AP Calculus Teachers Solution Manual

TEACHER’S

SOLUTIONS MANUAL

ROGER LIPSETT

Brandeis University

CALCULUS

EDITION

William Briggs

University of Colorado at Denver

Lyle Cochran

Whitworth University

Bernard Gillett

University of Colorado at Boulder

With the assistance of

Eric Schulz

Walla Walla Community College

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ISBN-13: 978-0-13-356357-3

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Contents

1 Functions 5

1.1 Review of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Representing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.3 Inverse, Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.4 Trigonometric Functions and Their Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

AP Practice Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

2 Limits 89

2.1 The Idea of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

2.2 Definition of a Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

2.3 Techniques of Computing Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

2.4 Infinite Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

2.5 Limits at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

2.6 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

2.7 Precise Definitions of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

AP Practice Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

3 Derivatives 163

3.1 Introducing the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

3.2 Working with the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

3.3 Rules of Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

3.4 The Product and Quotient Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

3.5 Derivatives of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

3.6 Derivatives as Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

3.7 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

3.8 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

3.9 Derivatives of Logarithmic and Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 254

3.10 Derivatives of Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 263

3.11 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

AP Practice Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

4 Applications of the Derivative 295

4.1 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

4.2 What Derivatives Tell Us . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

4.3 Graphing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

4.4 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

4.5 Linear Approximation and Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

4.6 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396

4.7 L’Hˆopital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

4.8 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

2014 Pearson Education, Inc.

361

AP Calculus AB Scoring Guide Unit 1 MCQs Part B Answers

AP Calculus AB Unit 1 Progress Check: MCQ Part C Answers

AP Biology Practice Exam

AP Calculus Ab 2018 International Practice Exam FRQ Scoring Guidelines

AP Calculus AB Scoring Guide Unit 1 Progress Check PDF

AP CLASSROOM Frequently Asked Questions

FAQs of AP Calculus Teachers Solution Manual

What topics are covered in the AP Calculus Teacher's Solutions Manual?

The AP Calculus Teacher's Solutions Manual covers a wide range of topics essential for the AP Calculus curriculum. Key subjects include functions, limits, derivatives, and integrals, with each section designed to enhance understanding and teaching effectiveness. The manual provides solutions to AP practice questions and detailed explanations of problem-solving techniques. This comprehensive approach ensures that educators can effectively guide their students through the complexities of AP Calculus.

Who is the author of the AP Calculus Teacher's Solutions Manual?

The AP Calculus Teacher's Solutions Manual is authored by Roger Lipsett, a recognized educator associated with Brandeis University. Alongside Lipsett, contributions have been made by William Briggs, Lyle Cochran, and Bernard Gillett, who are affiliated with various universities. Their collective expertise in mathematics education provides a solid foundation for the solutions and instructional strategies presented in the manual.

How does this manual assist teachers in preparing students for the AP Calculus exam?

This manual assists teachers by providing comprehensive solutions and methodologies that align with the AP Calculus exam format. Each solution is crafted to not only show the correct answer but also to explain the reasoning and techniques behind it, which is crucial for student understanding. By using this manual, teachers can enhance their instructional methods, ensuring that their students are well-prepared for the exam's challenges. Additionally, the inclusion of AP practice questions allows for targeted review and assessment.

What is the significance of the topics covered in the manual for AP Calculus students?

The topics covered in the AP Calculus Teacher's Solutions Manual are significant as they form the core concepts necessary for success in AP Calculus. Understanding functions, limits, derivatives, and integrals is essential for students to grasp advanced calculus concepts and apply them effectively. Mastery of these topics not only prepares students for the AP exam but also lays the groundwork for future studies in mathematics and related fields. The manual's structured approach helps students develop critical thinking and problem-solving skills.

AP Calculus Teachers Solution Manual

Key Points

AP Calculus AB Scoring Guide Unit 1 MCQs Part B Answers

AP Calculus AB Unit 1 Progress Check: MCQ Part C Answers

AP Biology Practice Exam

AP Calculus Ab 2018 International Practice Exam FRQ Scoring Guidelines

AP Calculus AB Scoring Guide Unit 1 Progress Check PDF

AP CLASSROOM Frequently Asked Questions

FAQs of AP Calculus Teachers Solution Manual

Related of AP Calculus Teachers Solution Manual

AP Calculus AB Scoring Guide Unit 1 MCQs Part B Answers

AP Calculus AB Unit 1 Progress Check: MCQ Part C Answers

AP Biology Practice Exam

AP Calculus Ab 2018 International Practice Exam FRQ Scoring Guidelines

AP Calculus AB Scoring Guide Unit 1 Progress Check PDF

AP CLASSROOM Frequently Asked Questions