Calculus Know-It-ALL by Gibilisco, Stan -- Read -- Imperial Library of Trantor

Index

Cover Page Calculus Know-It-ALL Copyright Page About the Author Content Preface Acknowledgments Part 1 Differentiation in One Variable

1 Single-Variable Functions

Mappings Linear Functions Nonlinear Functions “Broken” Functions Practice Exercises

2 Limits and Continuity

Concept of the Limit Continuity at a Point Continuity of a Function Practice Exercises

3 What’s a Derivative?

Vanishing Increments Basic Linear Functions Basic Quadratic Functions Basic Cubic Functions Practice Exercises

4 Derivatives Don’t Always Exist

Let’s Look at the Graph When We Can Differentiate When We Can’t Differentiate Practice Exercises

5 Differentiating Polynomial Functions

Power Rule Sum Rule Summing the Powers Practice Exercises

6 More Rules for Differentiation

Multiplication-by-Constant Rule Product Rule Reciprocal Rule Quotient Rule Chain Rule Practice Exercises

7 A Few More Derivatives

Real-Power Rule Sine and Cosine Functions Natural Exponential Function Natural Logarithm Function Practice Exercises

8 Higher Derivatives

Second Derivative Third Derivative Beyond the Third Derivative Practice Exercises

9 Analyzing Graphs with Derivatives

Three Common Traits Graph of a Quadratic Function Graph of a Cubic Function Graph of the Sine Function Practice Exercises

10 Review Questions and Answers

Part 2 Integration in One Variable

11 What’s an Integral?

Summation Notation Area Defined by a Curve Three Applications Practice Exercises

12 Derivatives in Reverse

Concept of the Antiderivative Some Simple Antiderivatives Indefinite Integral Definite Integral Practice Exercises

13 Three Rules for Integration

Reversal Rule Split-Interval Rule Substitution Rule Practice Exercises

14 Improper Integrals

Variable Bounds Singularity in the Interval Infinite Intervals Practice Exercises

15 Integrating Polynomial Functions

Three Rules Revisited Indefinite-Integral Situations Definite-Integral Situations Practice Exercises

16 Areas between Graphs

Line and Curve Two Curves Singular Curves Practice Exercises

17 A Few More Integrals

Sine and Cosine Functions Natural Exponential Function Reciprocal Function Practice Exercises

18 How Long Is the Arc?

A Chorus of Chords A Monomial Curve A More Exotic Curve Practice Exercises

19 Special Integration Tricks

Principle of Linearity Integration by Parts Partial Fractions Practice Exercises

20 Review Questions and Answers

Part 3 Advanced Topics

21 Differentiating Inverse Functions

A General Formula Derivative of the Arcsine Derivative of the Arccosine Practice Exercises

22 Implicit Differentiation

Two-Way Relations Two-Way Derivatives Practice Exercises

23 The L’Hôpital Principles

Expressions That Tend Toward 0/0 Expressions That Tend Toward ±∞/±∞ Other Indeterminate Limits Practice Exercises

24 Partial Derivatives

Multi-Variable Functions Two Independent Variables Three Independent Variables Practice Exercises

25 Second Partial Derivatives

Two Variables, Second Partials Two Variables, Mixed Partials Three Variables, Second Partials Three Variables, Mixed Partials Practice Exercises

26 Surface-Area and Volume Integrals

A Cylinder A Cone A Sphere Practice Exercises

27 Repeated, Double, and Iterated Integrals

Repeated Integrals in One Variable Double Integrals in Two Variables Iterated Integrals in Two Variables Practice Exercises

28 More Volume Integrals

Slicing and Integrating Base Bounded by Curve and x Axis Base Bounded by Curve and Line Base Bounded by Two Curves Practice Exercises

29 What’s a Differential Equation?

Elementary First-Order ODEs Elementary Second-Order ODEs Practice Exercises

30 Review Questions and Answers

Final Exam Appendix A Worked-Out Solutions to Exercises: Chapters 1 to 9 Appendix B Worked-Out Solutions to Exercises: Chapters 11 to 19 Appendix C Worked-Out Solutions to Exercises: Chapters 21 to 29 Appendix D Answers to Final Exam Questions Appendix E Special Characters in Order of Appearance Appendix F Table of Derivatives Appendix G Table of Integrals Suggested Additional Reading Index

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