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Calculus and Analytic Geometry

Sherman K. Stein and Anthony Barcellos
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We do not plan to review this book.


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 Derivative of a Vector Function

13.3 The Acceleration Vector

13.4 The Components of Acceleration

13.5 Newton's Law Implies Kepler's Laws

13.S Summary

14. Partial Derivatives.

14.1 Graphs

14.2 Quadratic Surfaces

14.3 Functions and Their Level Curves

14.4 Limits and Continuity

14.5 Partial Derivatives

14.6 The Chain Rule

14.7 Directional Derivatives and the Gradient

14.8 Normals and the Tangent Plane

14.9 Critical Points and Extrema

14.10 Lagrange Multipliers

14.11 The Chain Rule Revisited

14.S Summary

15. Definite Integrals over Plane and Solid Regions.

15.1 The Definite Integral of a Function over a Region in the Plane

15.2 Computing |R f(P) dA Using Rectangular Coordinates

15.3 Moments and Centers of Mass

15.4 Computing |R f(P) dA Using Polar Coordinates

15.5 The Definite Integral of a Function over a Region in Space

15.6 Computing |R f(P) dV Using Cylindrical Coordinates

15.7 Computing |R f(P) dV Using Spherical Coordinates

15.S Summary

16. Green's Theorem.

16.1 Vector and Scalar Fields

16.2 Line Integrals

16.3 Four Applications of Line Integrals

16.4 Green's Theorem

16.5 Applications of Green's Theorem

16.6 Conservative Vector Fields

16.S Summary

17. The Divergence Theorem and Stokes' Theorem.

17.1 Surface Integrals

17.2 The Divergence Theorem

17.3 Stokes' Theorem

17.4 Applications of Stokes' Theorem

17.S Summary


A. Real Numbers.

B. Graphs and Lines.

C. Topics in Algebra.

D. Exponents.

E. Mathematical Induction.

F. The Converse of a Statement.

G. Conic Sections.

H. Logarithms and Exponentials Defined through Calculus.

I. The Taylor Series for f(x,y).

J. Theory of Limits.

K. The Interchange of Limits.

L. The Jacobian.

M. Linear Differential Equations with Constant Coefficients.

Answers to Selected Odd-Numbered Problems and to Guide Quizzes

List of Symbols