Synopses & Reviews
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.
Table of Contents
Chapter 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
Chapter 1: Limits and Continuity
1.1 A First Look at Calculus
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
Chapter 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
Chapter 3: Applications of Differentiation
3.1 Linear Approximations and Newtons Method
3.2 Indeterminate Forms and LHopitals 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
Chapter 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
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 Surface Area
5.5 Projectile Motion
5.6 Applications of Integration to Economics and the Sciences
5.7 Probability
Chapter 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
General Strategies for Integration Techniques
6.5 Integration Tables and Computer Algebra Systems
6.6 Improper Integrals
A Comparison Test
Chapter 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
Chapter 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 Taylors Theorem
8.8 Applications of Taylor Series
The Binomial Series
8.9 Fourier Series
Chapter 9: 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
Chapter 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
Chapter 11: Vector-Valued Functions
11.1 Vector-Valued Functions
11.2 The Calculus of Vector-Valued Functions
11.3 Motion in Space
11.4 Curvature
11.5 Tangent and Normal Vectors
Tangential and Normal Components of Acceleration
Keplers Laws
11.6 Parametric Surfaces
Chapter 12: Functions of Several Variables and Differentiation
12.1 Functions of Several Variables
12.2 Limits and Continuity
12.3 Partial Derivatives
12.4 Tangent Planes and Linear Approximations
Increments and Differentials
12.5 The Chain Rule
12.6 The Gradient and Directional Derivatives
12.7 Extrema of Functions of Several Variables
12.8 Constrained Optimization and Lagrange Multipliers
Chapter 13: Multiple Integrals
13.1 Double Integrals
13.2 Area, Volume, and Center of Mass
13.3 Double Integrals in Polar Coordinates
13.4 Surface Area
13.5 Triple Integrals
Mass and Center of Mass
13.6 Cylindrical Coordinates
13.7 Spherical Coordinates
13.8 Change of Variables in Multiple Integrals
Chapter 14: Vector Calculus
14.1 Vector Fields
14.2 Line Integrals
14.3 Independence of Path and Conservative Vector Fields
14.4 Green's Theorem
14.5 Curl and Divergence
14.6 Surface Integrals
14.7 The Divergence Theorem
14.8 Stokes' Theorem
14.9 Applications of Vector Calculus
Chapter 15: Second Order Differential Equations
15.1 Second-Order Equations with Constant Coefficients
15.2 Nonhomogeneous Equations: Undetermined Coefficients
15.3 Applications of Second Order Equations
15.4 Power Series Solutions of Differential Equations
Appendix A: Proofs of Selected Theorems
Appendix B: Answers to Odd-Numbered Exercises