Calculus, Single Variable: Early Transcendental Functions

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 best elements of reform with the most reliable aspects of mainstream calculus teaching, resulting in a motivating, challenging book. Smith/Minton also provide exceptional, reality-based applications that appeal to students’ interests and demonstrate the elegance of math in the world around us. New features include: • A new organization placing all transcendental functions early in the book and consolidating the introduction to L'Hôpital's Rule in a single section. • More concisely written explanations in every chapter. • Many new exercises (for a total of 7,000 throughout the book) that require additional rigor not found in the 2nd Edition. • New exploratory exercises in every section that challenge students to synthesize key concepts to solve intriguing projects. • New commentaries (“Beyond Formulas”) that encourage students to think mathematically beyond the procedures they learn. • New counterpoints to the historical notes, “Today in Mathematics,” that 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.

0 Preliminaries

0.1 Polynomials 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

Hyperbolic Functions

Fitting a Curve to Data

0.6 Transformations of Functions

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

Asymptotes

1.6 Formal Definition of the Limit

Exploring the Definition of Limit Graphically

1.7 Limits and Loss-of-Significance Errors

Computer Representation of Real Numbers

2 Differentiation

2.1 Tangent Lines and Velocity

2.2 The Derivative

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 Derivatives of the Exponential and Logarithmic Functions

2.8 Implicit Differentiation and Inverse Trigonometric Functions

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 the Second Derivative Test

3.6 Overview of Curve Sketching

3.7 Optimization

3.8 Related Rates

3.9 Rates of Change in Economics and the Sciences

4 Integration

4.1 Antiderivatives

4.2 Sums and Sigma Notation

Principle of Mathematical Induction

4.3 Area

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

4.8 The Natural Logarithm as an Integral

The Exponential Function as the Inverse of the Natural Logarithm

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 Surface Area

5.5 Projectile Motion

5.6 Applications of Integration to Physics and Engineering

5.7 Probability

6 Integration Techniques

6.1 Review of Formulas and Techniques

6.2 Integration by Parts

6.3 Trigonometric Techniques of Integration

Integrals Involving Powers of Trigonometric Functions

Trigonometric Substitution

6.4 Integration of Rational Functions Using Partial Fractions

Brief Summary of Integration Techniques

6.5 Integration Tables and Computer Algebra Systems

6.6 Improper Integrals

A Comparison Test

7 First-Order Differential Equations

7.1 Growth and Decay Problems

Compound Interest

Modeling with Differential Equations

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

8 Infinite Series

8.1 Sequences of Real Numbers

8.2 Infinite Series

8.3 The Integral Test and Comparison Tests

8.4 Alternating Series

Estimating the Sum of an Alternating Series

8.5 Absolute Convergence and the Ratio Test

The Root Test

Summary of Convergence Tests

8.6 Power Series

8.7 Taylor Series

Representations of Functions as Series

Proof of Taylor’s Theorem

8.8 Applications of Taylor Series

The Binomial Series

8.9 Fourier Series

Parametric Equations and Polar Coordinates

9.1 Plane Curves and Parametric Equations

9.2 Calculus and Parametric Equations

9.3 Arc Length and Surface Area in Parametric Equations

9.4 Polar Coordinates

9.5 Calculus and Polar Coordinates

9.6 Conic Sections

9.7 Conic Sections in Polar Coordinates

10 Vectors and the Geometry of Space

10.1 Vectors in the Plane

10.2 Vectors in Space

10.3 The Dot Product

Components and Projections

10.4 The Cross Product

10.5 Lines and Planes in Space

10.6 Surfaces in Space