Conceptual Calculus: A Grand Review Guide for the Ap® Calculus Exam

By Jerry A. Yang

Conceptual Calculus, initially written as an AP Calculus Grand Review, reorients the focus of calculus away from the formulas toward understanding their underlying meanings and implications. Not only does this book give the whys to the hows, it also makes connections between seemingly disparate ideas and simplifies concepts to where even a seventh grader can understand. As a compendium for crammers, advanced students, and new teachers alike, every important topic is fully explained, with appendices included for a quick pocket review. Grouped into six big ideas, Conceptual Calculus is here to answer all of your AP Calculus conceptual needs.

• Language: English
• Publisher: Xlibris US
• Release date: Oct 7, 2015
• ISBN: 9781514408698

Jerry A. Yang

As a leading mathematician in his high school, Jerry A. Yang continues to strive for excellence in both mathematics and math education. Not only did he take AP Calculus AB and BC in his freshman and sophomore year of high school, he also scored a 5 on both exams. Since then, Jerry maintains his mathematical prowess as well as teaching passion by tutoring PAP Pre-calculus, AP Calculus, and AP Statistics students regularly. His impact on the school has been well-recognized by both staff and students alike, and he constantly inspires others to adopt his motto: Learn to Live, Live to Learn.

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Copyright © 2015 by Jerry A. Yang.

All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the copyright owner.

This is a work of fiction. Names, characters, places and incidents either are the product of the author’s imagination or are used fictitiously, and any resemblance to any actual persons, living or dead, events, or locales is entirely coincidental.

Any people depicted in stock imagery provided by Thinkstock are models, and such images are being used for illustrative purposes only.

Certain stock imagery © Thinkstock.

Rev. date: 10/05/2015

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CONTENTS

An asterisk (*) indicates that the corresponding topic is only covered in Calculus BC.

Introduction

Chapter 1 Basic Concepts

I. Limits and Continuity

II. Trig Topics to Review

III. The Polar Coordinate System*

IV. Parametric Equations and Vector Functions*

Chapter 2 Differential Calculus

I. Defining the Derivative

II. Differentiability

III. Basic Differentiation Rules

IV. Mean Value Theorem and Rolle’s Theorem

V. Derivative Tests

VI. Advanced Differentiation

VII. Polar Differentiation*

VIII. Applications of the Derivative: Optimization

IX. Applications of the Derivative: Related Rates

Chapter 3 Integral Calculus

I. The Indefinite Integral

II. The Definite Integral

III. Fundamental Theorem of Calculus (FTC)

IV. Rules of Integration

V. Integration Techniques

VI. Improper Integrals*

VII. Applications of the Integral

VIII. Average Value and the Mean Value Theorem for Integrals

IX. Numerical Approximations of Integrals

Chapter 4 Differential Equations

I. Slope Fields

II. Euler’s Method

III. Solving Separable Differential Equations (SICSE)

IV. Exponential Growth

V. Logistic Growth*

Chapter 5 Series*

I. Special Series

II. Properties of Series

III. Absolute and Conditional Convergence

IV. Convergence Tests

V. Guidelines for Testing for Convergence

VI. Power Series

VII. Series Manipulation

VIII. Taylor/Maclaurin Series

Chapter 6 Applications of Calculus

I. Particle Motion

II. Tabular Data

Chapter 7 About the Ap® Exam

Appendix A Formula Chart

Appendix B Integration Tables

Appendix C Important Theorems and Symbols

Bibliography

DEDICATION

To all AP Calculus students past, present, and future

EPIGRAPH

Learn to Live; Live to Learn

INTRODUCTION

The formal language and structure of mathematics place a large burden on you as the student. Not only do you need to first decode the presentation to reveal the essential ideas but you must also be able to utilize the proper syntax and formal register to communicate your ideas to the rest of the mathematical world. I advise you to remember these three things:

1. Do NOT get caught up in the formulas. Formulas are merely mathematical statements expressing an idea that stems from first principles. Simply grasping the concept that the statement is expressing is not enough to gain a deeper understanding of the formula. Always ask yourself: WHY is this formula the way it is? WHAT is it saying?

2. Make a picture. Pictures provide a way to visualize a concept at hand. In this review, I have provided pictures whenever and wherever appropriate as much as I can. A picture involves translating your ideas onto paper in a meaningful way that is also understandable to other people. Then, your job becomes simply explaining your picture, diagram, or product and how it represents a big concept. Use pictures as much as possible; they provide visual guides to complex concepts.

3. Teach what you learn. Science has proven time and again that the best way to learn something new is to teach it. I encourage you to place yourself in the mind of a teacher and explain each concept step-by-step, as logically as possible, either to yourself or to a willing friend. Learning is a two-way street—by teaching, you are inadvertently forging new paths and strengthening older ones in your understanding of the subject matter.

This work has been written for an audience with previous exposure to the material and is intended for those who wish to review and/or supplement big ideas and concepts that they may have missed in their AP® Calculus course (hence the name). Thus, there are no exercises provided in the text for you to practice with, only examples illustrating big ideas. By no means is this a self-contained work. No proofs are given; all explanations and discussions are geared toward AP® topics. There is nothing on the exam that will not be covered here, and everything in here will be covered on the exam.

Lastly, this review is, of course, a work in progress. As such, there most likely are errors; humans are imperfect, and this author is human. If any said errors (grammatical and/or mathematical) are found, please do not hesitate to contact me. Also, all comments and recommendations for future modifications are always welcome.

Without the support of many people, this work would never have come to fruition. I dedicate this work to them and all my students and friends, past, present, and future. I would also like to thank Dr. Douglas A. Sharp, whose guidance and novel approaches never cease to be an adventure, and Mrs. Nancy Stephenson, whose encouragement has never wavered. Finally, I would like to thank Dr. Marilupe Hren and Mr. Gabriel Torres for placing me on the path of passionate curiosity, learning and discovery that has made this, and many other achievements, possible.

Higher-level mathematics is certainly not for the fainthearted. I fully support every user in his/her endeavors and hope that this document serves its purpose in the long run. Onward!

—J. Yang

September 14, 2015

Chapter 1

BASIC CONCEPTS

I. Limits and Continuity

Up until this point in your mathematical career, you have probably concerned yourself with