Calculus: Early Transcendental Functions Book

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 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

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

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

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

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

Kepler’s Laws

11.6 Parametric Surfaces

12 Functions of Several Variables and Partial 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

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

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

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 Differential Equations

15.4 Power Series Solutions of Differential Equations

Appendix A: Proofs of Selected Theorems

Appendix B: Answers to Odd-Numbered Exercises