5 Steps to a 5: AP Calculus BC 2020

855 Pages·2019·37.845 MB·English

by  William Ma

Copyright © 2019, 2018, 2017, 2016, 2015, 2013 by McGraw-Hill Education.
All rights reserved. Except as permitted under the United States Copyright Act
of 1976, no part of this publication may be reproduced or distributed in any form
or by any means, or stored in a database or retrieval system, without the prior
written permission of the publisher.
ISBN: 978-1-26-045565-6
MHID: 1-26-045565-3
The material in this eBook also appears in the print version of this title: ISBN:
978-1-26-045564-9,
MHID: 1-26-045564-5.
eBook conversion by codeMantra
Version 1.0
All trademarks are trademarks of their respective owners. Rather than put a
trademark symbol after every occurrence of a trademarked name, we use names
in an editorial fashion only, and to the benefit of the trademark owner, with no
intention of infringement of the trademark. Where such designations appear in
this book, they have been printed with initial caps.
McGraw-Hill Education eBooks are available at special quantity discounts to use
as premiums and sales promotions or for use in corporate training programs. To
contact a representative, please visit the Contact Us page at
www.mhprofessional.com.
McGraw-Hill Education, the McGraw-Hill Education logo, 5 Steps to a 5, and
related trade dress are trademarks or registered trademarks of McGraw-Hill
Education and/or its affiliates in the United States and other countries and may
not be used without written permission. All other trademarks are the property of
their respective owners. McGraw-Hill Education is not associated with any
product or vendor mentioned in this book.
AP, Advanced Placement Program, and College Board are registered
trademarks of the College Board, which was not involved in the production of,
and does not endorse, this product.
The series editor was Grace Freedson, and the project editor was Del Franz.
Series design by Jane Tenenbaum.
TERMS OF USE
This is a copyrighted work and McGraw-Hill Education and its licensors reserve
all rights in and to the work. Use of this work is subject to these terms. Except as
permitted under the Copyright Act of 1976 and the right to store and retrieve one
copy of the work, you may not decompile, disassemble, reverse engineer,
reproduce, modify, create derivative works based upon, transmit, distribute,
disseminate, sell, publish or sublicense the work or any part of it without
McGraw-Hill Education’s prior consent. You may use the work for your own
noncommercial and personal use; any other use of the work is strictly prohibited.
Your right to use the work may be terminated if you fail to comply with these
terms.
THE WORK IS PROVIDED “AS IS.” McGRAW-HILL EDUCATION AND
ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO
THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS
TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY
INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA
HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY
WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED
TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR
A PARTICULAR PURPOSE. McGraw-Hill Education and its licensors do not
warrant or guarantee that the functions contained in the work will meet your
requirements or that its operation will be uninterrupted or error free. Neither
McGraw-Hill Education nor its licensors shall be liable to you or anyone else for
any inaccuracy, error or omission, regardless of cause, in the work or for any
damages resulting therefrom. McGraw-Hill Education has no responsibility for
the content of any information accessed through the work. Under no
circumstances shall McGraw-Hill Education and/or its licensors be liable for any
indirect, incidental, special, punitive, consequential or similar damages that
result from the use of or inability to use the work, even if any of them has been
advised of the possibility of such damages. This limitation of liability shall apply
to any claim or cause whatsoever whether such claim or cause arises in contract,
tort or otherwise.
CONTENTS
Dedication and Acknowledgments
Preface
About the Authors
Introduction: The Five-Step Program
STEP 1 Set Up
Your Study Plan
1 What You Need to Know About the AP Calculus BC Exam
1.1 What Is Covered on the AP Calculus BC Exam?
1.2 What Is the Format of the AP Calculus BC Exam?
1.3 What Are the Advanced Placement Exam Grades?
How Is the AP Calculus BC Exam Grade Calculated?
1.4 Which Graphing Calculators Are Allowed for the Exam?
Calculators and Other Devices Not Allowed for the AP
Calculus BC Exam
Other Restrictions on Calculators
2 How to Plan Your Time
2.1 Three Approaches to Preparing for the AP Calculus BC
Exam
Overview of the Three Plans
2.2 Calendar for Each Plan
Summary of the Three Study Plans
STEP 2 Determine Your Test Readiness
3 Take a Diagnostic Exam
3.1 Getting Started!
3.2 Diagnostic Test
3.3 Answers to Diagnostic Test
3.4 Solutions to Diagnostic Test
3.5 Calculate Your Score
Short-Answer Questions
AP Calculus BC Diagnostic Exam
STEP 3 Develop Strategies for Success
4 How to Approach Each Question Type
4.1 The Multiple-Choice Questions
4.2 The Free-Response Questions
4.3 Using a Graphing Calculator
4.4 Taking the Exam
What Do I Need to Bring to the Exam?
Tips for Taking the Exam
STEP 4 Review the Knowledge You Need to Score High
Big Idea 1: Limits
5 Limits and Continuity
5.1 The Limit of a Function
Definition and Properties of Limits
Evaluating Limits
One-Sided Limits
Squeeze Theorem
5.2 Limits Involving Infinities
Infinite Limits (as x → a)
Limits at Infinity (as x → ±∞)
Horizontal and Vertical Asymptotes
5.3 Continuity of a Function
Continuity of a Function at a Number
Continuity of a Function over an Interval
Theorems on Continuity
5.4 Rapid Review
5.5 Practice Problems
5.6 Cumulative Review Problems
5.7 Solutions to Practice Problems
5.8 Solutions to Cumulative Review Problems
Big Idea 2: Derivatives
6 Differentiation
6.1 Derivatives of Algebraic Functions
Definition of the Derivative of a Function
Power Rule
The Sum, Difference, Product, and Quotient Rules
The Chain Rule
6.2 Derivatives of Trigonometric, Inverse Trigonometric,
Exponential, and Logarithmic Functions
Derivatives of Trigonometric Functions
Derivatives of Inverse Trigonometric Functions
Derivatives of Exponential and Logarithmic Functions
6.3 Implicit Differentiation
Procedure for Implicit Differentiation
6.4 Approximating a Derivative
6.5 Derivatives of Inverse Functions
6.6 Higher Order Derivatives
L’Hôpital’s Rule for Indeterminate Forms
6.7 Rapid Review
6.8 Practice Problems
6.9 Cumulative Review Problems
6.10 Solutions to Practice Problems
6.11 Solutions to Cumulative Review Problems
7 Graphs of Functions and Derivatives
7.1 Rolle’s Theorem, Mean Value Theorem, and Extreme
Value Theorem
Rolle’s Theorem
Mean Value Theorem
Extreme Value Theorem
7.2 Determining the Behavior of Functions
Test for Increasing and Decreasing Functions
First Derivative Test and Second Derivative Test for
Relative Extrema
Test for Concavity and Points of Inflection
7.3 Sketching the Graphs of Functions
Graphing without Calculators
Graphing with Calculators
7.4 Graphs of Derivatives
7.5 Parametric, Polar, and Vector Representations
Parametric Curves
Polar Equations
Types of Polar Graphs
Symmetry of Polar Graphs
Vectors
Vector Arithmetic
7.6 Rapid Review
7.7 Practice Problems
7.8 Cumulative Review Problems
7.9 Solutions to Practice Problems
7.10 Solutions to Cumulative Review Problems
8 Applications of Derivatives
8.1 Related Rate
General Procedure for Solving Related Rate Problems
Common Related Rate Problems
Inverted Cone (Water Tank) Problem
Shadow Problem
Angle of Elevation Problem
8.2 Applied Maximum and Minimum Problems
General Procedure for Solving Applied Maximum and
Minimum Problems
Distance Problem
Area and Volume Problem
Business Problems
8.3 Rapid Review
8.4 Practice Problems
8.5 Cumulative Review Problems
8.6 Solutions to Practice Problems
8.7 Solutions to Cumulative Review Problems
9 More Applications of Derivatives
9.1 Tangent and Normal Lines
Tangent Lines
Normal Lines
9.2 Linear Approximations
Tangent Line Approximation (or Linear Approximation)
Estimating the nth Root of a Number
Estimating the Value of a Trigonometric Function of an
Angle
9.3 Motion Along a Line
Instantaneous Velocity and Acceleration
Vertical Motion
Horizontal Motion
9.4 Parametric, Polar, and Vector Derivatives
Derivatives of Parametric Equations
Position, Speed, and Acceleration
Derivatives of Polar Equations
Velocity and Acceleration of Vector Functions
9.5 Rapid Review
9.6 Practice Problems
9.7 Cumulative Review Problems
9.8 Solutions to Practice Problems
9.9 Solutions to Cumulative Review Problems
Big Idea 3: Integrals and the Fundamental Theorems of Calculus
10 Integration
10.1 Evaluating Basic Integrals
Antiderivatives and Integration Formulas
Evaluating Integrals
10.2 Integration by U-Substitution
The U-Substitution Method
U-Substitution and Algebraic Functions
U-Substitution and Trigonometric Functions
U-Substitution and Inverse Trigonometric Functions
U-Substitution and Logarithmic and Exponential
Functions
10.3 Techniques of Integration
Integration by Parts
Integration by Partial Fractions
10.4 Rapid Review
10.5 Practice Problems