Calculus – Soo T. Tan – 1st Edition

0: PRELIMINARIES

Lines

Functions and Their Graphs

The Trigonometric Functions

Combining Functions

Graphing Calculators and Computers

Mathematical Models

Chapter Review

1: LIMITS

An Intuitive Introduction to Limits

Techniques for Finding Limits

A Precise Definition of a Limit

Continuous Functions

Tangent Lines and Rates of Change

Chapter Review

Problem-Solving Techniques

Challenge Problems

2: THE DERIVATIVE

The Derivative

Basic Rules of Differentiation

The Product and Quotient Rules

The Role of the Derivative in the Real World

Derivatives of Trigonometric Functions

The Chain Rule

Implicit Differentiation

Related Rates

Differentials and Linear Approximations

Chapter Review

Problem-Solving Techniques

Challenge Problems

3: APPLICATIONS OF THE DERIVATIVE

Extrema of Functions

The Mean Value Theorem

Increasing and Decreasing Functions and the First

Derivative Test

Concavity and Inflection Points

Limits Involving Infinity

Asymptotes

Curve Sketching

Optimization Problems

Newton's Method

Chapter Review

Problem-Solving Techniques

Challenge Problems

4: INTEGRATION

Indefinite Integrals

Integration by Substitution

Area

The Definite Integral

The Fundamental Theorem of Calculus

Numerical Integration

Chapter Review

Problem-Solving Techniques

Challenge Problems

5: APPLICATIONS OF THE DEFINITE INTEGRAL

Areas Between Curves

Volumes: Disks, Washers, and Cross Sections

Volumes Using Cylindrical Shells

Arc Length and Areas of Surfaces of Revolution

Work

Fluid Pressure and Force

Moments and Centers of Mass

Chapter Review

Problem-Solving Techniques

Challenge Problems

6: THE TRANSCENDENTAL FUNCTIONS

The Natural Logarithmic Function

Inverse Functions

Exponential Functions

General Exponential and Logarithmic Functions

Inverse Trigonometric Functions

Hyperbolic Functions

Indeterminate Forms and L'Hôpital's Rule

Chapter Review

Challenge Problems

7: TECHNIQUES OF INTEGRATION

Integration by Parts

Trigonometric Integrals

Trigonometric Substitutions

The Method of Partial Fractions

Integration Using Tables of Integrals and CAS

Improper Integrals

Chapter Review

Problem-Solving Techniques

Challenge Problems

8: DIFFERENTIAL EQUATIONS

Differential Equations: Separable Equations

Direction Fields and Euler's Method

The Logistic Equation

First-Order Linear Differential Equations

Chapter Review

Challenge Problems

9: INFINITE SEQUENCES AND SERIES

Sequences

Series

The Integral Test

The Comparison Tests

Alternating Series

Absolute Convergence

The Ratio and Root Tests

Power Series

Taylor and Maclaurin Series

Approximation by Taylor Polynomials

Chapter Review

Problem-Solving Techniques

Challenge Problems

10: CONIC SECTIONS, PARAMETRIC EQUATIONS, AND POLAR COORDINATES

Conic Sections

Plane Curves and Parametric Equations

The Calculus of Parametric Equations

Polar Coordinates

Areas and Arc Lengths in Polar Coordinates

Conic Sections in Polar Coordinates

Chapter Review

Challenge Problems

11: VECTORS AND THE GEOMETRY OF SPACE

Vectors in the Plane

Coordinate Systems and Vectors in Three-Space

The Dot Product

The Cross Product

Lines and Planes in Space

Surfaces in Space

Cylindrical and Spherical Coordinates

Chapter Review

Challenge Problems

12: VECTOR-VALUED FUNCTIONS

Vector-Valued Functions and Space Curves

Differentiation and Integration of Vector- Valued

Functions

Arc Length and Curvature

Velocity and Acceleration

Tangential and Normal Components of Acceleration

Chapter Review

Challenge Problems

13: FUNCTIONS OF SEVERAL VARIABLES

Functions of Two or More Variables

Limits and Continuity

Partial Derivatives

Differentials

The Chain Rule

Directional Derivatives and Gradient Vectors

Tangent Planes and Normal Lines

Extrema of Functions of Two Variables

Lagrange Multipliers

Chapter Review

Challenge Problems

14: MULTIPLE INTEGRAL.S Double Integrals

Iterated Integrals

Double Integrals in Polar Coordinates

Applications of Double Integrals

Surface Area

Triple Integrals

Triple Integrals in Cylindrical and Spherical Coordinates

Change of Variables in Multiple Integrals

Chapter Review

Challenge Problems

15: VECTOR ANALYSIS

Vector Fields

Divergence and Curl

Line Integrals

Independence of Path and Conservative Vector Fields

Green's Theorem

Parametric Surfaces

Surface Integrals

The Divergence Theorem

Stoke's Theorem

Chapter Review

Challenge Problems

APPENDICES

A The Real Number Line, Inequalities, and Absolute Value

B Proofs of Selected Theorems.