Diagnostic Tests | |

A Preview of Calculus | |

Functions And Models | |

Four Ways to Represent a Function | |

Mathematical Models: A Catalog of Essential Functions | |

New Functions from Old Functions | |

Graphing Calculators and Computers | |

Exponential Functions | |

Inverse Functions and Logarithms | |

Review | |

Principles of Problem Solving | |

Limits And Derivatives | |

The Tangent and Velocity Problems | |

The Limit of a Function | |

Calculating Limits Using the Limit Laws | |

The Precise Definition of a Limit | |

Continuity | |

Limits at Infinity | |

Horizontal Asymptotes | |

Derivatives and Rates of Change | |

Writing Project: Early Methods for Finding Tangents | |

The Derivative as a Function | |

Review | |

Problems Plus | |

Differentiation Rules | |

Derivatives of Polynomials and Exponential Functions | |

Applied Project: Building a Better Roller Coaster | |

The Product and Quotient Rules | |

Derivatives of Trigonometric Functions | |

The Chain Rule | |

Applied Project: Where Should a Pilot Start Descent? | |

Implicit Differentiation | |

Derivatives of Logarithmic Functions | |

Rates of Change in the Natural and Social Sciences | |

Exponential Growth and Decay | |

Related Rates | |

Linear Approximations and Differentials | |

Laboratory Project: Taylor Polynomials | |

Hyperbolic Functions | |

Review | |

Problems Plus | |

Applications Of Differentiation | |

Maximum and Minimum Values | |

Applied Project: The Calculus of Rainbows | |

The Mean Value Theorem | |

How Derivatives Affect the Shape of a Graph | |

Indeterminate Forms and L'Hospital's Rule | |

Writing Project: The Origins of l'Hospital's Rule | |

Summary of Curve Sketching | |

Graphing with Calculus and Calculators | |

Optimization Problems | |

Applied Project: The Shape of a Can | |

Newton's Metho | |

Antiderivatives | |

Review | |

Problems Plus | |

Integrals | |

Areas and Distances | |

The Definite Integral | |

Discovery Project: Area Functions | |

The Fundamental Theorem of Calculus | |

Indefinite Integrals and the Net Change Theorem | |

Writing Project: Newton, Leibniz, and the Invention of Calculus | |

The Substitution Rule | |

Review | |

Problems Plus | |

Applications Of Integration | |

Areas between Curves | |

Volume | |

Volumes by Cylindrical Shells | |

Work | |

Average Value of a Function | |

Applied Project: Where to Sit at the Movies | |

Review | |

Problems Plus | |

Techniques Of Integration | |

Integration by Parts | |

Trigonometric Integrals | |

Trigonometric Substitution | |

Integration of Rational Functions by Partial Fractions | |

Strategy for Integration | |

Integration Using Tables and Computer Algebra Systems | |

Discovery Project: Patterns in Integrals | |

Approximate Integration | |

Improper Integrals | |

Review | |

Problems Plus | |

Further Applications Of Integration | |

Arc Length | |

Discovery Project: Arc Length ConteSt. Area of a Surface of Revolution | |

Discovery Project: Rotating on a Slant | |

Applications to Physics and Engineering | |

Discovery Project: Complementary Coffee Cups | |

Probability | |

Review | |

Problems Plus | |

Differential Equations | |

Modeling with Differential Equations | |

Direction Fields and Euler's Method | |

Separable Equations | |

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

Models for Population Growth | |

Applied Project: Calculus and Baseball | |

Linear Equations | |

Predator-Prey Systems | |

Review | |

Problems Plus | |

Parametric Equations And Polar Coordinates | |

Curves Defined by Parametric Equations | |

Laboratory Project: Families of Hypocycloids | |

Calculus with Parametric Curves | |

Laboratory Project: Bezier Curves | |

Polar Coordinates | |

Areas and Lengths in Polar Coordinates | |

Conic Sections | |

Conic Sections in Polar Coordinates | |

Review | |

Problems Plus | |

Infinite Sequences And Series | |

Sequences | |

Laboratory Project: Logistic Sequences | |

Series | |

The Integral Test and Estimates of Sums | |

The Comparison Tests | |

Alternating Series | |

Absolute Convergence and the Ratio and Root Tests | |

Strategy for Testing Series | |

Power Series | |

Representations of Functions as Power Series | |

Taylor and Maclaurin Series | |

Laboratory Project: An Elusive Limit | |

Writing Project: How Newton Discovered the Binomial Series | |

Applications of Taylor Polynomials | |

Applied Project: Radiation from the Stars | |

Review | |

Problems Plus | |

Vectors And The Geometry Of Space | |

Three-Dimensional Coordinate Systems | |

Vectors | |

The Dot Product | |

The Cross Product | |

Discovery Project: The Geometry of a Tetrahedron | |

Equations of Lines and Planes | |

Quadric Surfaces | |

Cylindrical and Spherical Coordinates | |

Laboratory Project: Families of Surfaces | |

Review | |

Problems Plus | |

Vector Functions | |

Vector Functions and Space Curves | |

Derivatives and Integrals of Vector Functions | |

Arc Length and Curvature | |

Motion in Space: Velocity and Acceleration | |

Applied Project: Kepler's Laws | |

Review | |

Problems Plus | |

Partial Derivatives | |

Functions of Several Variables | |

Limits and Continuity | |

Partial Derivatives | |

Tangent Planes and Differentials | |

The Chain Rule | |

Directional Derivatives and the Gradient Vector | |

Maximum and Minimum Values | |

Applied Project: Designing a Dumpster | |

Discovery Project: Quadratic Approximations and Critical Points | |

Lagrange Multipliers | |

Applied Project: Rocket Science | |

Applied Project: Hydro-Turbine Optimization | |

Review | |

Problems Plus | |

Multiple Integrals | |

Double Integrals over Rectangles | |

Iterated Integrals | |

Double Integrals over General Regions | |

Double Integrals in Polar Coordinates | |

Applications of Double Integrals | |

Surface Area | |

Triple Integrals | |

Discovery Project: Volumes of Hyperspheres | |

Triple Integrals in Cylindrical and Spherical Coordinates | |

Applied Project: Roller Derby | |

Discovery Project: The Intersection of Three Cylinders | |

Change of Variables in Multiple Integrals | |

Review | |

Problems Plus | |

Vector Calculu | |

Line Integrals | |

The Fundamental Theorem for Line Integrals | |

Green's Theorem | |

Curl and Divergence | |

Parametric Surfaces and Their Areas | |

Surface Integrals | |

Stokes' Theorem | |

Writing Project: Three Men and Two Theorems | |

The Divergence Theorem | |

Summary | |

Review | |

Problems Plus | |

Second-Order Differential Equations | |

Second-Order Linear Equations | |

Nonhomogeneous Linear Equations | |

Applications of Second-Order Differential Equations | |

Series Solutions | |

Review | |

Problems Plus | |

Appendixes | |

Numbers, Inequalities, and Absolute Values | |

Coordinate Geometry and Lines | |

Graphs of Second-Degree Equations | |

Trigonometry | |

Sigma Notation | |

Proofs of Theorems | |

The Logarithm Defined as an Integral | |

Complex Numbers | |

Answers to Odd-Numbered Exercises | |

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