Math Handbook of Formulas, Processes and Tricks

Math Handbook of Formulas, Processes and Tricks

(mathguy.us)

Calculus

Prepared by: Earl L. Whitney, FSA, MAAA Version 5.6 April 8, 2023

Copyright 2008-23, Earl Whitney, Reno NV. All Rights Reserved

Note to Students

This Calculus Handbook was developed primarily through work with a number of AP Calculus classes, so it contains what most students need to prepare for the AP Calculus Exam (AB or BC) or a first-year college Calculus course. In addition, a number of more advanced topics have been added to the handbook to whet the student's appetite for higher level study.

It is important to note that some of the tips and tricks noted in this handbook, while generating valid solutions, may not be acceptable to the College Board or to the student's instructor. The student should always check with their instructor to determine if a particular technique that they find useful is acceptable.

Why Make this Handbook?

One of my main purposes for writing this handbook is to encourage the student to wonder, to ask "what about ... ?" or "what if ... ?" I find that students are so busy today that they don't have the time, or don't take the time, to find the beauty and majesty that exists within Mathematics. And, it is there, just below the surface. So be curious and seek it out.

The answers to all of the questions below are inside this handbook, but are seldom taught.

What is oscillating behavior and how does it affect a limit? Is there a generalized rule for the derivative of a product of multiple functions? What's the partial derivative shortcut to implicit differentiation? What are the hyperbolic functions and how do they relate to the trigonometric

functions? When can I simplify a difficult definite integral by breaking it into its even and odd

components? What is Vector Calculus?

Additionally, ask yourself:

Why ... ? Always ask "why?" Can I come up with a simpler method of doing things than I am being taught? What problems can I come up with to stump my friends?

Those who approach math in this manner will be tomorrow's leaders. Are you one of them?

Please feel free to contact me at mathguy.us@ if you have any comments.

Thank you and best wishes! Earl

Cover art by Rebecca Williams, Twitter handle: @jolteonkitty

Version 5.6

Page 2 of 242

April 8, 2023

Calculus Handbook Table of Contents

Page Description

Chapter 1: Functions and Limits

8

Functions

10

Continuity Examples

11

Limits

12

Techniques for Finding Limits

14

Indeterminate Forms

16

When Limits Fail to Exist

Chapter 2: Differentiation

17

Definition, Basic Rules, Product Rule

18

Quotient, Chain and Power Rules; Exponential and Logarithmic Functions

19

Trigonometric and Inverse Trigonometric Functions

23

Generalized Product Rule

25

Inverse Function Rule

26

Partial Differentiation

27

Implicit Differentiation

30

Logarithmic Differentiation

Chapter 3: Applications of Derivatives

31

Maxima and Minima (i.e., Extrema)

33

Inflection Points

34

Special Case: Extrema and Inflection Points of Polynomials

35

Relationship Among f(x), f'(x) and f''(x)

39

Curve Sketching

44

Determining the Shape of a Curve Based On Its Derivatives

45

Rolles's Theorem and the Mean Value Theorem (MVT)

46

Related Rates

49

Kinematics (Particle Motion)

52

Differentials

53

Curvature

54

Newton's Method

Chapter 4: Integration

56

Indefinite Integration (Antiderivatives)

57

Exponential and Logarithmic Functions

57

Trigonometric Functions

60

Inverse Trigonometric Functions

62

Selecting the Right Function for an Intergral

Version 5.6

Page 3 of 242

April 8, 2023

Calculus Handbook Table of Contents

Page Description

Chapter 5: Techniques of Integration

63

u -Substitution

65

Integration by Partial Fractions

68

Integration by Parts

72

Integration by Trigonometric Substitution

74

Gamma Function

76

Beta Function

78

Impossible Integrals

Chapter 6: Hyperbolic Functions

79

Definitions

80

Identities

81

Relationship to Trigonometric Functions

82

Inverse Hyperbolic Functions

83

Graphs of Hyperbolic Functions and Their Inverses

84

Derivatives

85

Integrals

Chapter 7: Definite Integrals

87

Riemann Sums

92

Rules of Definite Integration

92

Fundamental Theorems of Calculus

94

Properties of Definite Integrals

95

Solving Definite Integrals with Directed Line Segments

96

u -Subsitution

98

Special Techniques for Evaluation

100

Derivative of an Integral

Chapter 8: Applications of Integration

101

Area Under a Curve

102

Area Between Curves

103

Area in Polar Form

105

Areas of Limacons

107

Arc Length

110

Comparison of Formulas for Rectangular, Polar and Parametric Forms

111

Area of a Surface of Revolution

112

Volumes of Solids of Revolution

Chapter 9: Improper Integrals

118

Definite Integrals with Infinite Limits of Integration

119

Definite Integrals with Discontinuous Integrands

Version 5.6

Page 4 of 242

April 8, 2023

Calculus Handbook Table of Contents

Page Description

Chapter 10: Differential Equations

120

Definitions

121

Separable First Order Differential Equations

123

Slope Fields

124

Logistic Function

125

Numerical Methods

Chapter 11: Vector Calculus

129

Introduction

129

Special Unit Vectors

129

Vector Components

130

Properties of Vectors

131

Dot Product

132

Cross Product

134

Triple Products

135

Kinematics (Particle Motion)

136

Gradient

137

Divergence

138

Curl

139

Laplacian

Chapter 12: Sequences

140

Definitions and Types of Sequences

141

More Definitions and Theorems

142

Limits (Convergence and Divergence)

143

Basic Recursive Sequence Theory

Chapter 13: Series

147

Introduction

148

Key Properties

148

n-th Term Convergence Theorems

148

Power Series

149

Telescoping Series

150

Geometric Series

151

Estimating the Value of Series with Positive Terms

152

Riemann Zeta Function (p -Series)

156

Bernoulli Numbers

158

Convergence Tests

163

Alternating Series

165

Radius and Interval of Convergence of Power Series

168

Summary of Convergence/Divergence Tests

Version 5.6

Page 5 of 242

April 8, 2023

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download