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1 Analytic Geometry

• Lines
• Distance Between Two Points; Circles
• Functions
• Shifts and Dilations

2 Instantaneous Rate of Change:
The Derivative

• The slope of a function
• An example
• Limits
• The Derivative Function
• Adjectives For Functions

3 Rules for Finding Derivatives

• The Power Rule
• Linearity of the Derivative
• The Product Rule
• The Quotient Rule
• The Chain Rule

4 Transcendental Functions

• Trigonometric Functions
• The Derivative of $\sin x$
• A hard limit
• The Derivative of $\sin x$, continued
• Derivatives of the Trigonometric Functions
• Exponential and Logarithmic functions
• Derivatives of the exponential and logarithmic functions
• Implicit Differentiation
• Inverse Trigonometric Functions
• Limits revisited
• Hyperbolic Functions

5 Curve Sketching

• Maxima and Minima
• The first derivative test
• The second derivative test
• Concavity and inflection points
• Asymptotes and Other Things to Look For

6 Applications of the Derivative

• Optimization
• Related Rates
• Newton's Method
• Linear Approximations
• The Mean Value Theorem

7 Integration

• Two examples
• The Fundamental Theorem of Calculus
• Some Properties of Integrals

8 Techniques of Integration

• Substitution
• Powers of sine and cosine
• Trigonometric Substitutions
• Integration by Parts
• Rational Functions
• Additional exercises

9 Applications of Integration

• Area between curves
• Distance, Velocity, Acceleration
• Volume
• Average value of a function
• Work
• Center of Mass
• Kinetic energy; improper integrals
• Probability
• Arc Length
• Surface Area

10 Polar Coordinates,
Parametric Equations

• Polar Coordinates
• Slopes in polar coordinates
• Areas in polar coordinates
• Parametric Equations
• Calculus with Parametric Equations

11 Sequences and Series

• Sequences
• Series
• The Integral Test
• Alternating Series
• Comparison Tests
• Absolute Convergence
• The Ratio and Root Tests
• Power Series
• Calculus with Power Series
• Taylor Series
• Taylor's Theorem
• Additional exercises

12 Three Dimensions

• The Coordinate System
• Vectors
• The Dot Product
• The Cross Product
• Lines and Planes
• Other Coordinate Systems

13 Vector Functions

• Space Curves
• Calculus with vector functions
• Arc length
• Motion along a curve

14 Partial Differentiation

• Functions of Several Variables
• Limits and Continuity
• Partial Differentiation
• The Chain Rule
• Directional Derivatives
• Higher order derivatives
• Maxima and minima
• Lagrange Multipliers

15 Multiple Integration

• Volume and Average Height
• Double Integrals in Cylindrical Coordinates
• Moment and Center of Mass
• Surface Area
• Triple Integrals
• Cylindrical and Spherical Coordinates
• Change of Variables

16 Vector Calculus

• Vector Fields
• Line Integrals
• The Fundamental Theorem of Line Integrals
• Green's Theorem
• Divergence and Curl
• Vector Equations of Surfaces
• Surface Integrals
• Stokes's Theorem
• The Divergence Theorem

17 Differential Equations

• First Order Differential Equations
• First Order Homogeneous Linear Equations
• First Order Linear Equations
• Approximation
• Second Order Homogeneous Equations
• Second Order Linear Equations
• Second Order Linear Equations, take two