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Table of Contents

I Differential Calculus

0 Functions and Precalculus
0.1 Numbers and Sets
0.2 Equations
0.3 Inequalities
0.4 Functions and Graphs
0.5 A Basic Library of Functions
0.6 Operations, Transformations, and Inverses
0.7 Logic and Mathematical Thinking*
Chapter Review, Self-Test, and Capstones

1 Limits
1.1 An Intuitive Introduction to Limits
1.2 Formal Definition of a Limit
1.3 Delta-Epsilon Proofs*
1.4 Continuity and Its Consequences
1.5 Limit Rules and Calculating Basic Limits
1.6 Infinite Limits and Indeterminate Forms
Chapter Review, Self-Test, and Capstones

2 Derivatives
2.1 An Intuitive Introduction to Derivatives
2.2 Formal Definition of the Derivative
2.3 Rules for Calculating Basic Derivatives
2.4 The Chain Rule and Implicit Differentiation
Chapter Review, Self-Test, and Capstones

3 Applications of the Derivative
3.1 The Mean Value Theorem
3.2 The First Derivative and Curve Sketching
3.3 The Second Derivative and Curve Sketching
3.4 Optimization
3.5 Related Rates
Chapter Review, Self-Test, and Capstones

II Exploring Functions with Calculus

4 Calculus with Power, Polynomial, and Rational Functions
4.1 Advanced Algebraic Techniques
4.2 Power Functions
4.3 Polynomial Functions
4.4 Rational Functions
Chapter Review, Self-Test, and Capstones

5 Calculus with Exponential and Logarithmic Functions
5.1 Defining Exponential and Logarithmic Functions
5.2 Limits of Exponential and Logarithmic Functions
5.3 Derivatives of Exponential and Logarithmic Functions
5.4 Applications of Exponential Functions
5.5 L’Hopital’s Rule
Chapter Review, Self-Test, and Capstones

6 Calculus with Trignometric and Inverse Trigonometric Functions
6.1 Defining the Trigonometric Functions
6.2 Trigonometric Identities
6.3 Limits and Derivatives of Trigonometric Functions
6.4 Inverse Trigonometric Functions
Chapter Review, Self-Test, and Capstones

III Integral Calculus

7 Definite Integrals
7.1 Addition and Accumulation
7.2 Riemann Sums
7.3 Definite Integrals
7.4 Indefinite Integrals
7.5 The Fundamental Theorem of Calculus
7.6 Areas and Average Values
7.7 Functions Defined by Integrals
Chapter Review, Self-Test, and Capstones

8 Techniques of Integration
8.1 Integration by Substitution
8.2 Integration by Parts
8.3 Partial Fractions and Other Algebraic Techniques
8.4 Trigonometric Integrals
8.5 Trigonometric Substitution
8.6 Improper Integrals
8.7 Numerical Integration*
Chapter Review, Self-Test, and Capstones

9 Applications of Integration
9.1 Volumes by Slicing
9.2 Volumes by Shells
9.3 Arc Length and Surface Area
9.4 Real-World Applications of Integration
9.5 Differential Equations*
Chapter Review, Self-Test, and Capstones

Answers to Odd Problems
Index

Authors

Laura Taalman

Laura Taalman and Peter Kohn are professors of mathematics at James Madison University, where they have taught calculus for a combined total of over 30 years.

 

Laura Taalman received her undergraduate degree from the University of Chicago
and master’s and Ph.D. degrees in mathematics from Duke University. Her research includes singular algebraic geometry, knot theory, and the mathematics of games and puzzles.  She is a recipient of both the Alder Award and the Trevor Evans award from the Mathematical Association of America, and the author of five books on Sudoku and the mathematics of Sudoku. In her spare time, she enjoys being a geek.

 

Peter Kohn received his undergraduate degree from Antioch College, a master’s
degree from San Francisco State University, and a Ph.D. in mathematics from the University of Texas at Austin. His main areas of research are low-dimensional topology and knot theory. He has been a national judge for MathCounts since 2001. In his spare time, he enjoys hiking and riding his bicycle in the beautiful Shenandoah Valley.

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