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 | |
Predator-Prey 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 Odd-Numbered Exercises | |
Index |