Contents
Introduction
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. Trigonometric Functions
Trigonometric Functions
The Derivative of $mathop { am z@ enrm sin}olimits x$
A hard limit
The Derivative of $mathop { am z@ enrm sin}olimits x$, continued
Derivatives of the Trigonometric Functions
Implicit Differentiation
Limits revisited
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
Substitution
8. Applications of Integration
Area between curves
Distance, Velocity, Acceleration
Volume
Average value of a function
Work
9. Transcendental Functions
Inverse functions
The natural logarithm
The exponential function
Other bases
Inverse Trigonometric Functions
Hyperbolic Functions
10. Techniques of Integration
Powers of sine and cosine
Trigonometric Substitutions
Integration by Parts
Rational Functions
Additional exercises
11. More Applications of Integration
Center of Mass
Kinetic energy; improper integrals
Probability
Arc Length
Surface Area
12. Polar Coordinates, Parametric Equations
Polar Coordinates
Slopes in polar coordinates
Areas in polar coordinates
Parametric Equations
Calculus with Parametric Equations
13. 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
A. Selected Answers
B. Useful Formulas
Index