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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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Principles of Problem Solving | |

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Limits | |

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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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Derivatives | |

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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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Differentiation Formulas | |

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Applied Project: Building a Better Roller Coaster | |

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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? | |

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Imlicit Differentiation | |

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Rates of Change in the Natural and Social Sciences | |

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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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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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Limits at Infinity | |

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Horizontal Asymptotes | |

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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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Newton's Method | |

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Antiderivatives | |

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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 Net Change | |

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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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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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Inverse Functions: Exponential, Logarithmic, and Inverse Trigonometric Functions | |

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Inverse Functions | |

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(Instructors may cover either Sections 7.2? 7.4 or Sections 7.2*?7.4*) | |

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Exponential Functions and Their Derivatives | |

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Logarithmic Functions | |

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Derivatives of Logarithmic Functions | |

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The Natural Logarithmic Function | |

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The Natural Exponential Function | |

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General Logarithmic and Exponential Functions | |

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Exponential Growth and Decay | |

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Inverse Trigonometric Functions | |

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Applied Project: Where to Sit at the Movies | |

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Hyperbolic Functions | |

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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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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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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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Discovery Project: Complementary Coffee Cups | |

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Applications to Economics and Biology | |

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Probability | |

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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? Models for Population Growth | |

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Applied Project: Calculus | |

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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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Calculus with Parametric Curves | |

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Laboratory Project: Bezier Curves | |

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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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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 | |

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Writing Project: How Newton Discovered the Binomial Series | |

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Applications of Taylor Polynomials | |

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Applied Pro | |