Chapter 1: Introduction to Calculus

Content

1.1 Velocity and Distance
Summary

1.2 Calculus Without Limits
Summary

1.3 The Velocity at an Instant
Summary

1.4 Circular Motion
Summary

1.5 A Review of Trigonometry
Summary

 1.6 A Thousand Points of Light
Summary

1.7 Computing in Calculus
Summary

Chapter 2: Derivatives

Content

2.1 The Derivative of a Function
Summary

2.2 Powers and Polynomials
Summary

2.3 The Slope and the Tangent Line
Summary

2.4 Derivative of the Sine and Cosine
Summary

2.5 The Product and Quotient and Power Rules
Summary

2.6 Limits
Summary

2.7 Continuous Functions
Summary

Chapter 3: Applications of the Derivative

Content

3.1 Linear Approximation
Summary

3.2 Maximum and Minimum Problems
Summary

3.3 Second Derivatives: Minimum vs. Maximum
Summary

3.4 Graphs
Summary

3.5 Ellipses, Parabolas, and Hyperbolas
Summary

3.6 Iterations x_n+1 = F(x_n)
Summary

3.7 Newton's Method and Chaos
Summary

3.8 The Mean Value Theorem and l'Hopital's Rule
Summary

Chapter 4: The Chain Rule

Content

4.1 Derivatives by the Chain Rule
Summary

4.2 Implicit Differentiation and Related Rates
Summary

4.3 Inverse Functions and Their Derivatives
Summary

4.4 Inverses of Trigonometric Functions
Summary

Chapter 5: Integrals

Content

5.1 The Idea of the Integral
Summary

5.2 Antiderivatives
Summary

5.3 Summation vs. Integration
Summary

5.4 Indefinite Integrals and Substitutions
Summary

5.5 The Definite Integral
Summary

5.6 Properties of the Integral and the Average Value
Summary

5.7 The Fundamental Theorem and Its Consequences
Summary

5.8 Numerical Integration
Summary

Chapter 6: Exponentials and Logarithms

Content

6.1 An Overview
Summary

6.2 The Exponential e^x
Summary

6.3 Growth and Decay in Science and Economics
Summary

6.4 Logarithms
Summary

6.5 Separable Equations Including the Logistic Equation
Summary

6.6 Powers Instead of Exponentials
Summary

6.7 Hyperbolic Functions
Summary

Chapter 7: Techniques of Integration

Content

7.1 Integration by Parts
Summary

7.2 Trigonometric Integrals
Summary

7.3 Trigonometric Substitutions
Summary

7.4 Partial Fractions
Summary

7.5 Improper Integrals
Summary

Chapter 8: Applications of the Integral

Content

8.1 Areas and Volumes by Slices
Summary

8.2 Length of a Plane Curve
Summary

8.3 Area of a Surface of Revolution
Summary

8.4 Probability and Calculus
Summary

8.5 Masses and Moments
Summary

8.6 Force, Work, and Energy
Summary

Chapter 9: Polar Coordinates and Complex Numbers

Content

9.1 Polar Coordinates
Summary

9.2 Polar Equations and Graphs
Summary

9.3 Slope, Length, and Area for Polar Curves
Summary

9.4 Complex Numbers
Summary

Chapter 10: Infinite Series

Content

10.1 The Geometric Series
Summary

10.2 Convergence Tests: Positive Series
Summary

10.3 Convergence Tests: All Series
Summary

10.4 The Taylor Series for e^x, sin x, and cos x
Summary

10.5 Power Series
Summary

Chapter 11: Vectors and Matrices

Content

11.1 Vectors and Dot Products
Summary

11.2 Planes and Projections
Summary

11.3 Cross Products and Determinants
Summary

11.4 Matrices and Linear Equations
Summary

11.5 Linear Algebra in Three Dimensions
Summary

Chapter 12: Motion along a Curve

Content

12.1 The Position Vector
Summary

12.2 Plane Motion: Projectiles and Cycloids
Summary

12.3 Tangent Vector and Normal Vector
Summary

12.4 Polar Coordinates and Planetary Motion
Summary

Chapter 13: Partial Derivatives

Content

13.1 Surfaces and Level Curves
Summary

13.2 Partial Derivatives
Summary

13.3 Tangent Planes and Linear Approximations
Summary

13.4 Directional Derivatives and Gradients
Summary

13.5 The Chain Rule
Summary

13.6 Maxima, Minima, and Saddle Points
Summary

13.7 Constraints and Lagrange Multipliers
Summary

Chapter 14: Multiple Integrals

Content

14.1 Double Integrals
Summary

14.2 Changing to Better Coordinates
Summary

14.3 Triple Integrals
Summary

14.4 Cylindrical and Spherical Coordinates
Summary

Chapter 15: Vector Calculus

Content

15.1 Vector Fields
Summary

15.2 Line Integrals
Summary

15.3 Green's Theorem
Summary

15.4 Surface Integrals
Summary

15.5 The Divergence Theorem
Summary

15.6 Stokes' Theorem and the Curl of F
Summary

Chapter 16: Mathematics after Calculus

Content

16.1 Linear Alegbra
Summary

16.2 Differential Equations
Summary

16.3 Discrete Mathematics
Summary