Calculus and Analytic Geometry

1. An Overview of Calculus.

1.1 The Derivative

1.2 The Integral

1.3 Survey of the Text

2. Functions, Limits, and Continuity.

2.1 Functions

2.2 Composite Functions

2.3 The Limit of a Function

2.4 Computations of Limits

2.5 Some Tools for Graphing

2.6 A Review of Trigonometry

2.7 The Limit of (sin Ø)/Ø as Ø Approaches 0

2.8 Continuous Functions

2.9 Precise Definitions of "lim(x->infinity)f(x)=infinity" and more >>

Calculus and Analytic Geometry

1. An Overview of Calculus.

1.1 The Derivative

1.2 The Integral

1.3 Survey of the Text

2. Functions, Limits, and Continuity.

2.1 Functions

2.2 Composite Functions

2.3 The Limit of a Function

2.4 Computations of Limits

2.5 Some Tools for Graphing

2.6 A Review of Trigonometry

2.7 The Limit of (sin Ø)/Ø as Ø Approaches 0

2.8 Continuous Functions

2.9 Precise Definitions of "lim(x->infinity)f(x)=infinity" and "lim(x->infinity)f(x)=L"

2.10 Precise Definition of "lim(x->a)f(x)=L"

2.S Summary

3. The Derivative.

3.1 Four Problems with One Theme

3.2 The Derivative

3.3 The Derivative and Continuity

3.4 The Derivative of the Sum, Difference, Product, and Quotient

3.5 The Derivatives of the Trigonometric Functions

3.6 The Derivative of a Composite Function

3.S Summary

4. Applications of the Derivative.

4.1 Three Theorems about the Derivative

4.2 The First Derivative and Graphing

4.3 Motion and the Second Derivative

4.4 Related Rates

4.5 The Second Derivative and Graphing

4.6 Newton's Method for Solving an Equation

4.7 Applied Maximum and Minimum Problems

4.9 The Differential and Linearization

4.10 The Second Derivative and Growth of a Function

4.S Summary

5. The Definite Integral.

5.1 Estimates in Four Problems

5.2 Summation Notation and Approximating Sums

5.3 The Definite Integral

5.4 Estimating the Definite Integral

5.5 Properties of the Antiderivative and the Definite Integral

5.6 Background for the Fundamental Theorems of Calculus

5.7 The Fundamental Theorems of Calculus

5.S Summary

6. Topics in Differential Calculus.

6.1 Logarithms

6.2 The Number e

6.3 The Derivative of a Logarithmic Function

6.4 One-to-One Functions and Their Inverses

6.5 The Derivative of b^x

6.6 The Derivatives of the Inverse Trigonometric Functions

6.7 The Differential Equation of Natural Growth and Decay

6.8 l'Hopital's Rule

6.9 The Hyperbolic Functions and Their Inverses

6.S Summary

7. Computing Antiderivatives.

7.1 Shortcuts, Integral Tables, and Machines

7.2 The Substitution Method

7.3 Integration by Parts

7.4 How to Integrate Certain Rational Functions

7.5 Integration of Rational Functions by Partial Fractions

7.6 Special Techniques

7.7 What to Do in the Face of an Integral

7.S Summary

8. Applications of the Definite Integral.

8.1 Computing Area by Parallel Cross Sections

8.2 Some Pointers on Drawing

8.3 Setting Up a Definite Integral

8.4 Computing Volumes

8.5 The Shell Method

8.6 The Centroid of a Plane Region

8.7 Work

8.8 Improper Integrals

8.S Summary

9. Plane Curves and Polar Coordinates.

9.1 Polar Coordinates

9.2 Area in Polar Coordinates

9.3 Parametric Equations

9.4 Arc Length and Speed on a Curve

9.5 The Area of a Surface of Revolution

9.6 Curvature

9.7 The Reflection Properties of the Conic Sections

9.S Summary

10. Series.

10.1 An Informal Introduction to Series

10.2 Sequences

10.3 Series

10.4 The Integral Test

10.5 Comparison Tests

10.6 Ratio Tests

10.7 Tests for Series with Both Positive and Negative Terms

10.S Summary

11. Power Series and Complex Numbers.

11.1 Taylor Series

11.2 The Error in Taylor Series

11.3 Why the Error in Taylor Series Is Controlled by a Derivative

11.4 Power Series and Radius of Convergence

11.5 Manipulating Power Series

11.6 Complex Numbers

11.7 The Relation between the Exponential and the Trigonometric Functions

11.S Summary

12. Vectors.

12.1 The Algebra of Vectors

12.2 Projections

12.3 The Dot Product of Two Vectors

12.4 Lines and Planes

12.5 Determinants

12.6 The Cross Product of Two Vectors

12.7 More on Lines and Planes

12.S Summary

13. The Derivative of a Vector Function.

13.1 The Derivative of a Vector Function

13.2 Properties of the Der << less