Math Handbook of Formulas, Processes and Tricks (www.us) Calculus Prepared by: Earl L. Whitney, FSA, MAAA Version 4.5 January 2, 2019 Copyright 2008‐19, 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.com if you have any comments.
Thank you and best wishes! Cover art by Rebecca Williams, Earl Twitter handle: @jolteonkitty Version 4.5 Page 2 of 236 January 2, 2019 www.com 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., Extrema) 33 Inflection Points 34 Special Case: Extrema and Inflection Points of Polynomials 35 Key Points on f(x), f'(x) and f''(x) 38 Curve Sketching 43 Determining the Shape of a Curve Based On Its Derivatives 44 Rolles's Theorem and the Mean Value Theorem (MVT) 45 Related Rates 48 Kinematics (Particle Motion) 50 Differentials 51 Curvature 52 Newton's Method Chapter 4: Integration 54 Indefinite Integration (Antiderivatives) 55 Exponential and Logarithmic Functions 55 Trigonometric Functions 58 Inverse Trigonometric Functions 60 Selecting the Right Function for an Intergral Version 4.5 Page 3 of 236 January 2, 2019 www.com Calculus Handbook Table of Contents Page Description Chapter 5: Techniques of Integration 61 u ‐Substitution 63 Integration by Partial Fractions 66 Integration by Parts 70 Integration by Parts ‐ Tabular Method 71 Integration by Trigonometric Substitution 72 Impossible Integrals Chapter 6: Hyperbolic Functions 73 Definitions 74 Identities 75 Relationship to Trigonometric Functions 76 Inverse Hyperbolic Functions 77 Graphs of Hyperbolic Functions and Their Inverses 78 Derivatives 79 Integrals Chapter 7: Definite Integrals 81 Riemann Sums 86 Rules of Definite Integration 86 Fundamental Theorems of Calculus 88 Properties of Definite Integrals 89 Solving Definite Integrals with Directed Line Segments 90 u ‐Subsitution 92 Special Techniques for Evaluation 94 Derivative of an Integral Chapter 8: Applications of Integration 95 Area Under a Curve 96 Area Between Curves 97 Area in Polar Form 99 Areas of Limacons 101 Arc Length 104 Comparison of Formulas for Rectangular, Polar and Parametric Forms 105 Area of a Surface of Revolution 106 Volumes of Solids of Revolution Chapter 9: Improper Integrals 112 Definite Integrals with Infinite Limits of Integration 113 Definite Integrals with Discontinuous Integrands Version 4.5 Page 4 of 236 January 2, 2019 www.com Calculus Handbook Table of Contents Page Description Chapter 10: Differential Equations 114 Definitions 115 Separable First Order Differential Equations 117 Slope Fields 118 Logistic Function 119 Numerical Methods Chapter 11: Vector Calculus 123 Introduction 123 Special Unit Vectors 123 Vector Components 124 Properties of Vectors 125 Dot Product 126 Cross Product 128 Triple Products 129 Kinematics (Particle Motion) 130 Gradient 131 Divergence 132 Curl 133 Laplacian Chapter 12: Sequences 134 Definitions and Types of Sequences 135 More Definitions and Theorems 136 Limits (Convergence and Divergence) 137 Basic Recursive Sequence Theory Chapter 13: Series 141 Introduction 142 Key Properties 142 n‐th Term Convergence Theorems 142 Power Series 143 Telescoping Series 144 Geometric Series 145 Estimating the Value of Series with Positive Terms 146 Riemann Zeta Function (p ‐Series) 150 Bernoulli Numbers 152 Convergence Tests 157 Alternating Series 159 Radius and Interval of Convergence of Power Series 162 Summary of Convergence/Divergence Tests Version 4.5 Page 5 of 236 January 2, 2019 www.com Calculus Handbook Table of Contents Page Description Chapter 14: Taylor and MacLaurin Series 163 Taylor Series 163 MacLaurin Series 165 LaGrange Remainder Chapter 15: Miscellaneous Cool Stuff 166 e 167 Derivation of Euler's Formula 169 Logarithms of Negative Real Numbers and Complex Numbers 170 What Is i i 171 z Derivative of e to a Complex Power (e ) 172 Derivatives of a Circle 173 Derivatives of a Ellipse 174 Derivatives of a Hyperbola 175 Derivative of: (x+y)3=x3+y3 176 Inflection Points of the PDF of the Normal Distribution Appendices 177 Appendix A: Key Definitions 197 Appendix B: Key Theorems 201 Appendix C: List of Key Derivatives and Integrals 208 Appendix D: Key Functions and Their Derivatives 212 Appendix E: Geometry and Trigonometry Formulas 217 Appendix F: Polar and Parametric Equations 228 Appendix G: Interesting Series 229 Index Useful Websites Mathguy.us – Developed specifically for math students from Middle School to College, based on the author's extensive experience in professional mathematics in a business setting and in math tutoring. Contains free downloadable handbooks, PC Apps, sample tests, and more.us Wolfram Math World – A premier site for mathematics on the Web. This site contains definitions, explanations and examples for elementary and advanced math topics.5 Page 6 of 236 January 2, 2019 www.com Calculus Handbook Table of Contents Schaum’s Outlines An important student resource for any high school math student is a Schaum’s Outline. Each book in this series provides explanations of the various topics in the course and a substantial number of problems for the student to try.
Many of the problems are worked out in the book, so the student can see how they can be solved. Schaum’s Outlines are available at Amazon.com, Barnes & Noble and other booksellers. Other Useful Books Version 4.5 Page 7 of 236 January 2, 2019 www.com Chapter 1 Functions and Limits Functions Definitions Expression: A meaningful arrangement of mathematical values, variables and operations. Relation: An expression that defines a connection between a set of inputs and a set of outputs.
The set of inputs is called the Domain of the relation. The set of outputs is called the Range of the relation. Function: A relation in which each element in the domain corresponds to exactly one element in the range. One‐to‐One Function: A function in which each element in the range is produced by exactly one element in the domain.
Continuity: A function, , is continuous at iff: o is defined, o lim exists, and Note: lim exists if and only if: → → o lim lim lim. → → → Continuity Rules If and are continuous functions at a point , , and if is a constant, then the following are also true at , : is continuous. Addition is continuous. Subtraction ∙ is continuous.
Scalar Multiplication ∙ is continuous. Multiplication is continuous if 0. Division is continuous if exists. Exponents is continuous if exists.
Roots Note: All polynomial functions are continuous on the interval ∞, ∞ .5 Page 8 of 236 January 2, 2019 www.com Chapter 1 Functions and Limits Types of Discontinuities A Discontinuity occurs at a location where the graph of a relation or function is not connected. A discontinuity that can be “repaired” by adding a single point to the graph. Typically, this will show up as a hole in a graph. In the function , a removable discontinuity exists at 1.
Mathematically, a removable discontinuity is a point at which the limit of at exists but does not equal. That is, lim lim → → Note: a removable discontinuity exists at whether or not exists. A discontinuity that is not removable. Mathematically, a removable discontinuity is a point at which the limit of at does not exist.
This includes: o Jump Discontinuity. A discontinuity at which the limit from the left does not equal the limit from the right. That is, lim lim → → In the function , a jump discontinuity exists at 1. These occur at vertical asymptotes.
In the function , infinite discontinuities exist at 3, 2 .5 Page 9 of 236 January 2, 2019 www.com Chapter 1 Functions and Limits Continuity Examples Case 1 Jump Discontinuity Not continuous Limit does not exist 5 may or may not exist (it does not exist in the graph shown) Case 2 Removable Discontinuity Not continuous Limit exists 5 does not exist Case 3 Removable Discontinuity Not continuous Limit exists 5 exists but does not equal the limit Case 4 No Discontinuity Continuous Limit exists 5 exists and is equal the limit Version 4.5 Page 10 of 236 January 2, 2019 www.com Chapter 1 Functions and Limits Limits Definitions Formal Definition: Let be a function defined on an open interval containing , except possibly at , and let be a real number. Then, the statement: lim → means that for each 0, there exists a 0 such that: 0 | | implies | |. Written using math symbols: ∀ 0 ∃ 0 ∋ 0 | | ⇒ | |. Informal Definition: The limit is the value that a function approaches as the value of the input variable approaches the desired value.
Limits may exist approaching from either the left lim or the right lim. → → If the limits from the left and right are the same (e., they are both equal to ), then the limit exists at and we say lim. → Limit Rules Assuming that each of the requisite limits exist, the following rules apply: lim lim lim Addition of Limits → → → lim lim lim Subtraction of Limits → → → lim ∙ ∙ lim Scalar Multiplication → → lim ∙ lim ∙ lim Multiplication of Limits → → → lim → Division of Limits → → lim lim Powers → → lim lim Roots → → Also, assuming that each of the requisite limits exists, the typical properties of addition and multiplication (e., commutative property, associative property, distributive property, inverse property, etc.) apply to limits.5 Page 11 of 236 January 2, 2019 www.com Chapter 1 Functions and Limits Techniques for Finding Limits Substitution The easiest method, when it works, for determining a limit is substitution.