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Introduction: Calculus: A New Horizon from Ancient Roots | |
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Functions | |
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Functions and the Analysis of Graphical Information | |
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Properties of Functions | |
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Graphing Functions on Calculators and Computers; Computer Algebra Systems | |
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New Functions from Old | |
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Lines | |
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Families of Functions | |
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Mathematical Models | |
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Parametric Equations | |
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Horizon Module: Iteration and Dynamical Systems | |
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Limits and Continuity | |
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Limits (An Intuitive Approach) | |
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Computing Limits | |
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Computing Limits: End Behavior | |
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Limits (Discussed More Rigorously) | |
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Continuity | |
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Limits and Continuity of Trigonometric Functions | |
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The Derivative | |
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Slopes and Rates of Change | |
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The Derivative | |
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Techniques of Differentiation | |
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Derivatives of Trigonometric Functions | |
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The Chain Rule | |
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Implicit Differentiation | |
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Related Rates | |
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Local Linear Approximation; Differentials | |
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Horizon Module: Robotics | |
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The Derivative in Graphing and Applications | |
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Analysis of Functions I: Increase, Decrease, and Concavity | |
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Analysis of Functions II: Relative Extrema; First and Second Derivative Tests | |
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Analysis of Functions III: Applying Technology and the Tools of Calculus | |
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Rectilinear Motion (Motion Along a Line) | |
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Absolute Maxima and Minima | |
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Applied Maximum and Minimum Problems | |
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Newton's Method | |
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Rolle's Theorem; Mean-Value Theorem | |
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Integration | |
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An Overview of the Area Problem | |
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The Indefinite Integral; Integral Curves and Direction Fields | |
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Integration by Substitution | |
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Sigma Notation; Area as a Limit | |
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The Definite Integral | |
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The Fundamental Theorem of Calculus | |
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Rectilinear Motion Revisited; Average Value | |
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Evaluating Definite Integrals by Substitution | |
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Horizon Module: Blammo the Human Cannonball | |
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Applications of the Definite Integral in Geometry, Science, and Engineering | |
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Area Between Two Curves | |
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Volumes by Slicing; Disks and Washers | |
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Volumes by Cylindrical Shells | |
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Length of a Plane Curve | |
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Area of a Surface of Revolution | |
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Work | |
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Fluid Pressure and Force | |
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Exponential, Logarithmic, and Inverse Trigonometric Functions | |
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Inverse Functions | |
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Exponential and Logarithmic Functions | |
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Derivatives and Integrals Involving Logarithmic and Exponential Functions | |
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Graphs and Applications Involving Logarithmic and Exponential Functions | |
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Logarithmic Functions from the Integral Point of View | |
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Derivatives and Integrals Involving Inverse Trigonometric Functions | |
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L'Hopital's Rule; Indeterminate Forms | |
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Hyperbolic Functions and Hanging Cables | |
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Principles of Integral Evaluation | |
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An Overview of Integration Methods | |
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Integration by Parts | |
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Trigonometric Integrals | |
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Trigonometric Substitutions | |
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Integrating Rational Functions by Partial Fractions | |
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Using Tables of Integrals and Computer Algebra Systems | |
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Numerical Integration; Simpson's Rule | |
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Improper Integrals | |
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Horizon Module: Railroad Design | |
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Mathematical Modeling with Differential Equations | |
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First-Order Differential Equations and Applications | |
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Direction Fields; Euler's Method | |
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Modeling with First-Order Differential Equations | |
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Second-Order Linear Homogeneous Differential Equations; The Vibrating Spring | |
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Infinite Series | |
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Maclaurin and Taylor Polynomial Approximations | |
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Sequences | |
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Monotone Sequences | |
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Infinite Series | |
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Convergence Tests | |
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The Comparison, Ratio, and Root Tests | |
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Alternating Series; Conditional Convergence | |
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Maclaurin and Taylor Series; Power Series | |
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Convergence of Taylor Series; Computational Methods | |
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Differentiating and Integrating Power Series; Modeling with Taylor Series | |
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Analytic Geometry in Calculus | |
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Polar Coordinates | |
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Tangent Lines and Arc Length for Parametric and Polar Curves | |
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Area in Polar Coordinates | |
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Conic Sections in Calculus | |
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Rotation of Axes; Second-Degree Equations | |
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Conic Sections in Polar Coordinates | |
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Horizon Module: Comet Collision | |
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Real Numbers, Intervals, and Inequalities | |
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Absolute Value | |
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Coordinate Planes and Lines | |
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Distance, Circles, and Quadratic Equations | |
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Trigonometry Review | |
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Solving Polynomial Equations | |
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Selected Proofs | |
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Answers | |
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Photo Credits | |
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Index | |