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Note: Each chapter concludes with Review Exercises and P.S. Problem Solving | |
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Preparation for Calculus | |
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Graphs and Models | |
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Linear Models and Rates of Change | |
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Functions and Their Graphs | |
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Fitting Models to Data | |
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Inverse Functions | |
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Exponential and Logarithmic Functions | |
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Limits and Their Properties | |
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A Preview of Calculus | |
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Finding Limits Graphically and Numerically | |
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Evaluating Limits Analytically | |
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Continuity and OneSided Limits | |
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Infinite Limits | |
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Section Project: Graphs and Limits of Trigonometric Functions | |
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Differentiation | |
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The Derivative and the Tangent Line Problem | |
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Basic Differentiation Rules and Rates of Change | |
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The Product and Quotient Rules and HigherOrder Derivatives | |
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The Chain Rule | |
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Implicit Differentiation | |
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Section Project: Optical Illusions | |
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Derivatives of Inverse Functions | |
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Related Rates | |
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Newton's Method | |
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Applications of Differentiation | |
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Extrema on an Interval | |
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Rolle's Theorem and the Mean Value Theorem | |
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Increasing and Decreasing Functions and the First Derivative Test | |
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Section Project: Rainbows | |
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Concavity and the Second Derivative Test | |
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Limits at Infinity | |
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A Summary of Curve Sketching | |
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Optimization Problems | |
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Section Project: Connecticut River | |
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Differentials | |
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Integration | |
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Antiderivatives and Indefinite Integration | |
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Area | |
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Riemann Sums and Definite Integrals | |
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The Fundamental Theorem of Calculus | |
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Section Project: Demonstrating the Fundamental Theorem | |
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Integration by Substitution | |
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Numerical Integration | |
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The Natural Logarithmic Function: Integration | |
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Inverse Trigonometric Functions: Integration | |
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Hyperbolic Functions | |
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Section Project: St. Louis Arch | |
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Differential Equations | |
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Slope Fields and Euler's Method | |
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Differential Equations: Growth and Decay | |
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Differential Equations: Separation of Variables | |
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The Logistic Equation | |
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FirstOrder Linear Differential Equations | |
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Section Project: Weight Loss | |
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PredatorPrey Differential Equations | |
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Applications of Integration | |
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Area of a Region Between Two Curves | |
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Volume: The Disk Method | |
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Volume: The Shell Method | |
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Section Project: Saturn | |
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Arc Length and Surfaces of Revolution | |
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Work | |
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Section Project: Tidal Energy | |
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Moments, Centers of Mass, and Centroids | |
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Fluid Pressure and Fluid Force | |
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Integration Techniques, L'Hopital's Rule, and Improper Integrals | |
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Basic Integration Rules | |
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Integration by Parts | |
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Trigonometric Integrals | |
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Section Project: Power Lines | |
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Trigonometric Substitution | |
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Partial Fractions | |
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Integration by Tables and Other Integration Techniques | |
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Indeterminate Forms and L'Hopital's Rule | |
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Improper Integrals | |
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Infinite Series | |
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Sequences | |
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Series and Convergence | |
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Section Project: Cantor's Disappearing Table | |
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The Integral Test andpSeries | |
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Section Project: The Harmonic Series | |
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Comparisons of Series | |
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Section Project: Solera Method | |
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Alternating Series | |
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The Ratio and Root Tests | |
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Taylor Polynomials and Approximations | |
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Power Series | |
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Representation of Functions by Power Series | |
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Taylor and Maclaurin Series | |
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Conics, Parametric Equations, and Polar Coordinates | |
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Conics and Calculus | |
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Plane Curves and Parametric Equations | |
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Section Project: Cycloids | |
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Parametric Equations and Calculus | |
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Polar Coordinates and Polar Graphs | |
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Section Project: Anamorphic Art | |
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Area and Arc Length in Polar Coordinates | |
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Polar Equations of Conics and Kepler's Laws | |
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Appendices | |
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Appendix | |