Calculus DeMYSTiFieD, Second Edition | DeMystified

Calculus DeMYSTiFieD, Second Edition

Authors: Steven Krantz

Published: November 2010

eISBN: 9780071743648 0071743642 | ISBN: 9780071743631

Contents

Preface

How to Use This Book

Chapter 1 Basics

1.0 Introductory Remarks

1.1 Number Systems

1.2 Coordinates in One Dimension

1.3 Coordinates in Two Dimensions

1.4 The Slope of a Line in the Plane

1.5 The Equation of a Line

1.6 Loci in the Plane

1.7 Trigonometry

1.8 Sets and Functions

1.8.1 Examples of Functions of a Real Variable

1.8.2 Graphs of Functions

1.8.3 Plotting the Graph of a Function

1.8.4 Composition of Functions

1.8.5 The Inverse of a Function

1.9 A Few Words About Logarithms and Exponentials

Quiz

Chapter 2 Foundations of Calculus

2.1 Limits

2.1.1 One-Sided Limits

2.2 Properties of Limits

2.3 Continuity

2.4 The Derivative

2.5 Rules for Calculating Derivatives

2.5.1 The Derivative of an Inverse

2.6 The Derivative as a Rate of Change

Quiz

Chapter 3 Applications of the Derivative

3.1 Graphing of Functions

3.2 Maximum/Minimum Problems

3.3 Related Rates

3.4 Falling Bodies

Quiz

Chapter 4 The Integral

4.0 Introduction

4.1 Antiderivatives and Indefinite Integrals

4.1.1 The Concept of Antiderivative

4.1.2 The Indefinite Integral

4.2 Area

4.3 Signed Area

4.4 The Area Between Two Curves

4.5 Rules of Integration

4.5.1 Linear Properties

4.5.2 Additivity

Quiz

Chapter 5 Indeterminate Forms

5.1 l'Hôpital's Rule

5.1.1 Introduction

5.1.2 l'Hôpital's Rule

5.2 Other Indeterminate Forms

5.2.1 Introduction

5.2.2 Writing a Product as a Quotient

5.2.3 The Use of the Logarithm

5.2.4 Putting Terms over a Common Denominator

5.2.5 Other Algebraic Manipulations

5.3 Improper Integrals: A First Look

5.3.1 Introduction

5.3.2 Integrals with Infinite Integrands

5.3.3 An Application to Area

5.4 More on Improper Integrals

5.4.1 Introduction

5.4.2 The Integral on an Infinite Interval

5.4.3 Some Applications

Quiz

Chapter 6 Transcendental Functions

6.0 Introductory Remarks

6.1 Logarithm Basics

6.1.1 A New Approach to Logarithms

6.1.2 The Logarithm Function and the Derivative

6.2 Exponential Basics

6.2.1 Facts about the Exponential Function

6.2.2 Calculus Properties of the Exponential

6.2.3 The Number e

6.3 Exponentials with Arbitrary Bases

6.3.1 Arbitrary Powers

6.3.2 Logarithms with Arbitrary Bases

6.4 Calculus with Logs and Exponentials to Arbitrary Bases

6.4.1 Differentiation and Integration of log[Sub(a)]X and a[Sup(x)]

6.4.2 Graphing of Logarithmic and Exponential Functions

6.4.3 Logarithmic Differentiation

6.5 Exponential Growth and Decay

6.5.1 A Differential Equation

6.5.2 Bacterial Growth

6.5.3 Radioactive Decay

6.5.4 Compound Interest

6.6 Inverse Trigonometric Functions

6.6.1 Introductory Remarks

6.6.2 Inverse Sine and Cosine

6.6.3 The Inverse Tangent Function

6.6.4 Integrals in Which Inverse Trigonometric Functions Arise

6.6.5 Other Inverse Trigonometric Functions

6.6.6 An Example Involving Inverse Trigonometric Functions

Quiz

Chapter 7 Methods of Integration

7.1 Integration by Parts

7.2 Partial Fractions

7.2.1 Introductory Remarks

7.2.2 Products of Linear Factors

7.2.3 Quadratic Factors

7.3 Substitution

7.4 Integrals of Trigonometric Expressions

Quiz

Chapter 8 Applications of the Integral

8.1 Volumes by Slicing

8.1.0 Introduction

8.1.1 The Basic Strategy

8.1.2 Examples

8.2 Volumes of Solids of Revolution

8.2.0 Introduction

8.2.1 The Method of Washers

8.2.2 The Method of Cylindrical Shells

8.2.3 Different Axes

8.3 Work

8.4 Averages

8.5 Arc Length and Surface Area

8.5.1 Arc Length

8.5.2 Surface Area

8.6 Hydrostatic Pressure

8.7 Numerical Methods of Integration

8.7.1 The Trapezoid Rule

8.7.2 Simpson's Rule

Quiz

Final Exam

Answers to Quizzes and Final Exam

Bibliography

Index