## Table of Contents, Volume 1

### Chapter 1: Functions in Calculus

- 1.1
- Functions, calculus-style
- 1.2
- Graphs
- 1.3
- Machine graphics
- 1.4
- What is a function?
- 1.5
- A field guide to elementary functions
- 1.6
- New functions from old
- 1.7
- Modeling with elementary functions
- 1.8
- Chapter summary

### Chapter 2: The Derivative

- 2.1
- Amount functions and rate functions: the idea of the derivative
- 2.2
- Estimating derivatives: a closer look
- 2.3
- The geometry of derivatives
- 2.4
- The geometry of higher-order derivatives
- 2.5
- Average and instantaneous rates: defining the derivative
- 2.6
- Limits and continuity
- 2.7
- Limits involving infinity; new limits from old
- 2.8
- Chapter summary

### Chapter 3: Derivatives of Elementary Functions

- 3.1
- Derivatives of power functions and polynomials
- 3.2
- Using derivative and antiderivative formulas
- 3.3
- Derivatives of exponential and logarithm functions
- 3.4
- Derivatives of trigonometric functions
- 3.5
- New derivatives from old: the product and quotient rules
- 3.6
- New derivatives from old: the chain rule
- 3.7
- Implicit differentiation
- 3.8
- Inverse trigonometric functions and their derivatives
- 3.9
- Chapter summary

### Chapter 4: Applications of the Derivative

- 4.1
- Differential equations and their solutions
- 4.2
- More differential equations: modeling growth
- 4.3
- Linear and quadratic approximation; Taylor polynomials
- 4.4
- Newton's method: finding roots
- 4.5
- Splines: connecting the dots
- 4.6
- Optimization
- 4.7
- Calculus for money: derivatives in economics
- 4.8
- Related rates
- 4.9
- Parametric equations, parametric curves
- 4.10
- Why continuity matters
- 4.11
- Why differentiability matters; the mean value theorem
- 4.12
- Chapter summary

### Chapter 5: The Integral

- 5.1
- Areas and integrals
- 5.2
- The area function
- 5.3
- The fundamental theorem of calculus
- 5.4
- Approximating sums: the integral as a limit
- 5.5
- Approximating sums: interpretations and applications
- 5.6
- Chapter summary

### Chapter 6: Finding Antiderivatives

- 6.1
- Antiderivatives: the idea
- 6.2
- Antidifferentiation by substitution
- 6.3
- Integral aids: tables and computers

### Appendices

- A
- Real numbers and the coordinate plane
- B
- Lines and linear functions
- C
- Polynomial algebra: a brisk review
- D
- Real-world calculus: from words to mathematics
- E
- Algebra of exponentials
- F
- Algebra of logarithms
- G
- Trigonometric functions
- H
- Selected proofs
- I
- A graphical glossary of functions