Math Handbook of Formulas, Processes and Tricks

1 Copyright 2008 22, Earl Whitney, Reno NV. All Rights Reserved math Handbook of Formulas, Processes and Tricks ( ) Calculus Prepared by: Earl L. Whitney, FSA, MAAA Version February 1, 2022 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.

2 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?

3 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 if you have any comments.

4 Thank you and best wishes! Earl Cover art by Rebecca Williams, Twitter handle: @jolteonkitty Version 2 of 236 February 1, 2022 Page DescriptionChapter 1: Functions and Limits8 Functions10 Continuity Examples11 Limits12 Techniques for Finding Limits14 Indeterminate Forms16 When Limits Fail to ExistChapter 2: Differentiation17 Definition, Basic Rules, Product Rule18 Quotient, Chain and Power Rules; Exponential and Logarithmic Functions19 Trigonometric and Inverse Trigonometric Functions23 Generalized Product Rule25 Inverse Function Rule26 Partial Differentiation27 Implicit Differentiation30 Logarithmic DifferentiationChapter 3: Applications of Derivatives31 Maxima and Minima ( , Extrema)33 Inflection Points34 Special Case.

5 Extrema and Inflection Points of Polynomials35 Key Points on f(x), f'(x) and f''(x)38 Curve Sketching43 Determining the Shape of a Curve Based On Its Derivatives44 Rolles's Theorem and the Mean Value Theorem (MVT)45 Related Rates48 Kinematics (Particle Motion)50 Differentials51 Curvature52 Newton's MethodChapter 4: Integration54 Indefinite Integration (Antiderivatives)55 Exponential and Logarithmic Functions55 Trigonometric Functions58 Inverse Trigonometric Functions60 Selecting the Right Function for an IntergralCalculus HandbookTable of ContentsVersion 3 of 236 February 1, 2022 Calculus HandbookTable of ContentsPage DescriptionChapter 5: Techniques of Integration61u Substitution63 Integration by Partial Fractions66 Integration by Parts70 Integration by Parts Tabular Method71 Integration by Trigonometric Substitution72 Impossible IntegralsChapter 6.

6 Hyperbolic Functions73 Definitions74 Identities75 Relationship to Trigonometric Functions76 Inverse Hyperbolic Functions77 Graphs of Hyperbolic Functions and Their Inverses78 Derivatives79 IntegralsChapter 7: Definite Integrals81 Riemann Sums86 Rules of Definite Integration86 Fundamental Theorems of Calculus88 Properties of Definite Integrals89 Solving Definite Integrals with Directed Line Segments90u Subsitution92 Special Techniques for Evaluation94 Derivative of an IntegralChapter 8: Applications of Integration95 Area Under a Curve96 Area Between Curves97 Area in Polar Form99 Areas of Limacons101 Arc Length104 Comparison of Formulas for Rectangular, Polar and Parametric Forms105 Area of a Surface of Revolution106 Volumes of Solids of RevolutionChapter 9: Improper Integrals112 Definite Integrals with Infinite Limits of Integration113 Definite Integrals with Discontinuous IntegrandsVersion 4 of 236 February 1, 2022 Calculus HandbookTable of ContentsPage DescriptionChapter 10.

7 Differential Equations114 Definitions115 Separable First Order Differential Equations117 Slope Fields118 Logistic Function119 Numerical MethodsChapter 11: Vector Calculus123 Introduction123 Special Unit Vectors123 Vector Components124 Properties of Vectors125 Dot Product126 Cross Product128 Triple Products129 Kinematics (Particle Motion)130 Gradient131 Divergence132 Curl133 LaplacianChapter 12: Sequences134 Definitions and Types of Sequences135 More Definitions and Theorems136 Limits (Convergence and Divergence)137 Basic Recursive Sequence TheoryChapter 13: Series141 Introduction142 Key Properties142 n th Term Convergence Theorems142 Power Series143 Telescoping Series144 Geometric Series145 Estimating the Value of series with Positive Terms146 Riemann Zeta Function (p series )150 Bernoulli Numbers152 Convergence Tests157 Alternating Series159 Radius and Interval of Convergence of Power Series162 Summary of Convergence/Divergence TestsVersion 5 of 236 February 1, 2022 Calculus HandbookTable of ContentsPage DescriptionChapter 14: taylor and MacLaurin Series163 taylor Series163 MacLaurin Series165 LaGrange RemainderChapter 15.

8 Miscellaneous Cool Stuff166e167 Derivation of Euler's Formula169 Logarithms of Negative Real Numbers and Complex Numbers170 What Is ii171 Derivative of e to a Complex Power (ez)172 Derivatives of a Circle173 Derivatives of a Ellipse174 Derivatives of a Hyperbola175 Derivative of: (x+y)3=x3+y3176 Inflection Points of the PDF of the Normal DistributionAppendices177 Appendix A: Key Definitions197 Appendix B: Key Theorems201 Appendix C: List of Key Derivatives and Integrals208 Appendix D: Key Functions and Their Derivatives212 Appendix E: Geometry and Trigonometry Formulas217 Appendix F: Polar and Parametric Equations228 Appendix G: Interesting series 229 IndexUseful math World A premier site for mathematics on the Web.

9 This site contains definitions, explanations and examples for elementary and advanced math topics. 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 6 of 236 February 1, 2022 Calculus HandbookTable of ContentsSchaum s Outlines Other Useful BooksAn 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.

10 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 , Barnes & Noble and other 7 of 236 February 1, 2022 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.