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Calculus for K-12 Textbooks

by Ron Larson and Elizabeth Farber

Edition: 8TH 06Copyright: 2006

Publisher: Houghton Mifflin Harcourt

Published: 2006

International: No

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This revision of the best-selling text for Advanced Placement Calculus incorporates feedback from dozens of previous-edition users and reflects the latest AP guidelines while retaining the solid, well-motivated mathematics that characterize earlier editions. The Eighth Edition continues to provide an evolving range of conceptual, technological, and creative tools that enable students to learn more effectively.

**1. Limits and Their Properties **

1.1 A Preview of Calculus

1.2 Finding Limits Graphically and Numerically

1.3 Evaluating Limits Analytically

1.4 Continuity and One-Sided Limits

1.5 Infinite Limits

Section Project: Graphs and Limits of Trigonometric Functions

**2. Differentiation **

2.1 The Derivative and the Tangent Line Problem

2.2 Basic Differentiation Rules and Rates of Change

2.3 The Product and Quotient Rules and Higher-Order Derivatives

2.4 The Chain Rule

2.5 Implicit Differentiation

Section Project: Optical Illusions

2.6 Related Rates

**3. Applications of Differentiation **

3.1 Extrema on an Interval

3.2 Rolle's Theorem and the Mean Value Theorem

3.3 Increasing and Decreasing Functions and the First Derivative Test

Section Project: Rainbows

3.4 Concavity and the Second Derivative Test

3.5 Limits at Infinity

3.6 A Summary of Curve Sketching

3.7 Optimization Problems

Section Project: Connecticut River

3.8 Newton's Method

3.9 Differentials

**4. Integration **

4.1 Antiderivatives and Indefinite Integration

4.2 Area

4.3 Riemann Sums and Definite Integrals

4.4 The Fundamental Theorem of Calculus

Section Project: Demonstrating the Fundamental Theorem

4.5 Integration by Substitution

4.6 Numerical Integration

**5. Logarithmic, Exponential, and Other Transcendental Functions **

5.1 The Natural Logarithmic Function: Differentiation

5.2 The Natural Logarithmic Function: Integration

5.3 Inverse Functions

5.4 Exponential Functions: Differentiation and Integration

5.5 Bases Other than e and Applications

Section Project: Using Graphing Utilities to Estimate Slope

5.6 Inverse Trigonometric Functions: Differentiation

5.7 Inverse Trigonometric Functions: Integration

5.8 Hyperbolic Functions

Section Project: St. Louis Arch

**6. Differential Equations **

6.1 Slope Fields and Euler's Method

6.2 Differential Equations: Growth and Decay

6.3 Separation of Variables and the Logistic Equation

6.4 First-Order Linear Differential Equations

Section Project: Weight Loss

**7. Applications of Integration **

7.1 Area of a Region Between Two Curves

7.2 Volume: The Disk Method

7.3 Volume: The Shell Method

Section Project: Saturn

7.4 Arc Length and Surfaces of Revolution

7.5 Work

Section Project: Tidal Energy

7.6 Moments, Centers of Mass, and Centroids

7.7 Fluid Pressure and Fluid Force

**8. Integration Techniques, L'Hopital's Rule, and Improper Integrals **

8.1 Basic Integration Rules

8.2 Integration by Parts

8.3 Trigonometric Integrals

Section Project: Power Lines

8.4 Trigonometric Substitution

8.5 Partial Fractions

8.6 Integration by Tables and Other Integration Techniques

8.7 Indeterminate Forms and L'Hopital's Rule

8.8 Improper Integrals

**9. Infinite Series **

9.1 Sequences

9.2 Series and Convergence

Section Project: Cantor's Disappearing Table

9.3 The Integral Test and p-Series

Section Project: The Harmonic Series

9.4 Comparisons of Series

Section Project: Solera Method

9.5 Alternating Series

9.6 The Ratio and Root Tests

9.7 Taylor Polynomials and Approximations

9.8 Power Series

9.9 Representation of Functions by Power Series

9.10 Taylor and Maclaurin Series

**10. Conics, Parametric Equations, and Polar Coordinates **

10.1 Conics and Calculus

10.2 Plane Curves and Parametric Equations

Section Project: Cycloids

10.3 Parametric Equations and Calculus

10.4 Polar Coordinates and Polar Graphs

Section Project: Anamorphic Art

10.5 Area and Arc Length in Polar Coordinates

10.6 Polar Equations of Conics and Kepler's Laws

**11. Vectors and the Geometry of Space **

11.1 Vectors in the Plane

11.2 Space Coordinates and Vectors in Space

11.3 The Dot Product of Two Vectors

11.4 The Cross Product of Two Vectors in Space

11.5 Lines and Planes in Space

Section Project: Distances in Space

11.6 Surfaces in Space

11.7 Cylindrical and Spherical Coordinates

**12. Vector-Valued Functions **

12.1 Vector-Valued Functions

Section Project: Witch of Agnesi

12.2 Differentiation and Integration of Vector-Valued Functions

12.3 Velocity and Acceleration

12.4 Tangent Vectors and Normal Vectors

12.5 Arc Length and Curvature

**13. Functions of Several Variables **

13.1 Introduction to Functions of Several Variables

13.2 Limits and Continuity

13.3 Partial Derivatives

Section Project: Moiré Fringes

13.4 Differentials

13.5 Chain Rules for Functions of Several Variables

13.6 Directional Derivatives and Gradients

13.7 Tangent Planes and Normal Lines

Section Project: Wildflowers

13.8 Extrema of Functions of Two Variables

13.9 Applications of Extrema of Functions of Two Variables

Section Project: Building a Pipeline

13.10 Lagrange Multipliers

**14. Multiple Integration **

14.1 Iterated Integrals and Area in the Plane

14.2 Double Integrals and Volume

14.3 Change of Variables: Polar Coordinates

14.4 Center of Mass and Moments of Inertia

Section Project: Center of Pressure on a Sail

14.5 Surface Area

Section Project: Capillary Action

14.6 Triple Integrals and Applications

14.7 Triple Integrals in Cylindrical and Spherical Coordinates

Section Project: Wrinkled and Bumpy Spheres

14.8 Change of Variables: Jacobians

**15. Vector Analysis **15.1 Vector Fields

15.2 Line Integrals

15.3 Conservative Vector Fields and Independence of Path

15.4 Green's Theorem

Section Project: Hyperbolic and Trigonometric Functions

15.5 Parametric Surfaces

15.6 Surface Integrals

Section Project: Hyperboloid of One Sheet

15.7 Divergence Theorem

15.8 Stokes's Theorem

**Section Project: The Planimeter**

Appendix

A. Proofs of Selected Theorems

B. Integration Tables

C. Additional Topics in Differential Equations (Web Only)

C.1 Exact First-Order Equations

C.2 Second-Order Homogeneous Linear Equations

C.3 Second-Order Nonhomogeneous Linear Equations

C.4 Series Solutions of Differential Equations

D. Precalculus Review (Web Only)

D.1 Real Numbers and the Real Number Line

D.2 The Cartesian Plane

D.3 Review of Trigonometric Functions

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Summary

This revision of the best-selling text for Advanced Placement Calculus incorporates feedback from dozens of previous-edition users and reflects the latest AP guidelines while retaining the solid, well-motivated mathematics that characterize earlier editions. The Eighth Edition continues to provide an evolving range of conceptual, technological, and creative tools that enable students to learn more effectively.

Table of Contents

**1. Limits and Their Properties **

1.1 A Preview of Calculus

1.2 Finding Limits Graphically and Numerically

1.3 Evaluating Limits Analytically

1.4 Continuity and One-Sided Limits

1.5 Infinite Limits

Section Project: Graphs and Limits of Trigonometric Functions

**2. Differentiation **

2.1 The Derivative and the Tangent Line Problem

2.2 Basic Differentiation Rules and Rates of Change

2.3 The Product and Quotient Rules and Higher-Order Derivatives

2.4 The Chain Rule

2.5 Implicit Differentiation

Section Project: Optical Illusions

2.6 Related Rates

**3. Applications of Differentiation **

3.1 Extrema on an Interval

3.2 Rolle's Theorem and the Mean Value Theorem

3.3 Increasing and Decreasing Functions and the First Derivative Test

Section Project: Rainbows

3.4 Concavity and the Second Derivative Test

3.5 Limits at Infinity

3.6 A Summary of Curve Sketching

3.7 Optimization Problems

Section Project: Connecticut River

3.8 Newton's Method

3.9 Differentials

**4. Integration **

4.1 Antiderivatives and Indefinite Integration

4.2 Area

4.3 Riemann Sums and Definite Integrals

4.4 The Fundamental Theorem of Calculus

Section Project: Demonstrating the Fundamental Theorem

4.5 Integration by Substitution

4.6 Numerical Integration

**5. Logarithmic, Exponential, and Other Transcendental Functions **

5.1 The Natural Logarithmic Function: Differentiation

5.2 The Natural Logarithmic Function: Integration

5.3 Inverse Functions

5.4 Exponential Functions: Differentiation and Integration

5.5 Bases Other than e and Applications

Section Project: Using Graphing Utilities to Estimate Slope

5.6 Inverse Trigonometric Functions: Differentiation

5.7 Inverse Trigonometric Functions: Integration

5.8 Hyperbolic Functions

Section Project: St. Louis Arch

**6. Differential Equations **

6.1 Slope Fields and Euler's Method

6.2 Differential Equations: Growth and Decay

6.3 Separation of Variables and the Logistic Equation

6.4 First-Order Linear Differential Equations

Section Project: Weight Loss

**7. Applications of Integration **

7.1 Area of a Region Between Two Curves

7.2 Volume: The Disk Method

7.3 Volume: The Shell Method

Section Project: Saturn

7.4 Arc Length and Surfaces of Revolution

7.5 Work

Section Project: Tidal Energy

7.6 Moments, Centers of Mass, and Centroids

7.7 Fluid Pressure and Fluid Force

**8. Integration Techniques, L'Hopital's Rule, and Improper Integrals **

8.1 Basic Integration Rules

8.2 Integration by Parts

8.3 Trigonometric Integrals

Section Project: Power Lines

8.4 Trigonometric Substitution

8.5 Partial Fractions

8.6 Integration by Tables and Other Integration Techniques

8.7 Indeterminate Forms and L'Hopital's Rule

8.8 Improper Integrals

**9. Infinite Series **

9.1 Sequences

9.2 Series and Convergence

Section Project: Cantor's Disappearing Table

9.3 The Integral Test and p-Series

Section Project: The Harmonic Series

9.4 Comparisons of Series

Section Project: Solera Method

9.5 Alternating Series

9.6 The Ratio and Root Tests

9.7 Taylor Polynomials and Approximations

9.8 Power Series

9.9 Representation of Functions by Power Series

9.10 Taylor and Maclaurin Series

**10. Conics, Parametric Equations, and Polar Coordinates **

10.1 Conics and Calculus

10.2 Plane Curves and Parametric Equations

Section Project: Cycloids

10.3 Parametric Equations and Calculus

10.4 Polar Coordinates and Polar Graphs

Section Project: Anamorphic Art

10.5 Area and Arc Length in Polar Coordinates

10.6 Polar Equations of Conics and Kepler's Laws

**11. Vectors and the Geometry of Space **

11.1 Vectors in the Plane

11.2 Space Coordinates and Vectors in Space

11.3 The Dot Product of Two Vectors

11.4 The Cross Product of Two Vectors in Space

11.5 Lines and Planes in Space

Section Project: Distances in Space

11.6 Surfaces in Space

11.7 Cylindrical and Spherical Coordinates

**12. Vector-Valued Functions **

12.1 Vector-Valued Functions

Section Project: Witch of Agnesi

12.2 Differentiation and Integration of Vector-Valued Functions

12.3 Velocity and Acceleration

12.4 Tangent Vectors and Normal Vectors

12.5 Arc Length and Curvature

**13. Functions of Several Variables **

13.1 Introduction to Functions of Several Variables

13.2 Limits and Continuity

13.3 Partial Derivatives

Section Project: Moiré Fringes

13.4 Differentials

13.5 Chain Rules for Functions of Several Variables

13.6 Directional Derivatives and Gradients

13.7 Tangent Planes and Normal Lines

Section Project: Wildflowers

13.8 Extrema of Functions of Two Variables

13.9 Applications of Extrema of Functions of Two Variables

Section Project: Building a Pipeline

13.10 Lagrange Multipliers

**14. Multiple Integration **

14.1 Iterated Integrals and Area in the Plane

14.2 Double Integrals and Volume

14.3 Change of Variables: Polar Coordinates

14.4 Center of Mass and Moments of Inertia

Section Project: Center of Pressure on a Sail

14.5 Surface Area

Section Project: Capillary Action

14.6 Triple Integrals and Applications

14.7 Triple Integrals in Cylindrical and Spherical Coordinates

Section Project: Wrinkled and Bumpy Spheres

14.8 Change of Variables: Jacobians

**15. Vector Analysis **15.1 Vector Fields

15.2 Line Integrals

15.3 Conservative Vector Fields and Independence of Path

15.4 Green's Theorem

Section Project: Hyperbolic and Trigonometric Functions

15.5 Parametric Surfaces

15.6 Surface Integrals

Section Project: Hyperboloid of One Sheet

15.7 Divergence Theorem

15.8 Stokes's Theorem

**Section Project: The Planimeter**

Appendix

A. Proofs of Selected Theorems

B. Integration Tables

C. Additional Topics in Differential Equations (Web Only)

C.1 Exact First-Order Equations

C.2 Second-Order Homogeneous Linear Equations

C.3 Second-Order Nonhomogeneous Linear Equations

C.4 Series Solutions of Differential Equations

D. Precalculus Review (Web Only)

D.1 Real Numbers and the Real Number Line

D.2 The Cartesian Plane

D.3 Review of Trigonometric Functions

Publisher Info

Publisher: Houghton Mifflin Harcourt

Published: 2006

International: No

Published: 2006

International: No

**1. Limits and Their Properties **

1.1 A Preview of Calculus

1.2 Finding Limits Graphically and Numerically

1.3 Evaluating Limits Analytically

1.4 Continuity and One-Sided Limits

1.5 Infinite Limits

Section Project: Graphs and Limits of Trigonometric Functions

**2. Differentiation **

2.1 The Derivative and the Tangent Line Problem

2.2 Basic Differentiation Rules and Rates of Change

2.3 The Product and Quotient Rules and Higher-Order Derivatives

2.4 The Chain Rule

2.5 Implicit Differentiation

Section Project: Optical Illusions

2.6 Related Rates

**3. Applications of Differentiation **

3.1 Extrema on an Interval

3.2 Rolle's Theorem and the Mean Value Theorem

3.3 Increasing and Decreasing Functions and the First Derivative Test

Section Project: Rainbows

3.4 Concavity and the Second Derivative Test

3.5 Limits at Infinity

3.6 A Summary of Curve Sketching

3.7 Optimization Problems

Section Project: Connecticut River

3.8 Newton's Method

3.9 Differentials

**4. Integration **

4.1 Antiderivatives and Indefinite Integration

4.2 Area

4.3 Riemann Sums and Definite Integrals

4.4 The Fundamental Theorem of Calculus

Section Project: Demonstrating the Fundamental Theorem

4.5 Integration by Substitution

4.6 Numerical Integration

**5. Logarithmic, Exponential, and Other Transcendental Functions **

5.1 The Natural Logarithmic Function: Differentiation

5.2 The Natural Logarithmic Function: Integration

5.3 Inverse Functions

5.4 Exponential Functions: Differentiation and Integration

5.5 Bases Other than e and Applications

Section Project: Using Graphing Utilities to Estimate Slope

5.6 Inverse Trigonometric Functions: Differentiation

5.7 Inverse Trigonometric Functions: Integration

5.8 Hyperbolic Functions

Section Project: St. Louis Arch

**6. Differential Equations **

6.1 Slope Fields and Euler's Method

6.2 Differential Equations: Growth and Decay

6.3 Separation of Variables and the Logistic Equation

6.4 First-Order Linear Differential Equations

Section Project: Weight Loss

**7. Applications of Integration **

7.1 Area of a Region Between Two Curves

7.2 Volume: The Disk Method

7.3 Volume: The Shell Method

Section Project: Saturn

7.4 Arc Length and Surfaces of Revolution

7.5 Work

Section Project: Tidal Energy

7.6 Moments, Centers of Mass, and Centroids

7.7 Fluid Pressure and Fluid Force

**8. Integration Techniques, L'Hopital's Rule, and Improper Integrals **

8.1 Basic Integration Rules

8.2 Integration by Parts

8.3 Trigonometric Integrals

Section Project: Power Lines

8.4 Trigonometric Substitution

8.5 Partial Fractions

8.6 Integration by Tables and Other Integration Techniques

8.7 Indeterminate Forms and L'Hopital's Rule

8.8 Improper Integrals

**9. Infinite Series **

9.1 Sequences

9.2 Series and Convergence

Section Project: Cantor's Disappearing Table

9.3 The Integral Test and p-Series

Section Project: The Harmonic Series

9.4 Comparisons of Series

Section Project: Solera Method

9.5 Alternating Series

9.6 The Ratio and Root Tests

9.7 Taylor Polynomials and Approximations

9.8 Power Series

9.9 Representation of Functions by Power Series

9.10 Taylor and Maclaurin Series

**10. Conics, Parametric Equations, and Polar Coordinates **

10.1 Conics and Calculus

10.2 Plane Curves and Parametric Equations

Section Project: Cycloids

10.3 Parametric Equations and Calculus

10.4 Polar Coordinates and Polar Graphs

Section Project: Anamorphic Art

10.5 Area and Arc Length in Polar Coordinates

10.6 Polar Equations of Conics and Kepler's Laws

**11. Vectors and the Geometry of Space **

11.1 Vectors in the Plane

11.2 Space Coordinates and Vectors in Space

11.3 The Dot Product of Two Vectors

11.4 The Cross Product of Two Vectors in Space

11.5 Lines and Planes in Space

Section Project: Distances in Space

11.6 Surfaces in Space

11.7 Cylindrical and Spherical Coordinates

**12. Vector-Valued Functions **

12.1 Vector-Valued Functions

Section Project: Witch of Agnesi

12.2 Differentiation and Integration of Vector-Valued Functions

12.3 Velocity and Acceleration

12.4 Tangent Vectors and Normal Vectors

12.5 Arc Length and Curvature

**13. Functions of Several Variables **

13.1 Introduction to Functions of Several Variables

13.2 Limits and Continuity

13.3 Partial Derivatives

Section Project: Moiré Fringes

13.4 Differentials

13.5 Chain Rules for Functions of Several Variables

13.6 Directional Derivatives and Gradients

13.7 Tangent Planes and Normal Lines

Section Project: Wildflowers

13.8 Extrema of Functions of Two Variables

13.9 Applications of Extrema of Functions of Two Variables

Section Project: Building a Pipeline

13.10 Lagrange Multipliers

**14. Multiple Integration **

14.1 Iterated Integrals and Area in the Plane

14.2 Double Integrals and Volume

14.3 Change of Variables: Polar Coordinates

14.4 Center of Mass and Moments of Inertia

Section Project: Center of Pressure on a Sail

14.5 Surface Area

Section Project: Capillary Action

14.6 Triple Integrals and Applications

14.7 Triple Integrals in Cylindrical and Spherical Coordinates

Section Project: Wrinkled and Bumpy Spheres

14.8 Change of Variables: Jacobians

**15. Vector Analysis **15.1 Vector Fields

15.2 Line Integrals

15.3 Conservative Vector Fields and Independence of Path

15.4 Green's Theorem

Section Project: Hyperbolic and Trigonometric Functions

15.5 Parametric Surfaces

15.6 Surface Integrals

Section Project: Hyperboloid of One Sheet

15.7 Divergence Theorem

15.8 Stokes's Theorem

**Section Project: The Planimeter**

Appendix

A. Proofs of Selected Theorems

B. Integration Tables

C. Additional Topics in Differential Equations (Web Only)

C.1 Exact First-Order Equations

C.2 Second-Order Homogeneous Linear Equations

C.3 Second-Order Nonhomogeneous Linear Equations

C.4 Series Solutions of Differential Equations

D. Precalculus Review (Web Only)

D.1 Real Numbers and the Real Number Line

D.2 The Cartesian Plane

D.3 Review of Trigonometric Functions