Calculus of a Single Variable

1. PREPARATION FOR CALCULUS. Graphs and Models. Linear Models and Rates of Change. Functions and Their Graphs. Review of Trigonometric Functions. 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: Polynomial Functions of Fourth Degree. Concavity and the Second Derivative Test. Limits at Infinity. A Summary of Curve Sketching. Optimization Problems. Section Project: Minimum Time. Differentials. Review Exercises. P.S. Problem Solving. 5. INTEGRATION. Antiderivatives and Indefinite Integration. Area. Riemann Sums and Definite Integrals. The Fundamental Theorem of Calculus. Integration by Substitution. Section Project: Probability. Indeterminate Forms and L'Hopital's Rule. The Natural Logarithmic Function: Integration. Inverse Trigonometric Functions: Integration. Hyperbolic Functions. Section Project: Mercator Map. Review Exercises. P.S. Problem Solving. 6. DIFFERENTIAL EQUATIONS. Slope Fields and Euler's Method. Growth and Decay. 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: Pyramid of Khufu. Moments, Centers of Mass, and Centroids. Fluid Pressure and Fluid Force. Review Exercises. P.S. Problem Solving. 8. INTEGRATION TECHNIQUES AND IMPROPER INTEGRALS. Basic Integration Rules. Integration by Parts. Trigonometric Integrals. Section Project: The Wallis Product. Trigonometric Substitution. Partial Fractions. Numerical Integration. Integration by Tables and Other Integration Techniques. 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. 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: Cassini Oval. Area and Arc Length in Polar Coordinates. Polar Equations of Conics and Kepler's Laws. Review Exercises. P.S. Problem Solving. APPENDIX. A. Proofs of Selected Theorems. B. Integration Tables. C. Precalculus Review. C.1 Real Numbers and the Real Number Line. C.2 The Cartesian Plane. D. Rotation and the General Second-Degree Equation (Online). E. Complex Numbers. (Online). F. Business and Economic Applications. (Online). G. Fitting Models to Data. (Online).