1. FUNCTIONS AND MODELS 

Four Ways to Represent a Function 

Mathematical Models 

New Functions from Old Functions 

Graphing Calculators and Computers 

Exponential Functions 

Inverse Functions and Logarithms 

Parametric Curves 

Review 

Principles of Problem Solving 

2. LIMITS AND DERIVATIVES 

The Tangent and Velocity Problems 

The Limit of a Function 

Calculating Limits Using the Limit Laws 

Continuity 

Limits Involving Infinity 

Tangents, Velocities, and Other Rates of Change 

Derivatives 

The Derivative as a Function 

Linear Approximations 

What does f' say about f? 

Review 

Focus on Problem Solving 

3. DIFFERENTIATION RULES 

Derivatives of Polynomials and Exponential Functions 

The Product and Quotient Rules 

Rates of Change in the Natural and Social Sciences 

Derivatives of Trigonometric Functions 

The Chain Rule 

Implicit Differentiation 

Derivatives of Logarithmic Functions 

Linear Approximations and Differentials 

Review 

Focus on Problem Solving 

4. APPLICATIONS OF DIFFERENTIATION 

Related Rates 

Maximum and Minimum Values 

Derivatives and the Shapes of Curves 

Graphing with Calculus and Calculators 

Indeterminate Forms and l'Hospital's Rule 

Optimization Problems 

Applications to Economics 

Newton's Method 

Antiderivatives 

Review 

Focus on Problem Solving 

5. INTEGRALS 

Areas and Distances 

The Definite Integral 

Evaluating Definite Integrals 

The Fundamental Theorem of Calculus 

The Substitution Rule 

Integration by Parts 

Additional Techniques of Integration 

Integration Using Tables and Computer Algebra Systems 

Approximate Integration 

Improper Integrals 

Review 

Focus on Problem Solving 

6. APPLICATIONS OF INTEGRATION 

More about Areas 

Volumes 

Arc Length 

Average Value of a Function 

Applications to Physics and Engineering 

Applications to Economics and Biology 

Probability 

Review 

Focus on Problem Solving 

7. DIFFERENTIAL EQUATIONS 

Modeling with Differential Equations 

Direction Fields and Euler's Method 

Separable Equations 

Exponential Growth and Decay 

The Logistic Equation 

PredatorPrey Systems 

Review 

Focus on Problem Solving 

8. INFINITE SEQUENCES AND SERIES 

Sequences 

Series 

The Integral and Comparison Tests; Estimating Sums 

Other Convergence Tests 

Power Series 

Representation of Functions as Power Series 

Taylor and Maclaurin Series 

The Binomial Series 

Applications of Taylor Polynomials 

Using Series to Solve Differential Equations 

Review 

Focus on Problem Solving 

9. VECTORS AND THE GEOMETRY OF SPACE 

Three Dimensional Coordinate Systems 

Vectors 

The Dot Product 

The Cross Product 

Equations of Lines and Planes 

Functions and Surfaces 

Cylindrical and Spherical Coordinates 

Review 

Focus on Problem Solving 

10. VECTOR FUNCTIONS 

Vector Functions and Space Curves 

Derivatives and Integrals of Vector Functions 

Arc Length and Curvature 

Motion in Space 

Parametric Surfaces 

Review 

Focus on Problem Solving 

11. PARTIAL DERIVATIVES 

Functions of Several Variables 

Limits and Continuity 

Partial Derivatives 

Tangent Planes and Linear Approximations 

The Chain Rule 

Directional Derivatives and the Gradient Vector 

Maximum and Minimum Values 

Lagrange Multipliers 

Review 

Focus on Problem Solving 

12. MULTIPLE INTEGRALS 

Double Integrals over Rectangles 

Integrated Integrals 

Double Integrals over General Regions 

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 

Review 

Focus on Problem Solving 

13. VECTOR CALCULUS 

Vector Fields 

Line Integrals 

The Fundamental Theorem for Line Integrals 

Green's Theorem 

Curl and Divergence 

Surface Integrals 

Stokes' Theorem 

The Divergence Theorem 

Summary 

Review 

Focus on Problem Solving 

Appendix A: Intervals, Inequalities, And Absolute Values 

Appendix B: Coordinate Geometry 

Appendix C: Trigonometry 

Appendix D: Precise Definitions Of Limits 

Appendix E: A Few Proofs 

Appendix F: Sigma Notation 

Appendix G: Integration Of Rational Functions By Partial Fractions 

Appendix H: Polar Coordinates 

Appendix I: Complex Numbers 

Appendix J: Answers To OddNumbered Exercises 

Index 
