Math Handbook
of Formulas, Processes and Tricks
Calculus
, 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 if you have any comments.
Thank you and best wishes!
Earl
pg. 2
, 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 245 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 245 April 8, 2023
of Formulas, Processes and Tricks
Calculus
, 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 if you have any comments.
Thank you and best wishes!
Earl
pg. 2
, 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 245 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 245 April 8, 2023
