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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 | |