AP Calculus Premium, 2022-2023: 12 Practice Tests + Comprehensive Review + Online Practice
Book Preface
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.
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.
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
1Functions
A.Definitions
B.Special Functions
C.Polynomial and Other Rational Functions
D.Trigonometric Functions
E.Exponential and Logarithmic Functions
*F.Parametrically Defined Functions
*G.Polar Functions
Practice Exercises
2Limits 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
3Differentiation
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
4Applications 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
L.Tangent-Line Approximations
M.Related Rates
*N.Slope of a Polar Curve
Practice Exercises
5Antidifferentiation
A.Antiderivatives
B.Basic Formulas
*C.Integration by Partial Fractions
*D.Integration by Parts
E.Applications of Antiderivatives; Differential Equations
Practice Exercises
6Definite 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
F.Graphing a Function from Its Derivative; Another Look
G.Interpreting ln x as an Area
H.Average Value
Practice Exercises
7Applications 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
8Further 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
9Differential 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
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
11Miscellaneous Multiple-Choice Practice Questions
12Miscellaneous 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
Index
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