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by Ron Larson

Edition: 6TH 98Copyright: 1998

Publisher: Houghton Mifflin Harcourt

Published: 1998

International: No

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This traditional text continues to offer a balanced approach that combines the theoretical instruction of calculus with the best aspects of reform, including creative teaching and learning techniques such as the integration of technology, the use of real data, real-life applications, and mathematical models. The *Calculus with Analytic Geometry Alternate,* 6/e, offers a late approach to trigonometry for those instructors who wish to introduce it later in their courses.

- Before students are exposed to selected topics, optional boxed
*Explorations*allow them to discover concepts on their own, making them more likely to remember the material. - Extended lab projects using real-life applications are found in the new lab series. The supplementary lab manuals use Maple, Mathematica, Derive, Mathcad, and the TI-92 graphing calculator.
- The art program in the Sixth Edition was computer-generated for accuracy, clarity, and realism to
- Although students are not required to use technology, they are encouraged to use a graphing utility or computer algebra system as a tool for exploration, discovery, and problem solving. Calculus Alternate, 6/e, offers more opportunities for students to execute complicated computations, to visualize theoretical concepts, to discover alternative approaches, and to verify the results of other solution methods using technology.

Note: Each chapter concludes with Review Exercises.

**1. The Cartesian Plane and Functions**

1.1 Real Numbers and the Real Line

1.2 The Cartesian Plane

1.3 Graphs of Equations

1.4 Lines in the Plane

1.5 Functions

**2. Limits and Their Properties**

2.1 An Introduction to Limits

2.2 Techniques for Evaluating Limits

2.3 Continuity

2.4 Infinite Limits

2.5 e-d Definition of Limits

**3. Differentiation**

3.1 The Derivative and the Tangent Line Problem

3.2 Velocity, Acceleration, and Other Rates of Change

3.3 Differentiation Rules for Powers, Constant Multiples, and Sums

3.4 Differentiation Rules for Products and Quotients

3.5 The Chain Rule

3.6 Implicit Differentiation

3.7 Related Rates

**4. Applications of Differentiation**

4.1 Extrema on an Interval

4.2 Rolle's Theorem and the Mean Value Theorem

4.3 Increasing and Decreasing Functions and the First Derivative Test

4.4 Concavity and the Second Derivative Test

4.5 Limits at Infinity

4.6 A Summary of Curve Sketching

4.7 Optimization Problems

4.8 Newton's Method

4.9 Differentials

4.10 Business and Economics Applications

**5. Integration**

5.1 Antiderivatives and Indefinite Integration

5.2 Area

5.3 Riemann Sums and the Definite Integral

5.4 The Fundamental Theorem of Calculus

5.5 Integration by Substitution

5.6 Numerical Integration

**6. Applications of Integration**

6.1 Area of a Region Between Two Curves

6.2 Volume: The Disc Method

6.3 Volume: The Shell Method

6.4 Arc Length and Surfaces of Revolution

6.5 Work

6.6 Fluid Pressure and Fluid Force

6.7 Moments, Centers of Mass, and Centroids

**7. Exponential and Logarithmic Functions**

7.1 Exponential Functions

7.2 Differentiation and Integration of Exponential Functions

7.3 Inverse Functions

7.4 Logarithmic Functions

7.5 Logarithmic Functions and Differentiation

7.6 Logarithmic Functions and Integration

7.7 Growth and Decay

7.8 Indeterminate Forms and L

Hôpital's Rule

**8. Trigonometric Functions and Inverse Trigonometric Functions**

8.1 Review of Trigonometric Functions

8.2 Graphs and Limits of Trigonometric Functions

8.3 Derivatives of Trigonometric Functions

8.4 Integrals of Trigonometric Functions

8.5 Inverse Trigonometric Functions and Differentiation

8.6 Inverse Trigonometric Functions: Integration and Completing the Square

8.7 Hyperbolic Functions

**9. Integration Techniques and Improper Integrals**

9.1 Basic Integration Formulas

9.2 Integration by Parts

9.3 Trigonometric Integrals

9.4 Trigonometric Substitution

9.5 Partial Fractions

9.6 Integration by Tables and Other Integration Techniques

9.7 Improper Integrals

**10. Infinite Series**

10.1 Sequences

10.2 Series and Convergence

10.3 The Integral Test and p-Series

10.4 Comparisons of Series

10.5 Alternating Series

10.6 The Ratio and Root Tests

10.7 Taylor Polynomials and Approximations

10.8 Power Series

10.9 Representation of Functions by Power Series

10.10 Taylor and Maclaurin Series

**11. Conic Sections**

11.1 Parabolas

11.2 Ellipses

11.3 Hyperbolas

11.4 Rotation and the General Second-Degree Equation

**12. Plane Curves, Parametric Equations, and Polar Coordinates**

12.1 Plane Curves and Parametric Equations

12.2 Parametric Equations and Calculus

12.3 Polar Coordinates and Polar Graphs

12.4 Tangent Lines and Curve Sketching in Polar Coordinates

12.5 Area and Arc Length in Polar Coordinates

12.6 Polar Equations for Conics and Kepler's Laws

**13. Vectors and Curves in the Plane**

13.1 Vectors in the Plane

13.2 The Dot Product of Two Vectors

13.3 Vector-Valued Functions

13.4 Velocity and Acceleration

13.5 Tangent Vectors and Normal Vectors

13.6 Arc Length and Curvature

**14. Solid Analytic Geometry and Vectors in Space**

14.1 Space Coordinates and Vectors in Space

14.2 The Cross Product of Two Vectors in Space

14.3 Lines and Planes in Space

14.4 Surfaces in Space

14.5 Curves and Vector-Valued Functions in Space

14.6 Tangent Vectors, Normal Vectors, and Curvature in Space

**15. Functions of Several Variables**

15.1 Introduction to Functions of Several Variables

15.2 Limits and Continuity

15.3 Partial Derivatives

15.4 Differentials

15.5 Chain Rules for Functions of Several Variables

15.6 Directional Derivatives and Gradients

15.7 Tangent Planes and Normal Lines

15.8 Extrema of Functions of Two Variables

15.9 Applications of Extrema of Functions of Two Variables

15.10 Lagrange Multipliers

**16. Multiple Integration**

16.1 Iterated Integrals and Area in the Plane

16.2 Double Integrals and Volume

16.3 Change of Variables: Polar Coordinates

16.4 Center of Mass and Moments of Inertia

16.5 Surface Area

16.6 Triple Integrals and Applications

16.7 Cylindrical and Spherical Coordinates

16.8 Triple Integrals in Cylindrical and Spherical Coordinates

16.9 Change of Variables: Jacobians

**17. Vector Analysis**

17.1 Vector Fields

17.2 Line Integrals

17.3 Conservative Vector Fields and Independence of Path

17.4 Green's Theorem

17.5 Parametric Surfaces

17.6 Surface Integrals

17.7 Divergence Theorem

17.8 Stokes's Theorem

**18. Differential Equations**

18.1 Definitions and Basic Concepts

18.2 Separation of Variables in First-Order Equations

18.3 Exact First-Order Equations

18.4 First-Order Linear Differential Equations

18.5 Second-Order Homogeneous Linear Equations

18.6 Second-Order Nonhomogeneous Linear Equations

18.7 Series Solutions of Differential Equations

- Appendixes: A. Proofs of Selected Theorems; B. Basic Differentiation Rules for Elementary Functions; C. Integration Tables
- Answers to Odd-Numbered Exercises
- Index

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Summary

This traditional text continues to offer a balanced approach that combines the theoretical instruction of calculus with the best aspects of reform, including creative teaching and learning techniques such as the integration of technology, the use of real data, real-life applications, and mathematical models. The *Calculus with Analytic Geometry Alternate,* 6/e, offers a late approach to trigonometry for those instructors who wish to introduce it later in their courses.

- Before students are exposed to selected topics, optional boxed
*Explorations*allow them to discover concepts on their own, making them more likely to remember the material. - Extended lab projects using real-life applications are found in the new lab series. The supplementary lab manuals use Maple, Mathematica, Derive, Mathcad, and the TI-92 graphing calculator.
- The art program in the Sixth Edition was computer-generated for accuracy, clarity, and realism to
- Although students are not required to use technology, they are encouraged to use a graphing utility or computer algebra system as a tool for exploration, discovery, and problem solving. Calculus Alternate, 6/e, offers more opportunities for students to execute complicated computations, to visualize theoretical concepts, to discover alternative approaches, and to verify the results of other solution methods using technology.

Table of Contents

Note: Each chapter concludes with Review Exercises.

**1. The Cartesian Plane and Functions**

1.1 Real Numbers and the Real Line

1.2 The Cartesian Plane

1.3 Graphs of Equations

1.4 Lines in the Plane

1.5 Functions

**2. Limits and Their Properties**

2.1 An Introduction to Limits

2.2 Techniques for Evaluating Limits

2.3 Continuity

2.4 Infinite Limits

2.5 e-d Definition of Limits

**3. Differentiation**

3.1 The Derivative and the Tangent Line Problem

3.2 Velocity, Acceleration, and Other Rates of Change

3.3 Differentiation Rules for Powers, Constant Multiples, and Sums

3.4 Differentiation Rules for Products and Quotients

3.5 The Chain Rule

3.6 Implicit Differentiation

3.7 Related Rates

**4. Applications of Differentiation**

4.1 Extrema on an Interval

4.2 Rolle's Theorem and the Mean Value Theorem

4.3 Increasing and Decreasing Functions and the First Derivative Test

4.4 Concavity and the Second Derivative Test

4.5 Limits at Infinity

4.6 A Summary of Curve Sketching

4.7 Optimization Problems

4.8 Newton's Method

4.9 Differentials

4.10 Business and Economics Applications

**5. Integration**

5.1 Antiderivatives and Indefinite Integration

5.2 Area

5.3 Riemann Sums and the Definite Integral

5.4 The Fundamental Theorem of Calculus

5.5 Integration by Substitution

5.6 Numerical Integration

**6. Applications of Integration**

6.1 Area of a Region Between Two Curves

6.2 Volume: The Disc Method

6.3 Volume: The Shell Method

6.4 Arc Length and Surfaces of Revolution

6.5 Work

6.6 Fluid Pressure and Fluid Force

6.7 Moments, Centers of Mass, and Centroids

**7. Exponential and Logarithmic Functions**

7.1 Exponential Functions

7.2 Differentiation and Integration of Exponential Functions

7.3 Inverse Functions

7.4 Logarithmic Functions

7.5 Logarithmic Functions and Differentiation

7.6 Logarithmic Functions and Integration

7.7 Growth and Decay

7.8 Indeterminate Forms and L

Hôpital's Rule

**8. Trigonometric Functions and Inverse Trigonometric Functions**

8.1 Review of Trigonometric Functions

8.2 Graphs and Limits of Trigonometric Functions

8.3 Derivatives of Trigonometric Functions

8.4 Integrals of Trigonometric Functions

8.5 Inverse Trigonometric Functions and Differentiation

8.6 Inverse Trigonometric Functions: Integration and Completing the Square

8.7 Hyperbolic Functions

**9. Integration Techniques and Improper Integrals**

9.1 Basic Integration Formulas

9.2 Integration by Parts

9.3 Trigonometric Integrals

9.4 Trigonometric Substitution

9.5 Partial Fractions

9.6 Integration by Tables and Other Integration Techniques

9.7 Improper Integrals

**10. Infinite Series**

10.1 Sequences

10.2 Series and Convergence

10.3 The Integral Test and p-Series

10.4 Comparisons of Series

10.5 Alternating Series

10.6 The Ratio and Root Tests

10.7 Taylor Polynomials and Approximations

10.8 Power Series

10.9 Representation of Functions by Power Series

10.10 Taylor and Maclaurin Series

**11. Conic Sections**

11.1 Parabolas

11.2 Ellipses

11.3 Hyperbolas

11.4 Rotation and the General Second-Degree Equation

**12. Plane Curves, Parametric Equations, and Polar Coordinates**

12.1 Plane Curves and Parametric Equations

12.2 Parametric Equations and Calculus

12.3 Polar Coordinates and Polar Graphs

12.4 Tangent Lines and Curve Sketching in Polar Coordinates

12.5 Area and Arc Length in Polar Coordinates

12.6 Polar Equations for Conics and Kepler's Laws

**13. Vectors and Curves in the Plane**

13.1 Vectors in the Plane

13.2 The Dot Product of Two Vectors

13.3 Vector-Valued Functions

13.4 Velocity and Acceleration

13.5 Tangent Vectors and Normal Vectors

13.6 Arc Length and Curvature

**14. Solid Analytic Geometry and Vectors in Space**

14.1 Space Coordinates and Vectors in Space

14.2 The Cross Product of Two Vectors in Space

14.3 Lines and Planes in Space

14.4 Surfaces in Space

14.5 Curves and Vector-Valued Functions in Space

14.6 Tangent Vectors, Normal Vectors, and Curvature in Space

**15. Functions of Several Variables**

15.1 Introduction to Functions of Several Variables

15.2 Limits and Continuity

15.3 Partial Derivatives

15.4 Differentials

15.5 Chain Rules for Functions of Several Variables

15.6 Directional Derivatives and Gradients

15.7 Tangent Planes and Normal Lines

15.8 Extrema of Functions of Two Variables

15.9 Applications of Extrema of Functions of Two Variables

15.10 Lagrange Multipliers

**16. Multiple Integration**

16.1 Iterated Integrals and Area in the Plane

16.2 Double Integrals and Volume

16.3 Change of Variables: Polar Coordinates

16.4 Center of Mass and Moments of Inertia

16.5 Surface Area

16.6 Triple Integrals and Applications

16.7 Cylindrical and Spherical Coordinates

16.8 Triple Integrals in Cylindrical and Spherical Coordinates

16.9 Change of Variables: Jacobians

**17. Vector Analysis**

17.1 Vector Fields

17.2 Line Integrals

17.3 Conservative Vector Fields and Independence of Path

17.4 Green's Theorem

17.5 Parametric Surfaces

17.6 Surface Integrals

17.7 Divergence Theorem

17.8 Stokes's Theorem

**18. Differential Equations**

18.1 Definitions and Basic Concepts

18.2 Separation of Variables in First-Order Equations

18.3 Exact First-Order Equations

18.4 First-Order Linear Differential Equations

18.5 Second-Order Homogeneous Linear Equations

18.6 Second-Order Nonhomogeneous Linear Equations

18.7 Series Solutions of Differential Equations

- Appendixes: A. Proofs of Selected Theorems; B. Basic Differentiation Rules for Elementary Functions; C. Integration Tables
- Answers to Odd-Numbered Exercises
- Index

Publisher Info

Publisher: Houghton Mifflin Harcourt

Published: 1998

International: No

Published: 1998

International: No

This traditional text continues to offer a balanced approach that combines the theoretical instruction of calculus with the best aspects of reform, including creative teaching and learning techniques such as the integration of technology, the use of real data, real-life applications, and mathematical models. The *Calculus with Analytic Geometry Alternate,* 6/e, offers a late approach to trigonometry for those instructors who wish to introduce it later in their courses.

- Before students are exposed to selected topics, optional boxed
*Explorations*allow them to discover concepts on their own, making them more likely to remember the material. - Extended lab projects using real-life applications are found in the new lab series. The supplementary lab manuals use Maple, Mathematica, Derive, Mathcad, and the TI-92 graphing calculator.
- The art program in the Sixth Edition was computer-generated for accuracy, clarity, and realism to
- Although students are not required to use technology, they are encouraged to use a graphing utility or computer algebra system as a tool for exploration, discovery, and problem solving. Calculus Alternate, 6/e, offers more opportunities for students to execute complicated computations, to visualize theoretical concepts, to discover alternative approaches, and to verify the results of other solution methods using technology.

Note: Each chapter concludes with Review Exercises.

**1. The Cartesian Plane and Functions**

1.1 Real Numbers and the Real Line

1.2 The Cartesian Plane

1.3 Graphs of Equations

1.4 Lines in the Plane

1.5 Functions

**2. Limits and Their Properties**

2.1 An Introduction to Limits

2.2 Techniques for Evaluating Limits

2.3 Continuity

2.4 Infinite Limits

2.5 e-d Definition of Limits

**3. Differentiation**

3.1 The Derivative and the Tangent Line Problem

3.2 Velocity, Acceleration, and Other Rates of Change

3.3 Differentiation Rules for Powers, Constant Multiples, and Sums

3.4 Differentiation Rules for Products and Quotients

3.5 The Chain Rule

3.6 Implicit Differentiation

3.7 Related Rates

**4. Applications of Differentiation**

4.1 Extrema on an Interval

4.2 Rolle's Theorem and the Mean Value Theorem

4.3 Increasing and Decreasing Functions and the First Derivative Test

4.4 Concavity and the Second Derivative Test

4.5 Limits at Infinity

4.6 A Summary of Curve Sketching

4.7 Optimization Problems

4.8 Newton's Method

4.9 Differentials

4.10 Business and Economics Applications

**5. Integration**

5.1 Antiderivatives and Indefinite Integration

5.2 Area

5.3 Riemann Sums and the Definite Integral

5.4 The Fundamental Theorem of Calculus

5.5 Integration by Substitution

5.6 Numerical Integration

**6. Applications of Integration**

6.1 Area of a Region Between Two Curves

6.2 Volume: The Disc Method

6.3 Volume: The Shell Method

6.4 Arc Length and Surfaces of Revolution

6.5 Work

6.6 Fluid Pressure and Fluid Force

6.7 Moments, Centers of Mass, and Centroids

**7. Exponential and Logarithmic Functions**

7.1 Exponential Functions

7.2 Differentiation and Integration of Exponential Functions

7.3 Inverse Functions

7.4 Logarithmic Functions

7.5 Logarithmic Functions and Differentiation

7.6 Logarithmic Functions and Integration

7.7 Growth and Decay

7.8 Indeterminate Forms and L

Hôpital's Rule

**8. Trigonometric Functions and Inverse Trigonometric Functions**

8.1 Review of Trigonometric Functions

8.2 Graphs and Limits of Trigonometric Functions

8.3 Derivatives of Trigonometric Functions

8.4 Integrals of Trigonometric Functions

8.5 Inverse Trigonometric Functions and Differentiation

8.6 Inverse Trigonometric Functions: Integration and Completing the Square

8.7 Hyperbolic Functions

**9. Integration Techniques and Improper Integrals**

9.1 Basic Integration Formulas

9.2 Integration by Parts

9.3 Trigonometric Integrals

9.4 Trigonometric Substitution

9.5 Partial Fractions

9.6 Integration by Tables and Other Integration Techniques

9.7 Improper Integrals

**10. Infinite Series**

10.1 Sequences

10.2 Series and Convergence

10.3 The Integral Test and p-Series

10.4 Comparisons of Series

10.5 Alternating Series

10.6 The Ratio and Root Tests

10.7 Taylor Polynomials and Approximations

10.8 Power Series

10.9 Representation of Functions by Power Series

10.10 Taylor and Maclaurin Series

**11. Conic Sections**

11.1 Parabolas

11.2 Ellipses

11.3 Hyperbolas

11.4 Rotation and the General Second-Degree Equation

**12. Plane Curves, Parametric Equations, and Polar Coordinates**

12.1 Plane Curves and Parametric Equations

12.2 Parametric Equations and Calculus

12.3 Polar Coordinates and Polar Graphs

12.4 Tangent Lines and Curve Sketching in Polar Coordinates

12.5 Area and Arc Length in Polar Coordinates

12.6 Polar Equations for Conics and Kepler's Laws

**13. Vectors and Curves in the Plane**

13.1 Vectors in the Plane

13.2 The Dot Product of Two Vectors

13.3 Vector-Valued Functions

13.4 Velocity and Acceleration

13.5 Tangent Vectors and Normal Vectors

13.6 Arc Length and Curvature

**14. Solid Analytic Geometry and Vectors in Space**

14.1 Space Coordinates and Vectors in Space

14.2 The Cross Product of Two Vectors in Space

14.3 Lines and Planes in Space

14.4 Surfaces in Space

14.5 Curves and Vector-Valued Functions in Space

14.6 Tangent Vectors, Normal Vectors, and Curvature in Space

**15. Functions of Several Variables**

15.1 Introduction to Functions of Several Variables

15.2 Limits and Continuity

15.3 Partial Derivatives

15.4 Differentials

15.5 Chain Rules for Functions of Several Variables

15.6 Directional Derivatives and Gradients

15.7 Tangent Planes and Normal Lines

15.8 Extrema of Functions of Two Variables

15.9 Applications of Extrema of Functions of Two Variables

15.10 Lagrange Multipliers

**16. Multiple Integration**

16.1 Iterated Integrals and Area in the Plane

16.2 Double Integrals and Volume

16.3 Change of Variables: Polar Coordinates

16.4 Center of Mass and Moments of Inertia

16.5 Surface Area

16.6 Triple Integrals and Applications

16.7 Cylindrical and Spherical Coordinates

16.8 Triple Integrals in Cylindrical and Spherical Coordinates

16.9 Change of Variables: Jacobians

**17. Vector Analysis**

17.1 Vector Fields

17.2 Line Integrals

17.3 Conservative Vector Fields and Independence of Path

17.4 Green's Theorem

17.5 Parametric Surfaces

17.6 Surface Integrals

17.7 Divergence Theorem

17.8 Stokes's Theorem

**18. Differential Equations**

18.1 Definitions and Basic Concepts

18.2 Separation of Variables in First-Order Equations

18.3 Exact First-Order Equations

18.4 First-Order Linear Differential Equations

18.5 Second-Order Homogeneous Linear Equations

18.6 Second-Order Nonhomogeneous Linear Equations

18.7 Series Solutions of Differential Equations

- Appendixes: A. Proofs of Selected Theorems; B. Basic Differentiation Rules for Elementary Functions; C. Integration Tables
- Answers to Odd-Numbered Exercises
- Index