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

Name: AP Calculus: With 8 Practice Tests, 15th Edition (2020)
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Evaluate your certification readiness with AP Calculus: With 8 Practice Tests, 15th Edition (2020), a well-structured revision test.

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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 T3 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.

© 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
10 9 8 7 6 5 4 3 2 1

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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
E1. 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
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

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7 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
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
10 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
11 Miscellaneous Multiple-Choice Practice Questions
12 Miscellaneous Free-Response Practice Exercises
AB PRACTICE EXAMINATIONS

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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:
1
Learn the basic facts:
● derivatives (p. 119) and antiderivatives (p. 217) of common
functions;
● the Product (p. 119), Quotient (p. 119), and Chain Rules (p.
120) for finding derivatives;
● the midpoint, left and right rectangle, and trapezoid
approximations for estimating definite integrals (p. 254);
● finding antiderivatives by substitution (p. 219);
● 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!)
2
Understand that a derivative is an instantaneous rate of
change, and be able to apply that concept to:
● using L’Hospital’s Rule to find limits of indeterminate forms
(only and ) (p. 136);
● find equations of tangent lines (p. 165);
0
0
∞
∞

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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.
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. 267);
● area (p. 287) and volume (p. 294);
● position of object in motion and distance traveled (p. 341);
● total amount when given the rate of accumulation (p. 346);
● differential equations, including solutions and slope fields (p.
359).
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;

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● 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.)
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:
1
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.
2
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);
● finding antiderivatives using integration by parts (p. 225) or
partial fractions (p. 224);
● finding length of curve (arc length) (p. 301).

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

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