailed solutions.In this book, review material on topics covered only in Calculus BC isindicated by a BC icon and a vertical lines, as are both multiple-choicequestions and free-response-type problems that are likely to only appear ona BC exam.

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Practice TestsThis 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 answerexplanations.Online PracticeThere are also six additional full-length practice tests online. You may takethese tests in practice (untimed) mode or in timed mode. Three of theseonline tests mirror the AB test, and the other three mirror the BC test. Allquestions are answered and explained.For StudentsStudents who use this book independentlywill improve theirperformance by studying the illustrative examples carefully and trying tocomplete practice problems before referring to the solutions.For TeachersThe teacher who uses this book with a classmay profitably do so inseveral ways. If the book is used throughout a year’s course, the teacher canassign all or part of each set of multiple-choice questions and somemiscellaneous exercises after the topic has been covered. These sets canalso be used for review purposes shortly before exam time. The practicetests will also be very helpful when reviewing toward the end of the year.

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BARRON’S ESSENTIAL 5As you review the content in this book to work toward earning that5onyour AP CALCULUSABexam, here are five things that youMUSTknowabove everything else:ImagesLearn 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) forfinding derivatives;the midpoint, left and right rectangle, and trapezoid approximationsfor estimating definite integrals (p. 224);finding antiderivatives by substitution (p. 202);the important theorems: Rolle’s Theorem (p. 115), the Mean ValueTheorem (p. 114), and especially the Fundamental Theorem ofCalculus (p. 217).ImagesUnderstand that a derivative is an instantaneous rate ofchange, and be able to apply that concept to:use L’Hospital’s Rule to find limits of indeterminate forms (onlyImages andImages) (p. 116);find equations of tangent lines (p. 141);determine where a function is increasing/decreasing (p. 143), isconcave up/down (p. 144), or has maxima, minima, or points ofinflection (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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ImagesUnderstand that integrals represent accumulation functionsbased on antiderivatives, and be able to apply those conceptsto: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).ImagesBe able to apply any of the aforementioned calculus concepts tofunctions defined algebraically, graphically, or in tables.ImagesBe able to maximize your score on the exam by:answeringallthe 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 aquestion requires that. (The free-response solutions in this book modelsuch answers.)

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BARRON’S ESSENTIAL 5As you review the content in this book to work toward earning that5onyour AP CALCULUSBCexam, here are five things that youMUSTknowabove everything else:ImagesReview and practice the Essential 5 listed for the AB CalculusExam.These form the core for questions that determine your ABsubscore and provide the essential knowledge base you’ll need forquestions related to the additional BC topics.ImagesUnderstand how to extend AB Calculus concepts to moreadvanced situations, including:•using limits to analyze improper integrals (p. 271);•solving logistic differential equations (p. 345) and estimatingsolutions using Euler’s method (p. 331);•finding antiderivatives using integration by parts (p. 199) orpartial fractions (p. 198);•finding length of curve (arc length) (p. 269).ImagesBe able to apply calculus concepts to parametrically definedfunctions (pp. 62, 109, 309) and polar functions (pp. 66, 170,260).ImagesKnow how to analyze the position, velocity, speed, acceleration,and distance traveled for an object in motion in twodimensions by applying calculus concepts to vectors (pp. 162,309).

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