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by Robert T. Smith and Roland B. Minton

Cover type: HardbackEdition: 06

Copyright: 2006

Publisher: McGraw-Hill Publishing Company

Published: 2006

International: No

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This modern calculus textbook places a strong emphasis on developing students' conceptual understanding and on building connections between key calculus topics and their relevance for the real world. It is written for the average student -- one who is mostly unfamiliar with the subject and who requires significant motivation. It follows a relatively standard order of presentation, with early coverage of transcendentals, and integrates thought-provoking applications, examples and exercises throughout. The text also provides balanced guidance on the appropriate role of technology in problem-solving, including its benefits and its potential pitfalls. Wherever practical, concepts are developed from graphical, numerical, algebraic and verbal perspectives (the "Rule of Four") to give students a complete understanding of calculus.

**New Features**

- EARLY TRANSCENDENTALS APPROACH: The text presents all transcendental functions and their derivatives as part of differential calculus so that students become familiar with these key topics early on and can utilize them in more advanced material later in the book.
- EMPHASIS ON PROBLEM SOLVING: The text strongly promotes problem-solving skills by frequently examining topics from graphical, numerical and algebraic points of view. This approach allows the authors to explore more realistic and complex problems than are often presented in calculus. As a result, students gain a better sense of the usefulness of calculus and can solve a wider variety of problems.
- EXPLANATIONS: The authors offer clear, intuitive explanations throughout the text to provide highly student-friendly introductions to complex topics in calculus.
- APPLICATIONS: The use of applications to relate calculus to the real world is a true strength of this text. Most concepts are developed with an applied focus in order to motivate the presentation of new topics, further illustrate topics already presented, and connect the conceptual development of calculus with students' everyday experiences.
- BALANCED EXERCISE SETS: This text contains thousands of exercises, found at the end of each section and chapter. Each exercise set has been carefully designed to provide a wide variety of routine, moderate, and challenging exercises. The authors have taken great care to create original and imaginative exercises that provide an appropriate review of the topics covered in each section and chapter.
- WRITING EXERCISES: Each exercise set begins with a set of writing exercises that are intended to encourage students to carefully consider important mathematical concepts and ideas and to express them in their own words. The writing exercises may also be used as springboards for class discussion.
- TECHNOLOGY GUIDANCE: The authors believe that technology is part of the coherent development of calculus and that students should learn to use it judiciously. Throughout the text, they discuss the appropriate role of technology in problem-solving and its benefits and drawbacks. The technology used in this text is shared by virtually all graphing calculators and computer algebra systems.

**0 Preliminaries**

0.1 Polynomial and Rational Functions

0.2 Graphing Calculators and Computer Algebra Systems

0.3 Inverse Functions

0.4 Trigonometric and Inverse Trigonometric Functions

0.5 Exponential and Logarithmic Functions

0.6 Transformations of Functions

0.7 Parametric Equations and Polar Coordinates

**1 Limits and Continuity**

1.1 A Brief Preview of Calculus

1.2 The Concept of Limit

1.3 Computation of Limits

1.4 Continuity and its Consequences

1.5 Limits Involving Infinity

1.6 Limits and Loss-of-Significance Errors

**2 Differentiation**

2.1 Tangent Lines and Velocity

2.2 The Derivative

2.3 Computation of Derivatives: The Power Rule

2.4 The Product and Quotient Rules

2.5 The Chain Rule

2.6 Derivatives of Trigonometric and Inverse Trigonometric Functions

2.7 Derivatives of Exponential and Logarithmic Functions

2.8 Implicit Differentiation

2.9 The Mean Value Theorem

**3 Applications of Differentiation**

3.1 Linear Approximations and Newton's Method

3.2 Indeterminate Forms and L'Hopital's Rule

3.3 Maximum and Minimum Values

3.4 Increasing and Decreasing Functions

3.5 Concavity and Overview of Curve Sketching

3.6 Optimization

3.7 Rates of Change in Economics and the Sciences

3.8 Related Rates and Parametric Equations

**4 Integration**

4.1 Area Under a Curve

4.2 The Definite Integral

4.3 Antiderivatives

4.4 The Fundamental Theorem of Calculus

4.5 Integration by Substitution

4.6 Integration by Parts

4.7 Other Techniques of Integration

4.8 Integration Tables and Computer Algebra Systems

4.9 Numerical Integration

4.10 Improper Integrals

**5 Applications of the Definite Integral**

5.1 Area Between Curves

5.2 Volume

5.3 Arc Length and Surface Area

5.4 Projectile Motion

5.5 Applications of Integration to Physics and Engineering

5.6 Probability

**6 Differential Equations**

6.1 Growth and Decay Problems

6.2 Separable Differential Equations

6.3 Euler's Method

6.4 Second Order Equations with Constant Coefficients

6.5 Nonhomogeneous Equations: Undetermined Coefficients

6.6 Applications of Differential Equations

**7 Infinite Series**

7.1 Sequences of Real Numbers

7.2 Infinite Series

7.3 The Integral Test and Comparison Tests

7.4 Alternating Series

7.5 Absolute Convergence and the Ratio Test

7.6 Power Series

7.7 Taylor Series

7.8 Applications of Taylor Series

7.9 Fourier Series

7.10 Power Series Solutions of Differential Equations

**8 Vectors and the Geometry of Space**

8.1 Vectors in the Plane

8.2 Vectors in Space

8.3 The Dot Product

8.4 The Cross Product

8.5 Lines and Planes in Space

8.6 Surfaces in Space

**9 Vector-Valued Functions**

9.1 Vector-Valued Functions

9.2 Parametric Surfaces

9.3 The Calculus of Vector-Valued Functions

9.4 Motion in Space

9.5 Curvature

9.6 Tangent and Normal Vectors

**10 Functions of Several Variables and Differentiation**

10.1 Functions of Several Variables

10.2 Limits and Continuity

10.3 Partial Derivatives

10.4 Tangent Planes and Linear Approximations

10.5 The Chain Rule

10.6 The Gradient and Directional Derivatives

10.7 Extrema of Functions of Several Variables

10.8 Constrained Optimization and Lagrange Multipliers

**11 Multiple Integrals**

11.1 Double Integrals

11.2 Area, Volume and Center of Mass

11.3 Double Integrals in Polar Coordinates

11.4 Surface Area

11.5 Triple Integrals

11.6 Cylindrical Coordinates

11.7 Spherical Coordinates

11.8 Change of Variables in Multiple Integrals

**12 Vector Calculus**

12.1 Vector Fields

12.2 Curl and Divergence

12.3 Line Integrals

12.4 Independence of Path and Conservative Vector Fields

12.5 Green's Theorem

12.6 Surface Integrals

12.7 The Divergence Theorem

12.8 Stokes' Theorem

12.9 Applications of Vector Calculus

Appendix A Graphs of Additional Polar Equations

Appendix B Formal Definition of Limit

Appendix C Complete Derivation of Derivatives of sin x and cos x

Appendix D Natural Logarithm Defined as an Integral; Exponential Defined as the Inverse of the Natural Logarithm

Appendix E Conic Sections in Polar Coordinates

Appendix F Proofs of Selected Theorems

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Summary

This modern calculus textbook places a strong emphasis on developing students' conceptual understanding and on building connections between key calculus topics and their relevance for the real world. It is written for the average student -- one who is mostly unfamiliar with the subject and who requires significant motivation. It follows a relatively standard order of presentation, with early coverage of transcendentals, and integrates thought-provoking applications, examples and exercises throughout. The text also provides balanced guidance on the appropriate role of technology in problem-solving, including its benefits and its potential pitfalls. Wherever practical, concepts are developed from graphical, numerical, algebraic and verbal perspectives (the "Rule of Four") to give students a complete understanding of calculus.

**New Features**

- EARLY TRANSCENDENTALS APPROACH: The text presents all transcendental functions and their derivatives as part of differential calculus so that students become familiar with these key topics early on and can utilize them in more advanced material later in the book.
- EMPHASIS ON PROBLEM SOLVING: The text strongly promotes problem-solving skills by frequently examining topics from graphical, numerical and algebraic points of view. This approach allows the authors to explore more realistic and complex problems than are often presented in calculus. As a result, students gain a better sense of the usefulness of calculus and can solve a wider variety of problems.
- EXPLANATIONS: The authors offer clear, intuitive explanations throughout the text to provide highly student-friendly introductions to complex topics in calculus.
- APPLICATIONS: The use of applications to relate calculus to the real world is a true strength of this text. Most concepts are developed with an applied focus in order to motivate the presentation of new topics, further illustrate topics already presented, and connect the conceptual development of calculus with students' everyday experiences.
- BALANCED EXERCISE SETS: This text contains thousands of exercises, found at the end of each section and chapter. Each exercise set has been carefully designed to provide a wide variety of routine, moderate, and challenging exercises. The authors have taken great care to create original and imaginative exercises that provide an appropriate review of the topics covered in each section and chapter.
- WRITING EXERCISES: Each exercise set begins with a set of writing exercises that are intended to encourage students to carefully consider important mathematical concepts and ideas and to express them in their own words. The writing exercises may also be used as springboards for class discussion.
- TECHNOLOGY GUIDANCE: The authors believe that technology is part of the coherent development of calculus and that students should learn to use it judiciously. Throughout the text, they discuss the appropriate role of technology in problem-solving and its benefits and drawbacks. The technology used in this text is shared by virtually all graphing calculators and computer algebra systems.

Table of Contents

**0 Preliminaries**

0.1 Polynomial and Rational Functions

0.2 Graphing Calculators and Computer Algebra Systems

0.3 Inverse Functions

0.4 Trigonometric and Inverse Trigonometric Functions

0.5 Exponential and Logarithmic Functions

0.6 Transformations of Functions

0.7 Parametric Equations and Polar Coordinates

**1 Limits and Continuity**

1.1 A Brief Preview of Calculus

1.2 The Concept of Limit

1.3 Computation of Limits

1.4 Continuity and its Consequences

1.5 Limits Involving Infinity

1.6 Limits and Loss-of-Significance Errors

**2 Differentiation**

2.1 Tangent Lines and Velocity

2.2 The Derivative

2.3 Computation of Derivatives: The Power Rule

2.4 The Product and Quotient Rules

2.5 The Chain Rule

2.6 Derivatives of Trigonometric and Inverse Trigonometric Functions

2.7 Derivatives of Exponential and Logarithmic Functions

2.8 Implicit Differentiation

2.9 The Mean Value Theorem

**3 Applications of Differentiation**

3.1 Linear Approximations and Newton's Method

3.2 Indeterminate Forms and L'Hopital's Rule

3.3 Maximum and Minimum Values

3.4 Increasing and Decreasing Functions

3.5 Concavity and Overview of Curve Sketching

3.6 Optimization

3.7 Rates of Change in Economics and the Sciences

3.8 Related Rates and Parametric Equations

**4 Integration**

4.1 Area Under a Curve

4.2 The Definite Integral

4.3 Antiderivatives

4.4 The Fundamental Theorem of Calculus

4.5 Integration by Substitution

4.6 Integration by Parts

4.7 Other Techniques of Integration

4.8 Integration Tables and Computer Algebra Systems

4.9 Numerical Integration

4.10 Improper Integrals

**5 Applications of the Definite Integral**

5.1 Area Between Curves

5.2 Volume

5.3 Arc Length and Surface Area

5.4 Projectile Motion

5.5 Applications of Integration to Physics and Engineering

5.6 Probability

**6 Differential Equations**

6.1 Growth and Decay Problems

6.2 Separable Differential Equations

6.3 Euler's Method

6.4 Second Order Equations with Constant Coefficients

6.5 Nonhomogeneous Equations: Undetermined Coefficients

6.6 Applications of Differential Equations

**7 Infinite Series**

7.1 Sequences of Real Numbers

7.2 Infinite Series

7.3 The Integral Test and Comparison Tests

7.4 Alternating Series

7.5 Absolute Convergence and the Ratio Test

7.6 Power Series

7.7 Taylor Series

7.8 Applications of Taylor Series

7.9 Fourier Series

7.10 Power Series Solutions of Differential Equations

**8 Vectors and the Geometry of Space**

8.1 Vectors in the Plane

8.2 Vectors in Space

8.3 The Dot Product

8.4 The Cross Product

8.5 Lines and Planes in Space

8.6 Surfaces in Space

**9 Vector-Valued Functions**

9.1 Vector-Valued Functions

9.2 Parametric Surfaces

9.3 The Calculus of Vector-Valued Functions

9.4 Motion in Space

9.5 Curvature

9.6 Tangent and Normal Vectors

**10 Functions of Several Variables and Differentiation**

10.1 Functions of Several Variables

10.2 Limits and Continuity

10.3 Partial Derivatives

10.4 Tangent Planes and Linear Approximations

10.5 The Chain Rule

10.6 The Gradient and Directional Derivatives

10.7 Extrema of Functions of Several Variables

10.8 Constrained Optimization and Lagrange Multipliers

**11 Multiple Integrals**

11.1 Double Integrals

11.2 Area, Volume and Center of Mass

11.3 Double Integrals in Polar Coordinates

11.4 Surface Area

11.5 Triple Integrals

11.6 Cylindrical Coordinates

11.7 Spherical Coordinates

11.8 Change of Variables in Multiple Integrals

**12 Vector Calculus**

12.1 Vector Fields

12.2 Curl and Divergence

12.3 Line Integrals

12.4 Independence of Path and Conservative Vector Fields

12.5 Green's Theorem

12.6 Surface Integrals

12.7 The Divergence Theorem

12.8 Stokes' Theorem

12.9 Applications of Vector Calculus

Appendix A Graphs of Additional Polar Equations

Appendix B Formal Definition of Limit

Appendix C Complete Derivation of Derivatives of sin x and cos x

Appendix D Natural Logarithm Defined as an Integral; Exponential Defined as the Inverse of the Natural Logarithm

Appendix E Conic Sections in Polar Coordinates

Appendix F Proofs of Selected Theorems

Publisher Info

Publisher: McGraw-Hill Publishing Company

Published: 2006

International: No

Published: 2006

International: No

This modern calculus textbook places a strong emphasis on developing students' conceptual understanding and on building connections between key calculus topics and their relevance for the real world. It is written for the average student -- one who is mostly unfamiliar with the subject and who requires significant motivation. It follows a relatively standard order of presentation, with early coverage of transcendentals, and integrates thought-provoking applications, examples and exercises throughout. The text also provides balanced guidance on the appropriate role of technology in problem-solving, including its benefits and its potential pitfalls. Wherever practical, concepts are developed from graphical, numerical, algebraic and verbal perspectives (the "Rule of Four") to give students a complete understanding of calculus.

**New Features**

- EARLY TRANSCENDENTALS APPROACH: The text presents all transcendental functions and their derivatives as part of differential calculus so that students become familiar with these key topics early on and can utilize them in more advanced material later in the book.
- EMPHASIS ON PROBLEM SOLVING: The text strongly promotes problem-solving skills by frequently examining topics from graphical, numerical and algebraic points of view. This approach allows the authors to explore more realistic and complex problems than are often presented in calculus. As a result, students gain a better sense of the usefulness of calculus and can solve a wider variety of problems.
- EXPLANATIONS: The authors offer clear, intuitive explanations throughout the text to provide highly student-friendly introductions to complex topics in calculus.
- APPLICATIONS: The use of applications to relate calculus to the real world is a true strength of this text. Most concepts are developed with an applied focus in order to motivate the presentation of new topics, further illustrate topics already presented, and connect the conceptual development of calculus with students' everyday experiences.
- BALANCED EXERCISE SETS: This text contains thousands of exercises, found at the end of each section and chapter. Each exercise set has been carefully designed to provide a wide variety of routine, moderate, and challenging exercises. The authors have taken great care to create original and imaginative exercises that provide an appropriate review of the topics covered in each section and chapter.
- WRITING EXERCISES: Each exercise set begins with a set of writing exercises that are intended to encourage students to carefully consider important mathematical concepts and ideas and to express them in their own words. The writing exercises may also be used as springboards for class discussion.
- TECHNOLOGY GUIDANCE: The authors believe that technology is part of the coherent development of calculus and that students should learn to use it judiciously. Throughout the text, they discuss the appropriate role of technology in problem-solving and its benefits and drawbacks. The technology used in this text is shared by virtually all graphing calculators and computer algebra systems.

**0 Preliminaries**

0.1 Polynomial and Rational Functions

0.2 Graphing Calculators and Computer Algebra Systems

0.3 Inverse Functions

0.4 Trigonometric and Inverse Trigonometric Functions

0.5 Exponential and Logarithmic Functions

0.6 Transformations of Functions

0.7 Parametric Equations and Polar Coordinates

**1 Limits and Continuity**

1.1 A Brief Preview of Calculus

1.2 The Concept of Limit

1.3 Computation of Limits

1.4 Continuity and its Consequences

1.5 Limits Involving Infinity

1.6 Limits and Loss-of-Significance Errors

**2 Differentiation**

2.1 Tangent Lines and Velocity

2.2 The Derivative

2.3 Computation of Derivatives: The Power Rule

2.4 The Product and Quotient Rules

2.5 The Chain Rule

2.6 Derivatives of Trigonometric and Inverse Trigonometric Functions

2.7 Derivatives of Exponential and Logarithmic Functions

2.8 Implicit Differentiation

2.9 The Mean Value Theorem

**3 Applications of Differentiation**

3.1 Linear Approximations and Newton's Method

3.2 Indeterminate Forms and L'Hopital's Rule

3.3 Maximum and Minimum Values

3.4 Increasing and Decreasing Functions

3.5 Concavity and Overview of Curve Sketching

3.6 Optimization

3.7 Rates of Change in Economics and the Sciences

3.8 Related Rates and Parametric Equations

**4 Integration**

4.1 Area Under a Curve

4.2 The Definite Integral

4.3 Antiderivatives

4.4 The Fundamental Theorem of Calculus

4.5 Integration by Substitution

4.6 Integration by Parts

4.7 Other Techniques of Integration

4.8 Integration Tables and Computer Algebra Systems

4.9 Numerical Integration

4.10 Improper Integrals

**5 Applications of the Definite Integral**

5.1 Area Between Curves

5.2 Volume

5.3 Arc Length and Surface Area

5.4 Projectile Motion

5.5 Applications of Integration to Physics and Engineering

5.6 Probability

**6 Differential Equations**

6.1 Growth and Decay Problems

6.2 Separable Differential Equations

6.3 Euler's Method

6.4 Second Order Equations with Constant Coefficients

6.5 Nonhomogeneous Equations: Undetermined Coefficients

6.6 Applications of Differential Equations

**7 Infinite Series**

7.1 Sequences of Real Numbers

7.2 Infinite Series

7.3 The Integral Test and Comparison Tests

7.4 Alternating Series

7.5 Absolute Convergence and the Ratio Test

7.6 Power Series

7.7 Taylor Series

7.8 Applications of Taylor Series

7.9 Fourier Series

7.10 Power Series Solutions of Differential Equations

**8 Vectors and the Geometry of Space**

8.1 Vectors in the Plane

8.2 Vectors in Space

8.3 The Dot Product

8.4 The Cross Product

8.5 Lines and Planes in Space

8.6 Surfaces in Space

**9 Vector-Valued Functions**

9.1 Vector-Valued Functions

9.2 Parametric Surfaces

9.3 The Calculus of Vector-Valued Functions

9.4 Motion in Space

9.5 Curvature

9.6 Tangent and Normal Vectors

**10 Functions of Several Variables and Differentiation**

10.1 Functions of Several Variables

10.2 Limits and Continuity

10.3 Partial Derivatives

10.4 Tangent Planes and Linear Approximations

10.5 The Chain Rule

10.6 The Gradient and Directional Derivatives

10.7 Extrema of Functions of Several Variables

10.8 Constrained Optimization and Lagrange Multipliers

**11 Multiple Integrals**

11.1 Double Integrals

11.2 Area, Volume and Center of Mass

11.3 Double Integrals in Polar Coordinates

11.4 Surface Area

11.5 Triple Integrals

11.6 Cylindrical Coordinates

11.7 Spherical Coordinates

11.8 Change of Variables in Multiple Integrals

**12 Vector Calculus**

12.1 Vector Fields

12.2 Curl and Divergence

12.3 Line Integrals

12.4 Independence of Path and Conservative Vector Fields

12.5 Green's Theorem

12.6 Surface Integrals

12.7 The Divergence Theorem

12.8 Stokes' Theorem

12.9 Applications of Vector Calculus

Appendix A Graphs of Additional Polar Equations

Appendix B Formal Definition of Limit

Appendix C Complete Derivation of Derivatives of sin x and cos x

Appendix D Natural Logarithm Defined as an Integral; Exponential Defined as the Inverse of the Natural Logarithm

Appendix E Conic Sections in Polar Coordinates

Appendix F Proofs of Selected Theorems