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Functions and Models | |
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Four Ways to Represent a Function | |
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Mathematical Models: A Catalog of Essential Functions | |
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New Functions from Old Functions | |
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Graphing Calculators and Computers | |
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Exponential Functions | |
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Inverse Functions and Logarithms | |
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Review | |
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Principles of Problem Solving | |
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Limits and Derivatives | |
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The Tangent and Velocity Problems | |
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The Limit of a Function | |
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Calculating Limits Using the Limit Laws | |
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The Precise Definition of a Limit | |
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Continuity | |
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Limits at Infinity | |
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Horizontal Asymptotes | |
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Derivatives and Rates of Change | |
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Writing Project: Early Methods for Finding Tangents | |
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The Derivative as a Function | |
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Review | |
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Problems Plus | |
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Differentiation Rules | |
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Derivatives of Polynomials and Exponential Functions | |
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Applied Project: Building a Better Roller Coaster | |
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The Product and Quotient Rules | |
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Derivatives of Trigonometric Functions | |
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The Chain Rule | |
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Applied Project: Where Should a Pilot Start Descent? Implicit Differentiation | |
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Derivatives of Logarithmic Functions | |
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Rates of Change in the Natural and Social Sciences | |
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Exponential Growth and Decay | |
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Related Rates | |
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Linear Approximations and Differentials | |
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Laboratory Project: Taylor Polynomials | |
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Hyperbolic Functions | |
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Review | |
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Problems Plus | |
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Applications of Differentiation | |
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Maximum and Minimum Values | |
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Applied Project: The Calculus of Rainbows | |
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The Mean Value Theorem | |
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How Derivatives Affect the Shape of a Graph | |
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Indeterminate Forms and L?Hospital?s Rule | |
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Writing Project: The Origins of L?Hospital?s Rule | |
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Summary of Curve Sketching | |
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Graphing with Calculus and Calculators | |
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Optimization Problems | |
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Applied Project: The Shape of a Can | |
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Applications to Business and Economics | |
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Newton?s Method | |
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Antiderivatives | |
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Review | |
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Problems Plus | |
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Integrals | |
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Areas and Distances | |
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The Definite Integral | |
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Discovery Project: Area Functions | |
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The Fundamental Theorem of Calculus | |
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Indefinite Integrals and the Total Change Theorem | |
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Writing Project: Newton, Leibniz, and the Invention of Calculus | |
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The Substitution Rule | |
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Review | |
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Problems Plus | |
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Applications of Integration | |
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Areas between Curves | |
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Volume | |
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Volumes by Cylindrical Shells | |
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Work | |
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Average Value of a Function | |
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Applied Project: Where to Sit at the Movies | |
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Review | |
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Problems Plus | |
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Techniques of Integration | |
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Integration by Parts | |
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Trigonometric Integrals | |
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Trigonometric Substitution | |
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Integration of Rational Functions by Partial Fractions | |
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Strategy for Integration | |
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Integration Using Tables and Computer Algebra Systems | |
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Discovery Project: Patterns in Integrals | |
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Approximate Integration | |
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Improper Integrals | |
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Review | |
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Problems Plus | |
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Further Applications of Integration | |
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Arc Length | |
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Discovery Project: Arc Length Contest | |
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Area of a Surface of Revolution | |
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Discovery Project: Rotating on a Slant | |
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Applications to Physics and Engineering | |
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Applications to Economics and Biology | |
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Probability | |
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Review | |
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Problems Plus | |
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Differential Equations | |
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Modeling with Differential Equations | |
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Direction Fields and Euler?s Method | |
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Separable Equations | |
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Applied Project: Which is Faster, Going Up or Coming Down? Exponential Growth and Decay | |
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Applied Project: Calculus and Baseball | |
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The Logistic Equation | |
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Linear Equations | |
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Predator-Prey Systems | |
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Review | |
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Problems Plus | |
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Parametric Equations and Polar Coordinates | |
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Curves Defined by Parametric Equations | |
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Laboratory Project: Families of Hypocycloids | |
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Tangents and Areas | |
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Laboratory Project: Bezier Curves | |
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Arc Length and Surface Area | |
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Polar Coordinates | |
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Areas and Lengths in Polar Coordinates | |
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Conic Sections | |
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Conic Sections in Polar Coordinates | |
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Applied Project: Transfer Orbits | |
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Review | |
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Problems Plus | |
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Infinite Sequences and Series | |
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Sequences | |
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Laboratory Project: Logistic Sequences | |
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Series | |
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The Integral Test and Estimates of Sums | |
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The Comparison Tests | |
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Alternating Series | |
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Absolute Convergence and the Ratio and Root Tests | |
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Strategy for Testing Series | |
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Power Series | |
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Representation of Functions as Power Series | |
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Taylor and Maclaurin Series The Binomial Series | |
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Writing Project: How Newton Discovered the Binomial Series | |
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Applications of Taylor Polynomials | |