Calculus

1. PREPARATION FOR CALCULUS. Graphs and Models. Linear Models and Rates of Change. Functions and Their Graphs. Fitting Models to Data. Inverse Functions. Exponential and Logarithmic Functions. Review Exercises. P.S. Problem Solving. 2. LIMITS AND THEIR PROPERTIES. A Preview of Calculus. Finding Limits Graphically and Numerically. Evaluating Limits Analytically. Continuity and One-Sided Limits. Infinite Limits. Section Project: Graphs and Limits of Trigonometric Functions. Review Exercises. P.S. Problem Solving. 3. DIFFERENTIATION. The Derivative and the Tangent Line Problem. Basic Differentiation Rules and Rates of Change. Product and Quotient Rules and Higher-Order Derivatives. The Chain Rule. Implicit Differentiation. Section Project: Optical Illusions. Derivatives of Inverse Functions, Related Rates. Newton's Method. Review Exercises. P.S. Problem Solving. 4. APPLICATIONS OF DIFFERENTIATION. Extrema on an Interval. Rolle's Theorem and the Mean Value Theorem. Increasing and Decreasing Functions and the First Derivative Test. Section Project: Rainbows. Concavity and the Second Derivative Test. Limits at Infinity. A Summary of Curve Sketching. Optimization Problems. Section Project: Connecticut River. Differentials. Review Exercises. P.S. Problem Solving. 5. INTEGRATION. Antiderivatives and Indefinite Integration. Area. Riemann Sums and Definite Integrals. The Fundamental Theorem of Calculus. Section Project: Demonstrating the Fundamental Theorem. Integration by Substitution. Numerical Integration. The Natural Logarithmic Function: Integration. Inverse Trigonometric Functions: Integration. Hyperbolic Functions. Section Project: St. Louis Arch. Review Exercises. P.S. Problem Solving. 6. DIFFERENTIAL EQUATIONS. Slope Fields and Euler's Method. Differential Equations: Growth and Decay. Differential Equations: Separation of Variables. The Logistic Equation. First-Order Linear Differential Equations. Section Project: Weight Loss. Predator-Prey Differential Equations. Review Exercises. P.S. Problem Solving. 7. APPLICATIONS OF INTEGRATION. Area of a Region Between Two Curves. Volume: The Disk Method. Volume: The Shell Method. Section Project: Saturn. Arc Length and Surfaces of Revolution. Work. Section Project: Tidal Energy. Moments, Centers of Mass, and Centroids. Fluid Pressure and Fluid Force. Review Exercises. P.S. Problem Solving. 8. INTEGRATION TECHNIQUES, L'HOPITAL'S RULE, AND IMPROPER INTEGRALS. Basic Integration Rules. Integration by Parts. Trigonometric Integrals. Section Project: Power Lines. Trigonometric Substitution. Partial Fractions. Integration by Tables and Other Integration Techniques. Indeterminate Forms and L'Hopital's Rule. Improper Integrals. Review Exercises. P.S. Problem Solving. 9. INFINITE SERIES. Sequences. Series and Convergence. Section Project: Cantor's Disappearing Table. The Integral Test and p-Series. Section Project: The Harmonic Series. Comparisons of Series. Section Project: Solera Method. Alternating Series. The Ratio and Root Tests. Taylor Polynomials and Approximations. Power Series. Representation of Functions by Power Series. Taylor and Maclaurin Series. Review Exercises. P.S. Problem Solving. 10. CONICS, PARAMETRIC EQUATIONS, AND POLAR COORDINATES. Conics and Calculus. Plane Curves and Parametric Equations. Section Project: Cycloids. Parametric Equations and Calculus. Polar Coordinates and Polar Graphs. Section Project: Anamorphic Art. Area and Arc Length in Polar Coordinates. Polar Equations of Conics and Kepler's Laws. Review Exercises. P.S. Problem Solving. 11. VECTORS AND THE GEOMETRY OF SPACE. Vectors in the Plane. Space Coordinates and Vectors in Space. The Dot Product of Two Vectors. The Cross Product of Two Vectors in Space. Lines and Planes in Space. Section Project: Distances in Space. Surfaces in Space. Cylindrical and Spherical Coordinates. Review Exercises. P.S. Problem Solving. 12. VECTOR-VALUED FUNCTIONS. Vector-Valued Functions. Section Project: Witch of Agnesi. Differentiation and Integration of Vector-Valued Functions. Velocity and Acceleration. Tangent Vectors and Normal Vectors. Arc Length and Curvature. Review Exercises. P.S. Problem Solving. 13. FUNCTIONS OF SEVERAL VARIABLES. Introduction to Functions of Several Variables. Limits and Continuity. Partial Derivatives. Section Project: Moire Fringes. Differentials. Chain Rules for Functions of Several Variables. Directional Derivatives and Gradients. Tangent Planes and Normal Lines. Section Project: Wildflowers. Extrema of Functions of Two Variables. Applications of Extrema of Functions of Two Variables. Section Project: Building a Pipeline. Lagrange Multipliers. Review Exercises. P.S. Problem Solving. 14. MULTIPLE INTEGRATION. Iterated Integrals and Area in the Plane. Double Integrals and Volume. Change of Variables: Polar Coordinates. Center of Mass and Moments of Inertia. Section Project: Center of Pressure on a Sail. Surface Area. Section Project: Capillary Action. Triple Integrals and Applications. Triple Integrals in Cylindrical and Spherical Coordinates. Section Project: Wrinkled and Bumpy Spheres. Change of Variables: Jacobians. Review Exercises. P.S. Problem Solving. 15. VECTOR ANALYSIS. Vector Fields. Line Integrals. Conservative Vector Fields and Independence of Path. Green's Theorem. Section Project: Hyperbolic and Trigonometric Functions. Parametric Surfaces. Surface Integrals. Section Project: Hyperboloid of One Sheet. Divergence Theorem. Stokes's Theorem. Review Exercises. Section Project: The Planimeter. P.S. Problem Solving. 16. ADDITIONAL TOPICS IN DIFFERENTIAL EQUATIONS (Web). Exact First-Order Equations. Second-Order Homogeneous Linear Equations. Second-Order Nonhomogeneous Linear Equations. Series Solutions of Differential Equations. Review Exercises. P.S. Problem Solving. APPENDIX. A. Proofs of Selected Theorems (Web). B. Integration Tables. C. Precalculus Review. (Web). C.1 Real Numbers and the Real Number Line. C.2 The Cartesian Plane. C.3 Review of Trigonometric Functions. D. Rotation and the General Second-Degree Equation (Web). E. Complex Numbers. (Web).