| |

| |

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 Method | |

| |

| |

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 | |

| |

| |

Applications to Economics and Biology | |

| |

| |

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 | |

| |

| |

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 | |