Calculus: Single Variable

Smith/Minton: Mathematically Precise. Student-Friendly. Superior Technology.Students who have used Smith/Minton's Calculus say it was easier to read than any other math book they've used. That testimony underscores the success of the authors’ approach which combines the most reliable aspects of mainstream Calculus teaching with the best elements of reform, resulting in a motivating, challenging book. Smith/Minton wrote the book for the students who will use it, in a language that they understand, and with the expectation that their backgrounds may have some gaps. Smith/Minton provide exceptional, reality-based applications that appeal to students’ interests and demonstrate the elegance of math in the world around us. New features include: • Many new exercises and examples (for a total of 7,000 exercises and 1000 examples throughout the book) provide a careful balance of routine, intermediate and challenging exercises • New exploratory exercises in every section that challenge students to make connections to previous introduced material. • New commentaries (“Beyond Formulas”) that encourage students to think mathematically beyond the procedures they learn. • New counterpoints to the historical notes, “Today in Mathematics,” stress the contemporary dynamism of mathematical research and applications, connecting past contributions to the present. • An enhanced discussion of differential equations and additional applications of vector calculus. • Exceptional Media Resources: Within MathZone, instructors and students have access to a series of unique Conceptual Videos that help students understand key Calculus concepts proven to be most difficult to comprehend, 248 Interactive Applets that help students master concepts and procedures and functions, 1600 algorithms , and 113 e-Professors.

Chapter 0: Preliminaries

0.1 The Real Numbers and the Cartesian Plane

0.2 Lines and Functions

0.3 Graphing Calculators and Computer Algebra Systems

0.4 Trigonometric Functions

0.5 Transformations of Functions

Chapter 1: Limits and Continuity

1.1 A Brief Preview of Calculus: Tangent Lines and the Length of a Curve

1.2 The Concept of Limit

1.3 Computation of Limits

1.4 Continuity and its Consequences

The Method of Bisections

1.5 Limits Involving Infinity

Asysmptotes

1.6 The Formal Definition of the Limit

1.7 Limits and Loss-of-Significance Errors

Computer Representation or Real Numbers

Chaper 2: Differentiation

2.1 Tangent Lines and Velocity

2.2 The Derivative

Alternative Derivative Notations

Numerical Differentiation

2.3 Computation of Derivatives: The Power Rule

Higher Order Derivatives

Acceleration

2.4 The Product and Quotient Rules

2.5 The Chain Rule

2.6 Derivatives of the Trigonometric Functions

2.7 Implicit Differentiation

2.8 The Mean Value Theorem

Chapter 3: Applications of Differentiation

3.1 Linear Approximations and Newton's Method

3.2 Maximum and Minimum Values

3.3 Increasing and Decreasing Functions

3.4 Concavity and the Second Derivative Test

3.5Overview of Curve Sketching

3.6Optimization

3.8Related Rates

3.8Rates of Change in Economics and the Sciences

Chapter 4: Integration

4.1 Antiderivatives

4.2 Sums and Sigma Notation

Principle of Mathematical Induction

4.3 Area under a Curve

4.4 The Definite Integral

Average Value of a Function

4.5 The Fundamental Theorem of Calculus

4.6 Integration by Substitution

4.7 Numerical Integration

Error bounds for Numerical Integration

Chapter 5: Applications of the Definite Integral

5.1 Area Between Curves

5.2 Volume: Slicing, Disks, and Washers

5.3 Volumes by Cylindrical Shells

5.4 Arc Length and Srface Area

5.5 Projectile Motion

5.6 Applications of Integration to Physics and Engineering

Chapter 6: Exponentials, Logarithms and other Transcendental Functions

6.1 The Natural Logarithm

6.2 Inverse Functions

6.3 Exponentials

6.4 The Inverse Trigonometric Functions

6.5 The Calculus of the Inverse Trigonometric Functions

6.6 The Hyperbolic Function

Chapter 7: First-Order Differential Equations

7.1 Modeling with Differential Equations

Growth and Decay Problems

Compound Interest

7.2 Separable Differential Equations

Logistic Growth

7.3 Direction Fields and Euler's Method

7.4 Systems of First-Order Differential Equations

Predator-Prey Systems

7.6 Indeterminate Forms and L'Hopital's Rule

Improper Integrals

A Comparison Test

7.8 Probability

Chapter 8: First-Order Differential Equations

8.1 modeling with Differential Equations

Growth and Decay Problems

Compound Interest

8.2 Separable Differential Equations

Logistic Growth

8.3 Direction Fields and Euler's Method

Systems of First Order Equations

Chapter 9: Infinite Series

9.1 Sequences of Real Numbers

9.2 Infinite Series

9.3 The Integral Test and Comparison Tests

9.4 Alternating Series

Estimating the Sum of an Alternating Series

9.5 Absolute Convergence and the Ratio Test

The Root Test

Summary of Convergence Test

9.6 Power Series

9.7 Taylor Series

Representations of Functions as Series

Proof of Taylor's Theorem

9.8 Applications of Taylor Series

The Binomial Series

9.9 Fourier Series

Chapter 10: Parametric Equations and Polar Coordinates

10.1 Plane Curves and Parametric Equations

10.2 Calculus and Parametric Equations

10.3 Arc Length and Surface Area in Parametric Equations

10.4 Polar Coordinates

10.5 Calculus and Polar Coordinates

10.6 Conic Sections

10.7 Conic Sections in Polar Coordinates

Chapter 11: Vectors and the Geometry of Space

11.1 Vectors in the Plane

11.2 Vectors in Space

11.3 The Dot Product

Components and Projections

11.4 The Cross Product

11.5 Lines and Planes in Space

11.6 Surfaces in Space

Chapter 12: Vector-Valued Functions

12.1 Vector-Valued Functions

12.2 The Calculus Vector-Valued Functions

12.3 Motion in Space

12.4 Curvature

12.5 Tangent and Normal Vectors

Components of Acceleration, Kepler's Laws

11.6 Parametric Surfaces

Chapter 13: Functions of Several Variables and Partial Differentiation

13.1 Functions of Several Variables

13.2 Limits and Continuity

13.3 Partial Derivatives

13.4 Tangent Planes and Linear Approximations

Increments and Differentials

13.5 The Chain Rule

Implicit Differentiation

13.6 The Gradient and Directional Derivatives

13.7 Extrema of Functions of Several Variables

13.8 Constrained Optimization and Lagrange Multipliers

Chapter 14: Multiple Integrals

14.1 Double Integrals

14.2 Area, Volume, and Center of Mass

14.3 Double Integrals in Polar Coordinates

14.4 Surface Area

14.5 Triple Integrals

Mass and Center of Mass

14.6 Cylindrical Coordinates

14.7 Spherical Coordinates

14.8 Change of Variables in Multiple Integrals

Chapter 15: Vector Calculus

15.1 Vector Fields

15.2 Line Integrals

15.3 Independence of Path and Conservative Vector Fields

15.4 Green's Theorem

15.5 Curl and Divergence

15.6 Surface Integrals

15.7 The Divergence Theorem

15.8 Stokes' Theorem

15.9 Applications of Vector Calculus

Chapter 16: Second-Order Differential Equations

16.1 Second-Order Equations with Constant Coefficients

16.2 Nonhomogeneous Equations: Undetermined Coefficients

16.3 Applications of Second-Order Differential Equations

16.4 Power Series Solutions of Differential Equations

Appendix A: Proofs of Selected Theorems

Appendix B: Answers to Odd-Numbered Exercises