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.