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Diagnostic Tests | |

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A Preview of Calculus | |

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Functions And Limits | |

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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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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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Review | |

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Principles of Problem Solving | |

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Derivatives | |

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Derivatives and Rates of Change | |

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Writing Project: 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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Implicit Differentiation | |

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Laboratory Project: Families of Implicit Curves | |

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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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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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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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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 Net 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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Applied Project: The Gini Index | |

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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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Review | |

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Problems Plus | |

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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 6.2-6.4 or Sections 6.2*-6.4* | |

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See the Preface | |

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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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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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Applied Project: Calculus and Baseball | |

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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. 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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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: How Fast Does a Tank Drain? | |

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Applied Project: Which is Faster, Going Up or Coming Down? | |

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Models for Population Growth | |

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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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Calculus with Parametric Curves | |

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Laboratory Project: B?zier Curves | |

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Polar Coordinates | |

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Laboratory Project: Families of Polar Curves | |

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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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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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Representations of Functions as Power Series | |

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Taylor and Maclaurin Series | |

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Laboratory Project: An Elusive Limit | |

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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 Project: Radiation from the Stars | |

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Review | |

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Problems Plus | |

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Vectors And The Geometry Of Space | |

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Three-Dimensional Coordinate Systems | |

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Vectors | |

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The Dot Product | |

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The Cross Product | |

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Discovery Project: The Geometry of a Tetrahedron | |

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Equations of Lines and Planes | |

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Cylinders and Quadric Surfaces | |

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Review | |

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Problems Plus | |

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Vector Functions | |

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Vector Functions and Space Curves | |

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Derivatives and Integrals of Vector Functions | |

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Arc Length and Curvature | |

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Motion in Space: Velocity and Acceleration | |

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Applied Project: Kepler's Laws | |

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Review | |

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Problems Plus | |

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Partial Derivatives | |

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Functions of Several Variables | |

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Limits and Continuity | |

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Partial Derivatives | |

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Tangent Planes and Linear Approximation | |

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The Chain Rule | |

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Directional Derivatives and the Gradient Vector | |

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Maximum and Minimum Values | |

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Applied Project: Designing a Dumpster | |

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Discovery Project: Quadratic Approximations and Critical Points | |

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Lagrange Multipliers | |

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Applied Project: Rocket Science | |

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Applied Project: Hydro-Turbine Optimization | |

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Review | |

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Problems Plus | |

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Multiple Integrals | |

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Double Integrals over Rectangles | |

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Iterated Integrals | |

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Double Integrals over General Regions | |

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Double Integrals in Polar Coordinates | |

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Applications of Double Integrals | |

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Triple Integrals | |

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Discovery Project: Volumes of Hyperspheres | |

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Triple Integrals in Cylindrical Coordinates | |

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Discovery Project: The Intersection of Three Cylinders | |

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Triple Integrals in Spherical Coordinates | |

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Applied Project: Roller Derby | |

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Change of Variables in Multiple Integrals | |

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Review | |

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Problems Plus | |

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Vector Calculu | |

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Line Integrals | |

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The Fundamental Theorem for Line Integrals | |

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Green's Theorem | |

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Curl and Divergence | |

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Parametric Surfaces and Their Areas | |

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Surface Integrals | |

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Stokes' Theorem | |

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Writing Project: Three Men and Two Theorems | |

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The Divergence Theorem | |

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Summary | |

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Review | |

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Problems Plus | |

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Second-Order Differential Equations | |

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Second-Order Linear Equations | |

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Nonhomogeneous Linear Equations | |

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Applications of Second-Order Differential Equations | |

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Series Solutions | |

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Review | |

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Problems Plus | |

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Appendixes | |

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Intervals, Inequalities, and Absolute Values | |

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Coordinate Geometry and Lines | |

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Graphs of Second-Degree Equations | |

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Trigonometry | |

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Sigma Notation | |

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Proofs of Theorems | |

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Graphing Calculators and Computers | |

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Complex Numbers | |

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Answers to Odd-Numbered Exercises | |