# Calculus with Analytic Geometry

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Calculus with Analytic Geometry presents the essentials of calculus with analytic geometry. The emphasis is on how to set up and solve calculus problems, that is, how to apply calculus. The initial approach to each topic is intuitive, numerical, and motivated by examples, with theory kept to a bare minimum. Later, after much experience in the use of the topic, an appropriate amount of theory is presented. Comprised of 18 chapters, this book begins with a review of some basic pre-calculus algebra and analytic geometry, paying particular attention to functions and graphs. The reader is then introduced to derivatives and applications of differentiation; exponential and trigonometric functions; and techniques and applications of integration. Subsequent chapters deal with inverse functions, plane analytic geometry, and approximation as well as convergence, and power series. In addition, the book considers space geometry and vectors; vector functions and curves; higher partials and applications; and double and multiple integrals. This monograph will be a useful resource for undergraduate students of mathematics and algebra.

## Table of Contents

Preface

1. Functions and Graphs

1. Introduction

2. Real Numbers

3. Coordinates

4. Functions and Graphs

5. Linear Functions

6. Quadratic Functions

7. More on Graphing

8. Polynomials and Rational Functions

9. Distance Formula and Applications

10. Tangents

11. Miscellaneous Exercises

2. Derivatives

1. The Slope Problem

2. Limits

3. The Derivative

4. Sums and Products

5. Quotients and Square Roots

6. The Chain Rule

7. The Tangent Line

8. Antiderivatives

9. Higher Derivatives

10. Limits and Continuity

11. Differentiable Functions

12. Miscellaneous Exercises

3. Applications of Differentiation

1. Curve Sketching

2. Rectilinear Motion

3. Related Rates

4. Maxima and Minima

5. Applications of Max and Min

6. Second Derivative Test

7. On Problem Solving

8. Extrema and Convexity

9. Miscellaneous Exercises

4. Exponential and Trigonometric Functions

1. The Exponential Function

2. Properties of Exponential Functions

3. Approximation and Growth Rates

4. Applications

5. Trigonometric Functions

6. Additional Trigonometric Functions

7. Derivatives

8. Applications

9. Miscellaneous Exercises

5. Integration

1. The Area Problem

2. Examples of Integrals

3. The Definite Integral and the Fundamental Theorem

4. Applications

5. Approximate Integration

6. Integration of Products

7. Symmetry

8. Inequalities and Estimates

9. Insights into Integration

10. Miscellaneous Exercises

6. Applications of Integration

1. Introduction

2. Area

3. Volume

4. Work

5. Fluid Pressure

6. Miscellaneous Applications

7. Miscellaneous Exercises

7. Inverse Functions

1. Inverse Functions and Their Derivatives

2. The Logarithm Function

3. Further Properties of Logarithms

4. Applications of Logarithms

5. Inverse Trigonometric Functions

6. Derivatives and Applications

7. Hyperbolic Functions

8. Basic Properties

9. Miscellaneous Exercises

8. Techniques of Integration

1. Indefinite Integrals

2. Substitutions and Differentials

3. Other Substitutions

4. Use of Identities

5. Partial Fractions

6. Trigonometric Substitutions

7. Integration by Parts

8. Reduction Formulas

9. Integral Tables

10. Miscellaneous Exercises

9. Plane Analytic Geometry

1. Translation and Circles

2. Locus

3. Parabola and Ellipse

4. Hyperbola

5. Polar Coordinates

6. Polar Graphs

7. Rotation of Axes

8. Calculus Applied to Conics

9. Miscellaneous Exercises

10. Approximation

1. Introduction

2. First and Second Degree Approximations

3. Taylor Approximations

4. Taylor's Formula

5. Rolle's Theorem

6. Mean Value Theorems and Lhospital's Rule

7. Interpolation

8. Lagrange Interpolation

9. Approximate Integration

10. Root Approximation and Hill Climbing

11. Iteration and Newton's Method

12. Miscellaneous Exercises

11. Convergence

1. Sequences and Limits

2. Properties of Limits

3. Infinite Series

4. Series with Positive Terms

5. Series with Positive and Negative Terms

6. Improper Integrals

7. Convergence and Divergence Tests

8. Relation between Integrals and Series

9. Other Improper Integrals

10. Miscellaneous Exercises

12. Power Series

1. Basic Properties

2. Taylor Series

3. Expansion of Functions

4. Further Techniques

5. Binomial Series

6. Numerical Applications

7. Sequences and Series of Functions

8. Miscellaneous Exercises

13. Space Geometry and Vectors

1. Rectangular Coordinates

2. Vector Algebra

3. Length and Inner Product

4. Lines and Planes

5. Linear Systems

6. Cross Poduct

7. Applications of the Cross Product

8. Miscellaneous Exercises

14. Vector Functions and Curves

1. Differentiation

2. Arc Length

3. Plane Curves

4. Tangent, Normal, and Curvature

5. Velocity and Acceleration

6. Curves in Polar Coordinates

7. Miscellaneous Exercises

15. Functions of Several Variables

1. Functions and Graphs

2. Partial Derivatives

3. Gradients and Directional Derivatives

4. Surfaces

5. Parametric Surfaces and Surfaces of Revolution

6. Quadric Surfaces

7. Optimization

8. Further Optimization Problems

9. Implicit Functions

10. Differentials and Approximation

11. Differentiable Functions

12. Miscellaneous Exercises

16. Higher Partials and Applications

1. Mixed Partials

2. Taylor Approximation

3. Stability

4. Constrained Optimization

5. Further Constraint Problems

6. Additional Topics

7. Miscellaneous Exercises

17. Double Integrals

1. Introduction

2. Rectangular Domains

3. Domain between Graphs

4. Arbitrary Domains

5. Polar Coordinates

6. Applications

7. Physical Applications

8. Approximate Integration

9. Miscellaneous Exercises

18. Multiple Integrals

1. Triple Integrals

2. Cylindrical Coordinates

3. Spherical Coordinates

4. Center of Gravity

5. Moments of Inertia

6. Line Integrals

7. Green's Theorem

8. Surface Integrals

9. Miscellaneous Exercises

Numerical Tables

1. Three-Place Mantissas for Common Logarithms

2. Exponential Functions

3. Natural Logarithms

4. Trigonometric Functions of Degrees

5. Trigonometric Functions of Radians

6. Inverse Trigonometric Functions to Radians

7. Trigonometric Functions of a = π · x Radians

Answers to Odd-Numbered Exercises

Index

## Product details

- No. of pages: 1054
- Language: English
- Copyright: © Academic Press 1978
- Published: January 1, 1978
- Imprint: Academic Press
- eBook ISBN: 9781483262406

## About the Authors

### Harley Flanders

#### Affiliations and Expertise

Purdue University Lafayette, Indiana

### Justin J. Price

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