Calculus, Single Variable: Early Transcendental Functions

by Smith, Robert T; Minton , Roland

• ISBN13: 9780073309439
• ISBN10: 0073309435
• Edition: 3rd
• Format: Hardcover
• Copyright: 2006-09-18
• Publisher: McGraw-Hill/Irwin

Summary

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: bull; A new organization placing all transcendental functions early in the book and consolidating the introduction to L'Hocirc;pital's Rule in a single section. bull; More concisely written explanations in every chapter. bull; Many new exercises (for a total of 7,000 throughout the book) that require additional rigor not found in the 2nd Edition. bull; New exploratory exercises in every section that challenge students to synthesize key concepts to solve intriguing projects. bull; New commentaries ("Beyond Formulas") that encourage students to think mathematically beyond the procedures they learn. bull; 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. bull; An enhanced discussion of differential equations and additional applications of vector calculus.

Table of Contents

Preliminaries

Polynomials and Rational Functions

Graphing Calculators and Computer Algebra Systems

Inverse Functions

Trigonometric and Inverse Trigonometric Functions

Exponential and Logarithmic Functions Hyperbolic Functions Fitting a Curve to Data

Transformations of Functions

Limits and Continuity

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

The Concept of Limit

Computation of Limits

Continuity and its Consequences The Method of Bisections

Limits Involving Infinity Asymptotes

Formal Definition of the Limit Exploring the Definition of Limit Graphically

Limits and Loss-of-Significance Errors Computer Representation of Real Numbers

Differentiation

Tangent Lines and Velocity

The Derivative Numerical Differentiation

Computation of Derivatives: The Power Rule Higher Order Derivatives Acceleration

The Product and Quotient Rules

The Chain Rule

Derivatives of the Trigonometric Functions

Derivatives of the Exponential and Logarithmic Functions

Implicit Differentiation and Inverse Trigonometric Functions

The Mean Value Theorem

Applications of Differentiation

Linear Approximations and Newton’s Method

Indeterminate Forms and L’Hopital’s Rule

Maximum and Minimum Values

Increasing and Decreasing Functions

Concavity and the Second Derivative Test

Overview of Curve Sketching

Optimization

Related Rates

Rates of Change in Economics and the Sciences

Integration

Antiderivatives

Sums and Sigma Notation Principle of Mathematical Induction

Area

The Definite Integral Average Value of a Function

The Fundamental Theorem of Calculus

Integration by Substitution

Numerical Integration Error Bounds for Numerical Integration

The Natural Logarithm as an Integral The Exponential Function as the Inverse of the Natural Logarithm

Applications of the Definite Integral

Area Between Curves

Volume: Slicing, Disks, and Washers

Volumes by Cylindrical Shells

Arc Length and Surface Area

Projectile Motion

Applications of Integration to Physics and Engineering

Probability

Integration Techniques

Review of Formulas and Techniques

Integration by Parts

Trigonometric Techniques of Integration Integrals Involving Powers of Trigonometric Functions Trigonometric Substitution

Integration of Rational Functions Using Partial Fractions Brief Summary of Integration Techniques

Integration Tables and Computer Algebra Systems

Improper Integrals A Comparison Test

First-Order Differential Equations

Growth and Decay Problems Compound Interest Modeling with Differential Equations

Separable Differential Equations Logistic Growth

Direction Fields and Euler's Method

Systems of First-Order Differential Equations Predator-Prey Systems

Infinite Series

Sequences of Real Numbers

Infinite Series

The Integral Test and Comparison Tests

Alternating Series Estimating the Sum of an Alternating Series

Absolute Convergence and the Ratio Test The Root Test Summary of Convergence Tests

Power Series

Taylor Series Representations of Functions as Series Proof of Taylor’s Theorem

Applications of Taylor Series The Binomial Series

Fourier Series Parametric Equations and Polar Coordinates

Plane Curves and Parametric Equations

Calculus and Parametric Equations

Arc Length and Surface Area in Parametric Equations

Polar Coordinates

Calculus and Polar Coordinates

Conic Sections

Conic Sections in Polar Coordinates

Vectors and the Geometry of Space

Vectors in the Plane

Vectors in Space

The Dot Product Components and Projections

The Cross Product

Lines and Planes in Space

Surfaces in Space

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