# Calculus

**Author(s):**
Karl J
Smith
,
Monty
Strauss
,
Magdalena
Toda

**Edition:
**
7

**Copyright:
**
2017

**Pages:
**
1292

**Edition:
**
7

**Copyright:
**
2017

**Choose Your Format**

** Calculus** blends the best aspects of calculus reform along with the goals and methodology of traditional calculus. The format of this text is enhanced, but is not dominated by new technology. Its innovative presentation includes:

- Conceptual Understanding through Verbalization
- Mathematical Communication
- Cooperative Learning Group Research Projects
- Integration of Technology
- Greater Text Visualization
- Supplementary Materials

The new seventh edition of ** Calculus**:

**Is an early transcendental book**, but does include an optional section that defines the logarithm as an integral.**Incorporates over 7,300 problems!**Most of these are available on WebAssign© which provides instant assessment.**Has been updated!**The new edition features drawing lessons that include hints on plotting points in three dimensions, drawing circles, ellipses, hyperbolas, trigonometric curves, and polar-form curves, as well as planes and lines in three dimensions.**Interactive art**-Many pieces of art in the book link online to dynamic art to illustrate such topics as limits, slopes, areas, and direction fields- An early presentation of transcendental functions: Logarithms, exponential functions, and trigonometric functions
- Differential equations in a natural and reasonable way
- Utilization of the
*humanness*of mathematics - Precalculus mathematics being taught at most colleges and universities correctly reflected
**Think Tank**problems to prove the proposition true or to find a counterexample to disprove the proposition**Exploration Problems**that go beyond the category of counterexample problem to provide opportunities for innovative thinking**Journal Problems**have been reprinted from leading mathematics journals in an effort to show that “mathematicians work problems too”**Modeling Problems**requires the reader to make assumptions about the real world in order to come up with the necessary formula or information to answer the question- A student solutions manual, instructor’s manual, and accompanying website

**1 Functions and Graphs **

1.1 What Is Calculus?

1.2 Preliminaries

1.3 Lines in the Plane; Parametric Equations

1.4 Functions and Graphs

1.5 Inverse Functions; Inverse Trigonometric Functions

Chapter 1 Review

**Book Report ***Ethnomathematics *by Marcia Ascher

Chapter 1 Group Research Project

**2 Limits and Continuity **

2.1 The Limit of a Function

2.2 Algebraic Computation of Limits

2.3 Continuity

2.4 Exponential and Logarithmic Functions

Chapter 2 Review

Chapter 2 Group Research Project

**3 Differentiation **

3.1 An Introduction to the Derivative: Tangents

3.2 Techniques of Differentiation

3.3 Derivatives of Trigonometric, Exponential, and Logarithmic Functions

3.4 Rates of Change: Modeling Rectilinear Motion

3.5 The Chain Rule

3.6 Implicit Differentiation

3.7 Related Rates and Applications

3.8 Linear Approximation and Differentials

Chapter 3 Review

**Book Report ***Fermat’s Enigma *by Simon Singh

Chapter 3 Group Research Project

**4 Additional Applications of the Derivative **

4.1 Extreme Values of a Continuous Function

4.2 The Mean Value Theorem

4.3 Using Derivatives to Sketch the Graph of a Function

4.4 Curve Sketching with Asymptotes: Limits Involving Infinity

4.5 Hopital’s Rule

4.6 Optimization in the Physical Sciences and Engineering

4.7 Optimization in Business, Economics, and the Life Sciences

Chapter 4 Review

Chapter 4 Group Research Project

**5 Integration **

5.1 Antidifferentiation

5.2 Area as the Limit of a Sum

5.3 Riemann Sums and the Definite Integral

5.4 The Fundamental Theorems of Calculus

5.5 Integration by Substitution

5.6 Introduction to Differential Equations

5.7 The Mean Value Theorem for Integrals; Average Value

5.8 Numerical Integration: The Trapezoidal Rule and Simpson’s Rule

5.9 An Alternative Approach: The Logarithm as an Integral

Chapter 5 Review

Chapter 5 Group Research Project

**Cumulative Review Problems—Chapters 1–5 **

**6 Additional Applications of the Integral **

6.1 Area Between Two Curves

6.2 Volume

6.3 Polar Forms and Area

6.4 Arc Length and Surface Area

6.5 Physical Applications: Work, Liquid Force, and Centroids

6.6 Applications to Business, Economics, and Life Sciences

Chapter 6 Review

**Book Report ***To Infinity and Beyond, A Cultural History*

*of the Infinite*, by Eli Maor

Chapter 6 Group Research Project

**7 Methods of Integration **

7.1 Review of Substitution and Integration by Table

7.2 Integration By Parts

7.3 Trigonometric Methods

7.4 Method of Partial Fractions

7.5 Summary of Integration Techniques

7.6 First-Order Differential Equations

7.7 Improper Integrals

7.8 Hyperbolic and Inverse Hyperbolic Functions

Chapter 7 Review

Chapter 7 Group Research Project

**8 Infinite Series **

8.1 Sequences and Their Limits

8.2 Introduction to Infinite Series; Geometric Series

8.3 The Integral Test; *p*-series

8.4 Comparison Tests

8.5 The Ratio Test and the Root Test

8.6 Alternating Series; Absolute and Conditional Convergence

8.7 Power Series

8.8 Taylor and Maclaurin Series

Chapter 8 Review

Chapter 8 Group Research Project

**Cumulative Review Problems—Chapters 6–8 **

**9 Vectors in the Plane and in Space **

9.1 Vectors in R2

9.2 Coordinates and Vectors in R3

9.3 The Dot Product

9.4 The Cross Product

9.5 Lines in R3

9.6 Planes in R3

9.7 Quadric Surfaces

Chapter 9 Review

Chapter 9 Group Research Project

**10 Vector-Valued Functions **

10.1 Introduction to Vector Functions

10.2 Differentiation and Integration of Vector Functions

10.3 Modeling Ballistics and Planetary Motion

10.4 Unit Tangent and Principal Unit Normal Vectors; Curvature

10.5 Tangential and Normal Components of Acceleration

Chapter 10 Review

Chapter 10 Group Research Project

**Cumulative Review Problems—Chapters 1–10 **

**11 Partial Differentiation **

11.1 Functions of Several Variables

11.2 Limits and Continuity

11.3 Partial Derivatives

11.4 Tangent Planes, Approximations, and Differentiability

11.5 Chain Rules

11.6 Directional Derivatives and the Gradient

11.7 Extrema of Functions of Two Variables

11.8 Lagrange Multipliers

Chapter 11 Review

**Book Report ***Hypatia’s Heritage *by Margaret Alic

Chapter 11 Group Research Project

**12 Multiple Integration **

12.1 Double Integration over Rectangular Regions

12.2 Double Integration over Nonrectangular Regions

12.3 Double Integrals in Polar Coordinates

12.4 Surface Area

12.5 Triple Integrals

12.6 Mass, Moments, and Probability Density Functions

12.7 Cylindrical and Spherical Coordinates

12.8 Jacobians: Change of Variables

Chapter 12 Review

Chapter 12 Group Research Project

**13 Vector Analysis **

13.1 Properties of a Vector Field: Divergence and Curl

13.2 Line Integrals

13.3 The Fundamental Theorem and Path Independence

13.4 Green’s Theorem

13.5 Surface Integrals

13.6 Stokes’ Theorem and Applications

13.7 Divergence Theorem and Applications

Chapter 13 Review

Chapter 13 Group Research Project

**Cumulative Review Problems—Chapters 11–13 **

**14 Introduction to Differential Equations **

14.1 First-Order Differential Equations

14.2 Second-Order Homogeneous Linear Differential Equations

14.3 Second-Order Nonhomogeneous Linear Differential Equations

Chapter 14 Review

**Book Report ***Mathematical Experience *by Philip J. Davis and

Reuben Hersh

Chapter 14 Group Research Project

**Appendices **

A: Introduction to the Theory of Limits

B: Selected Proofs

C: Significant Digits

D: Short Table of Integrals

E: Trigonometry

F: Parabolas

G: Ellipses

H: Hyperbolas

I: Determinants

J: Answers to Selected Problems

**Karl J Smith**

Karl J. Smith received his B.A. and M.A. degrees in mathematics from UCLA. In 1968, he moved to northern California to teach mathematics at Santa Rosa Junior College, where he taught until his retirement in 1993. Along the way, he served as department chair, and he received a Ph.D. in 1979 in mathematics education at Southeastern University. A past president of the *American Mathematical Association of Two-Year Colleges*, Professor Smith is active nationally in mathematics education. He was the founding editor of the *Western AMATYC News*, a chairperson of the Committee on Mathematics Excellence, and an NSF grant reviewer. In 1979 he received an *Outstanding Young Men of America Award*, in 1980 an *Outstanding Educator Award*, and in 1989 an *Outstanding Teacher Award*. Professor Smith is the author of over 60 successful textbooks. Over two million students have learned mathematics from his textbooks.

**Monty Strauss**

Monty Strauss has been on the mathematics faculty at Texas Tech University for almost forty years. He has a Ph.D. from the Courant Institute of Mathematical Sciences at New York University and has taught all levels of mathematics at Texas Tech, from precollege mathematics to doctoral level. He particularly has enjoyed working with honors students and with mathematics and engineering majors. Among his administrative assignments have been departmental undergraduate programs chair and departmental associate chair.

**Magdalena Toda**

Magdalena Toda holds a PhD in Mathematics from University of Kansas and a PhD in Applied Mathematics from University Politehnica Bucharest. She is employed as a Professor of Mathematics at Texas Tech University, in Lubbock, TX, where she has served as interim chairperson between 2015-2016, and as department chairperson since 2016.

She has authored and co-authored 35 refereed articles, in the areas of differential geometry and geometric PDEs, and has served as an editor for a research monograph that appeared in 2017. She has extensive experience in teaching Calculus, in both the traditional, face-to-face format (since 1995), and online (since 2011). She is a recipient of 6 teaching awards (2 at University of Kansas and 4 at Texas Tech University - including the *President's Award for Excellence in Teaching*, 2008).

** Calculus** blends the best aspects of calculus reform along with the goals and methodology of traditional calculus. The format of this text is enhanced, but is not dominated by new technology. Its innovative presentation includes:

- Conceptual Understanding through Verbalization
- Mathematical Communication
- Cooperative Learning Group Research Projects
- Integration of Technology
- Greater Text Visualization
- Supplementary Materials

The new seventh edition of ** Calculus**:

**Is an early transcendental book**, but does include an optional section that defines the logarithm as an integral.**Incorporates over 7,300 problems!**Most of these are available on WebAssign© which provides instant assessment.**Has been updated!**The new edition features drawing lessons that include hints on plotting points in three dimensions, drawing circles, ellipses, hyperbolas, trigonometric curves, and polar-form curves, as well as planes and lines in three dimensions.**Interactive art**-Many pieces of art in the book link online to dynamic art to illustrate such topics as limits, slopes, areas, and direction fields- An early presentation of transcendental functions: Logarithms, exponential functions, and trigonometric functions
- Differential equations in a natural and reasonable way
- Utilization of the
*humanness*of mathematics - Precalculus mathematics being taught at most colleges and universities correctly reflected
**Think Tank**problems to prove the proposition true or to find a counterexample to disprove the proposition**Exploration Problems**that go beyond the category of counterexample problem to provide opportunities for innovative thinking**Journal Problems**have been reprinted from leading mathematics journals in an effort to show that “mathematicians work problems too”**Modeling Problems**requires the reader to make assumptions about the real world in order to come up with the necessary formula or information to answer the question- A student solutions manual, instructor’s manual, and accompanying website

**1 Functions and Graphs **

1.1 What Is Calculus?

1.2 Preliminaries

1.3 Lines in the Plane; Parametric Equations

1.4 Functions and Graphs

1.5 Inverse Functions; Inverse Trigonometric Functions

Chapter 1 Review

**Book Report ***Ethnomathematics *by Marcia Ascher

Chapter 1 Group Research Project

**2 Limits and Continuity **

2.1 The Limit of a Function

2.2 Algebraic Computation of Limits

2.3 Continuity

2.4 Exponential and Logarithmic Functions

Chapter 2 Review

Chapter 2 Group Research Project

**3 Differentiation **

3.1 An Introduction to the Derivative: Tangents

3.2 Techniques of Differentiation

3.3 Derivatives of Trigonometric, Exponential, and Logarithmic Functions

3.4 Rates of Change: Modeling Rectilinear Motion

3.5 The Chain Rule

3.6 Implicit Differentiation

3.7 Related Rates and Applications

3.8 Linear Approximation and Differentials

Chapter 3 Review

**Book Report ***Fermat’s Enigma *by Simon Singh

Chapter 3 Group Research Project

**4 Additional Applications of the Derivative **

4.1 Extreme Values of a Continuous Function

4.2 The Mean Value Theorem

4.3 Using Derivatives to Sketch the Graph of a Function

4.4 Curve Sketching with Asymptotes: Limits Involving Infinity

4.5 Hopital’s Rule

4.6 Optimization in the Physical Sciences and Engineering

4.7 Optimization in Business, Economics, and the Life Sciences

Chapter 4 Review

Chapter 4 Group Research Project

**5 Integration **

5.1 Antidifferentiation

5.2 Area as the Limit of a Sum

5.3 Riemann Sums and the Definite Integral

5.4 The Fundamental Theorems of Calculus

5.5 Integration by Substitution

5.6 Introduction to Differential Equations

5.7 The Mean Value Theorem for Integrals; Average Value

5.8 Numerical Integration: The Trapezoidal Rule and Simpson’s Rule

5.9 An Alternative Approach: The Logarithm as an Integral

Chapter 5 Review

Chapter 5 Group Research Project

**Cumulative Review Problems—Chapters 1–5 **

**6 Additional Applications of the Integral **

6.1 Area Between Two Curves

6.2 Volume

6.3 Polar Forms and Area

6.4 Arc Length and Surface Area

6.5 Physical Applications: Work, Liquid Force, and Centroids

6.6 Applications to Business, Economics, and Life Sciences

Chapter 6 Review

**Book Report ***To Infinity and Beyond, A Cultural History*

*of the Infinite*, by Eli Maor

Chapter 6 Group Research Project

**7 Methods of Integration **

7.1 Review of Substitution and Integration by Table

7.2 Integration By Parts

7.3 Trigonometric Methods

7.4 Method of Partial Fractions

7.5 Summary of Integration Techniques

7.6 First-Order Differential Equations

7.7 Improper Integrals

7.8 Hyperbolic and Inverse Hyperbolic Functions

Chapter 7 Review

Chapter 7 Group Research Project

**8 Infinite Series **

8.1 Sequences and Their Limits

8.2 Introduction to Infinite Series; Geometric Series

8.3 The Integral Test; *p*-series

8.4 Comparison Tests

8.5 The Ratio Test and the Root Test

8.6 Alternating Series; Absolute and Conditional Convergence

8.7 Power Series

8.8 Taylor and Maclaurin Series

Chapter 8 Review

Chapter 8 Group Research Project

**Cumulative Review Problems—Chapters 6–8 **

**9 Vectors in the Plane and in Space **

9.1 Vectors in R2

9.2 Coordinates and Vectors in R3

9.3 The Dot Product

9.4 The Cross Product

9.5 Lines in R3

9.6 Planes in R3

9.7 Quadric Surfaces

Chapter 9 Review

Chapter 9 Group Research Project

**10 Vector-Valued Functions **

10.1 Introduction to Vector Functions

10.2 Differentiation and Integration of Vector Functions

10.3 Modeling Ballistics and Planetary Motion

10.4 Unit Tangent and Principal Unit Normal Vectors; Curvature

10.5 Tangential and Normal Components of Acceleration

Chapter 10 Review

Chapter 10 Group Research Project

**Cumulative Review Problems—Chapters 1–10 **

**11 Partial Differentiation **

11.1 Functions of Several Variables

11.2 Limits and Continuity

11.3 Partial Derivatives

11.4 Tangent Planes, Approximations, and Differentiability

11.5 Chain Rules

11.6 Directional Derivatives and the Gradient

11.7 Extrema of Functions of Two Variables

11.8 Lagrange Multipliers

Chapter 11 Review

**Book Report ***Hypatia’s Heritage *by Margaret Alic

Chapter 11 Group Research Project

**12 Multiple Integration **

12.1 Double Integration over Rectangular Regions

12.2 Double Integration over Nonrectangular Regions

12.3 Double Integrals in Polar Coordinates

12.4 Surface Area

12.5 Triple Integrals

12.6 Mass, Moments, and Probability Density Functions

12.7 Cylindrical and Spherical Coordinates

12.8 Jacobians: Change of Variables

Chapter 12 Review

Chapter 12 Group Research Project

**13 Vector Analysis **

13.1 Properties of a Vector Field: Divergence and Curl

13.2 Line Integrals

13.3 The Fundamental Theorem and Path Independence

13.4 Green’s Theorem

13.5 Surface Integrals

13.6 Stokes’ Theorem and Applications

13.7 Divergence Theorem and Applications

Chapter 13 Review

Chapter 13 Group Research Project

**Cumulative Review Problems—Chapters 11–13 **

**14 Introduction to Differential Equations **

14.1 First-Order Differential Equations

14.2 Second-Order Homogeneous Linear Differential Equations

14.3 Second-Order Nonhomogeneous Linear Differential Equations

Chapter 14 Review

**Book Report ***Mathematical Experience *by Philip J. Davis and

Reuben Hersh

Chapter 14 Group Research Project

**Appendices **

A: Introduction to the Theory of Limits

B: Selected Proofs

C: Significant Digits

D: Short Table of Integrals

E: Trigonometry

F: Parabolas

G: Ellipses

H: Hyperbolas

I: Determinants

J: Answers to Selected Problems

**Karl J Smith**

Karl J. Smith received his B.A. and M.A. degrees in mathematics from UCLA. In 1968, he moved to northern California to teach mathematics at Santa Rosa Junior College, where he taught until his retirement in 1993. Along the way, he served as department chair, and he received a Ph.D. in 1979 in mathematics education at Southeastern University. A past president of the *American Mathematical Association of Two-Year Colleges*, Professor Smith is active nationally in mathematics education. He was the founding editor of the *Western AMATYC News*, a chairperson of the Committee on Mathematics Excellence, and an NSF grant reviewer. In 1979 he received an *Outstanding Young Men of America Award*, in 1980 an *Outstanding Educator Award*, and in 1989 an *Outstanding Teacher Award*. Professor Smith is the author of over 60 successful textbooks. Over two million students have learned mathematics from his textbooks.

**Monty Strauss**

Monty Strauss has been on the mathematics faculty at Texas Tech University for almost forty years. He has a Ph.D. from the Courant Institute of Mathematical Sciences at New York University and has taught all levels of mathematics at Texas Tech, from precollege mathematics to doctoral level. He particularly has enjoyed working with honors students and with mathematics and engineering majors. Among his administrative assignments have been departmental undergraduate programs chair and departmental associate chair.

**Magdalena Toda**

Magdalena Toda holds a PhD in Mathematics from University of Kansas and a PhD in Applied Mathematics from University Politehnica Bucharest. She is employed as a Professor of Mathematics at Texas Tech University, in Lubbock, TX, where she has served as interim chairperson between 2015-2016, and as department chairperson since 2016.

She has authored and co-authored 35 refereed articles, in the areas of differential geometry and geometric PDEs, and has served as an editor for a research monograph that appeared in 2017. She has extensive experience in teaching Calculus, in both the traditional, face-to-face format (since 1995), and online (since 2011). She is a recipient of 6 teaching awards (2 at University of Kansas and 4 at Texas Tech University - including the *President's Award for Excellence in Teaching*, 2008).