AP Calculus Premium, 2022-2023 (2021)

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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
10 9 8 7 6 5 4 3 2 1
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.

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.

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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
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

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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
E1. Using Rectangles
E2. Using Trapezoids
E3. Comparing Approximating Sums

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F. Graphing a Function from Its Derivative; Another Look
G. Interpreting ln x as an Area
H. Average Value
Practice Exercises
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
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

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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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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 and )
(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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