# Study Guide

ChapterS FILES

1: Introduction to Calculus

1.1 Velocity and Distance
1.2 Calculus Without Limits
1.3 The Velocity at an Instant
1.4 Circular Motion
1.5 A Review of Trigonometry
1.6 A Thousand Points of Light
1.7 Computing in Calculus

2: Derivatives

2.1 The Derivative of a Function
2.2 Powers and Polynomials
2.3 The Slope and the Tangent Line
2.4 Derivative of the Sine and Cosine
2.5 The Product and Quotient and Power Rules
2.6 Limits
2.7 Continuous Functions

3: Applications of the Derivative

3.1 Linear Approximation
3.2 Maximum and Minimum Problems
3.3 Second Derivatives: Minimum vs. Maximum
3.4 Graphs
3.5 Ellipses, Parabolas, and Hyperbolas
3.6 Iterations x,+ ,= F(x,)
3.7 Newton's Method and Chaos
3.8 The Mean Value Theorem and l'Hôpital's Rule

4: The Chain Rule

4.1 Derivatives by the Charin Rule
4.2 Implicit Differentiation and Related Rates
4.3 Inverse Functions and Their Derivatives
4.4 Inverses of Trigonometric Functions

(PDF)

5: Integrals

5.1 The Idea of an Integral
5.2 Antiderivatives
5.3 Summation vs. Integration
5.4 Indefinite Integrals and Substitutions
5.5 The Definite Integral
5.6 Properties of the Integral and the Average Value
5.7 The Fundamental Theorem and Its Consequences
5.8 Numerical Integration

6: Exponentials and Logarithms

6.1 An Overview
6.2 The Exponential e^x
6.3 Growth and Decay in Science and Economics
6.4 Logarithms
6.5 Separable Equations Including the Logistic Equation
6.7 Hyperbolic Functions

7: Techniques of Integration

7.1 Integration by Parts
7.2 Trigonometric Integrals
7.3 Trigonometric Substitutions
7.4 Partial Fractions
7.5 Improper Integrals

8: Applications of the Integral

8.1 Areas and Volumes by Slices
8.2 Length of a Plane Curve
8.3 Area of a Surface of Revolution
8.4 Probability and Calculus
8.5 Masses and Moments
8.6 Force, Work, and Energy

9: Polar Coordinates and Complex Numbers

9.1 Polar Coordinates
9.2 Polar Equations and Graphs
9.3 Slope, Length, and Area for Polar Curves
9.4 Complex Numbers

(PDF)

10: Infinite Series

10.1 The Geometric Series
10.2 Convergence Tests: Positive Series
10.3 Convergence Tests: All Series
10.4 The Taylor Series for e^x, sin x, and cos x
10.5 Power Series

11: Vectors and Matrices

11.1 Vectors and Dot Products
11.2 Planes and Projections
11.3 Cross Products and Determinants
11.4 Matrices and Linear Equations
11.5 Linear Algebra in Three Dimensions

(PDF)

12: Motion along a Curve

12.1 The Position Vector
12.2 Plane Motion: Projectiles and Cycloids
12.3 Tangent Vector and Normal Vector
12.4 Polar Coordinates and Planetary Motion

(PDF)

13: Partial Derivatives

13.1 Surface and Level Curves
13.2 Partial Derivatives
13.3 Tangent Planes and Linear Approximations
13.5 The Chain Rule
13.6 Maxima, Minima, and Saddle Points
13.7 Constraints and Lagrange Multipliers

14: Multiple Integrals

14.1 Double Integrals
14.2 Changing to Better Coordinates
14.3 Triple Integrals
14.4 Cylindrical and Spherical Coordinates

(PDF)

15: Vector Calculus

15.1 Vector Fields
15.2 Line Integrals
15.3 Green's Theorem
15.4 Surface Integrals
15.5 The Divergence Theorem
15.6 Stokes' Theorem and the Curl of F