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

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