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Introduction

1 Analytic Geometry

• 1. Lines
• 2. Distance Between Two Points; Circles
• 3. Functions
• 4. Shifts and Dilations

2 Instantaneous Rate of Change:
The Derivative

• 1. The slope of a function
• 2. An example
• 3. Limits
• 4. The Derivative Function
• 5. Properties of Functions

3 Rules for Finding Derivatives

• 1. The Power Rule
• 2. Linearity of the Derivative
• 3. The Product Rule
• 4. The Quotient Rule
• 5. The Chain Rule

4 Transcendental Functions

• 1. Trigonometric Functions
• 2. The Derivative of $\sin x$
• 3. A hard limit
• 4. The Derivative of $\sin x$, continued
• 5. Derivatives of the Trigonometric Functions
• 6. Exponential and Logarithmic functions
• 7. Derivatives of the exponential and
  logarithmic functions
• 8. Implicit Differentiation
• 9. Inverse Trigonometric Functions
• 10. Limits revisited
• 11. Hyperbolic Functions

5 Curve Sketching

• 1. Maxima and Minima
• 2. The first derivative test
• 3. The second derivative test
• 4. Concavity and inflection points
• 5. Asymptotes and Other Things to Look For

6 Applications of the Derivative

• 1. Optimization
• 2. Related Rates
• 3. Newton's Method
• 4. Linear Approximations
• 5. The Mean Value Theorem

7 Integration

• 1. Two examples
• 2. The Fundamental Theorem of Calculus
• 3. Some Properties of Integrals

8 Techniques of Integration

• 1. Substitution
• 2. Powers of sine and cosine
• 3. Trigonometric Substitutions
• 4. Integration by Parts
• 5. Rational Functions
• 6. Numerical Integration
• 7. Additional exercises

9 Applications of Integration

• 1. Area between curves
• 2. Distance, Velocity, Acceleration
• 3. Volume
• 4. Average value of a function
• 5. Work
• 6. Center of Mass
• 7. Kinetic energy; improper integrals
• 8. Probability
• 9. Arc Length
• 10. Surface Area

10 Polar Coordinates,
Parametric Equations

• 1. Polar Coordinates
• 2. Slopes in polar coordinates
• 3. Areas in polar coordinates
• 4. Parametric Equations
• 5. Calculus with Parametric Equations

11 Sequences and Series

• 1. Sequences
• 2. Series
• 3. The Integral Test
• 4. Alternating Series
• 5. Comparison Tests
• 6. Absolute Convergence
• 7. The Ratio and Root Tests
• 8. Power Series
• 9. Calculus with Power Series
• 10. Taylor Series
• 11. Taylor's Theorem
• 12. Additional exercises

12 Three Dimensions

• 1. The Coordinate System
• 2. Vectors
• 3. The Dot Product
• 4. The Cross Product
• 5. Lines and Planes
• 6. Other Coordinate Systems

13 Vector Functions

• 1. Space Curves
• 2. Calculus with vector functions
• 3. Arc length and curvature
• 4. Motion along a curve

14 Partial Differentiation

• 1. Functions of Several Variables
• 2. Limits and Continuity
• 3. Partial Differentiation
• 4. The Chain Rule
• 5. Directional Derivatives
• 6. Higher order derivatives
• 7. Maxima and minima
• 8. Lagrange Multipliers

15 Multiple Integration

• 1. Volume and Average Height
• 2. Double Integrals in Cylindrical Coordinates
• 3. Moment and Center of Mass
• 4. Surface Area
• 5. Triple Integrals
• 6. Cylindrical and Spherical Coordinates
• 7. Change of Variables

16 Vector Calculus

• 1. Vector Fields
• 2. Line Integrals
• 3. The Fundamental Theorem of Line Integrals
• 4. Green's Theorem
• 5. Divergence and Curl
• 6. Vector Functions for Surfaces
• 7. Surface Integrals
• 8. Stokes's Theorem
• 9. The Divergence Theorem

17 Differential Equations

• 1. First Order Differential Equations
• 2. First Order Homogeneous Linear Equations
• 3. First Order Linear Equations
• 4. Approximation
• 5. Second Order Homogeneous Equations
• 6. Second Order Linear Equations
• 7. Second Order Linear Equations, take two

18 Useful formulas

19 Introduction to Sage

• 1. Basics
• 2. Differentiation
• 3. Integration