| |

| |

Introduction: Calculus: A New Horizon from Ancient Roots | |

| |

| |

| |

Functions | |

| |

| |

| |

Functions and the Analysis of Graphical Information | |

| |

| |

| |

Properties of Functions | |

| |

| |

| |

Graphing Functions on Calculators and Computers; Computer Algebra Systems | |

| |

| |

| |

New Functions from Old | |

| |

| |

| |

Lines | |

| |

| |

| |

Families of Functions | |

| |

| |

| |

Mathematical Models | |

| |

| |

| |

Parametric Equations | |

| |

| |

Horizon Module: Iteration and Dynamical Systems | |

| |

| |

| |

Limits and Continuity | |

| |

| |

| |

Limits (An Intuitive Approach) | |

| |

| |

| |

Computing Limits | |

| |

| |

| |

Computing Limits: End Behavior | |

| |

| |

| |

Limits (Discussed More Rigorously) | |

| |

| |

| |

Continuity | |

| |

| |

| |

Limits and Continuity of Trigonometric Functions | |

| |

| |

| |

The Derivative | |

| |

| |

| |

Slopes and Rates of Change | |

| |

| |

| |

The Derivative | |

| |

| |

| |

Techniques of Differentiation | |

| |

| |

| |

Derivatives of Trigonometric Functions | |

| |

| |

| |

The Chain Rule | |

| |

| |

| |

Implicit Differentiation | |

| |

| |

| |

Related Rates | |

| |

| |

| |

Local Linear Approximation; Differentials | |

| |

| |

Horizon Module: Robotics | |

| |

| |

| |

The Derivative in Graphing and Applications | |

| |

| |

| |

Analysis of Functions I: Increase, Decrease, and Concavity | |

| |

| |

| |

Analysis of Functions II: Relative Extrema; First and Second Derivative Tests | |

| |

| |

| |

Analysis of Functions III: Applying Technology and the Tools of Calculus | |

| |

| |

| |

Rectilinear Motion (Motion Along a Line) | |

| |

| |

| |

Absolute Maxima and Minima | |

| |

| |

| |

Applied Maximum and Minimum Problems | |

| |

| |

| |

Newton's Method | |

| |

| |

| |

Rolle's Theorem; Mean-Value Theorem | |

| |

| |

| |

Integration | |

| |

| |

| |

An Overview of the Area Problem | |

| |

| |

| |

The Indefinite Integral; Integral Curves and Direction Fields | |

| |

| |

| |

Integration by Substitution | |

| |

| |

| |

Sigma Notation; Area as a Limit | |

| |

| |

| |

The Definite Integral | |

| |

| |

| |

The Fundamental Theorem of Calculus | |

| |

| |

| |

Rectilinear Motion Revisited; Average Value | |

| |

| |

| |

Evaluating Definite Integrals by Substitution | |

| |

| |

Horizon Module: Blammo the Human Cannonball | |

| |

| |

| |

Applications of the Definite Integral in Geometry, Science, and Engineering | |

| |

| |

| |

Area Between Two Curves | |

| |

| |

| |

Volumes by Slicing; Disks and Washers | |

| |

| |

| |

Volumes by Cylindrical Shells | |

| |

| |

| |

Length of a Plane Curve | |

| |

| |

| |

Area of a Surface of Revolution | |

| |

| |

| |

Work | |

| |

| |

| |

Fluid Pressure and Force | |

| |

| |

| |

Exponential, Logarithmic, and Inverse Trigonometric Functions | |

| |

| |

| |

Inverse Functions | |

| |

| |

| |

Exponential and Logarithmic Functions | |

| |

| |

| |

Derivatives and Integrals Involving Logarithmic and Exponential Functions | |

| |

| |

| |

Graphs and Applications Involving Logarithmic and Exponential Functions | |

| |

| |

| |

Logarithmic Functions from the Integral Point of View | |

| |

| |

| |

Derivatives and Integrals Involving Inverse Trigonometric Functions | |

| |

| |

| |

L'Hopital's Rule; Indeterminate Forms | |

| |

| |

| |

Hyperbolic Functions and Hanging Cables | |

| |

| |

| |

Principles of Integral Evaluation | |

| |

| |

| |

An Overview of Integration Methods | |

| |

| |

| |

Integration by Parts | |

| |

| |

| |

Trigonometric Integrals | |

| |

| |

| |

Trigonometric Substitutions | |

| |

| |

| |

Integrating Rational Functions by Partial Fractions | |

| |

| |

| |

Using Tables of Integrals and Computer Algebra Systems | |

| |

| |

| |

Numerical Integration; Simpson's Rule | |

| |

| |

| |

Improper Integrals | |

| |

| |

Horizon Module: Railroad Design | |

| |

| |

| |

Mathematical Modeling with Differential Equations | |

| |

| |

| |

First-Order Differential Equations and Applications | |

| |

| |

| |

Direction Fields; Euler's Method | |

| |

| |

| |

Modeling with First-Order Differential Equations | |

| |

| |

| |

Second-Order Linear Homogeneous Differential Equations; The Vibrating Spring | |

| |

| |

| |

Infinite Series | |

| |

| |

| |

Maclaurin and Taylor Polynomial Approximations | |

| |

| |

| |

Sequences | |

| |

| |

| |

Monotone Sequences | |

| |

| |

| |

Infinite Series | |

| |

| |

| |

Convergence Tests | |

| |

| |

| |

The Comparison, Ratio, and Root Tests | |

| |

| |

| |

Alternating Series; Conditional Convergence | |

| |

| |

| |

Maclaurin and Taylor Series; Power Series | |

| |

| |

| |

Convergence of Taylor Series; Computational Methods | |

| |

| |

| |

Differentiating and Integrating Power Series; Modeling with Taylor Series | |

| |

| |

| |

Analytic Geometry in Calculus | |

| |

| |

| |

Polar Coordinates | |

| |

| |

| |

Tangent Lines and Arc Length for Parametric and Polar Curves | |

| |

| |

| |

Area in Polar Coordinates | |

| |

| |

| |

Conic Sections in Calculus | |

| |

| |

| |

Rotation of Axes; Second-Degree Equations | |

| |

| |

| |

Conic Sections in Polar Coordinates | |

| |

| |

Horizon Module: Comet Collision | |

| |

| |

| |

Real Numbers, Intervals, and Inequalities | |

| |

| |

| |

Absolute Value | |

| |

| |

| |

Coordinate Planes and Lines | |

| |

| |

| |

Distance, Circles, and Quadratic Equations | |

| |

| |

| |

Trigonometry Review | |

| |

| |

| |

Solving Polynomial Equations | |

| |

| |

| |

Selected Proofs | |

| |

| |

Answers | |

| |

| |

Photo Credits | |

| |

| |

Index | |