## Table of Contents

(*Practice Exercises, Additional Exercises, and Questions to Guide Your Review appear at the end of each chapter.*)

**Preliminaries.**

Real Numbers and the Real Line.

Lines, Circles, and Parabolas.

Functions and Their Graphs.

Identifying Functions; Mathematical Models.

Combining Functions; Shifting and Scaling Graphs.

Trigonometric Functions.

Graphing with Calculators and Computers.

**2. Limits and Derivatives.**

Rates of Change and Limits.

Calculating Limits Using the Limit Laws.

Precise Definition of a Limit.

One-Sided Limits and Limits at Infinity.

Infinite Limits and Vertical Asymptotes.

Continuity.

Tangents and Derivatives.

**3. Differentiation.**

The Derivative as a Function.

Differentiation Rules.

The Derivative as a Rate of Change.

Derivatives of Trigonometric Functions.

The Chain Rule and Parametric Equations.

Implicit Differentiation.

Related Rates.

Linearization and Differentials.

**4. Applications of Derivatives.**

Extreme Values of Functions.

The Mean Value Theorem.

Monotonic Functions and the First Derivative Test.

Concavity and Curve Sketching.

Applied Optimization Problems.

Indeterminate Forms and L'Hopital's Rule.

Newton's Method.

Antiderivatives.

**5. Integration.**

Estimating with Finite Sums.

Sigma Notation and Limits of Finite Sums.

The Definite Integral.

The Fundamental Theorem of Calculus.

Indefinite Integrals and the Substitution Rule.

Substitution and Area Between Curves.

**6. Applications of Definite Integrals.**

Volumes by Slicing and Rotation About an Axis.

Volumes by Cylindrical Shells.

Lengths of Plane Curves.

Moments and Centers of Mass.

Areas of Surfaces of Revolution and The Theorems of Pappus.

Work.

Fluid Pressures and Forces.

**7. Transcendental Functions.**

Inverse Functions and their Derivatives.

Natural Logarithms.

The Exponential Function.

*a*x and loga *x*.

Exponential Growth and Decay.

Relative Rates of Growth.

Inverse Trigonometric Functions.

Hyperbolic Functions.

**8. Techniques of Integration.**

Basic Integration Formulas.

Integration by Parts.

Integration of Rational Functions by Partial Fractions.

Trigonometric Integrals.

Trigonometric Substitutions.

Integral Tables and Computer Algebra Systems.

Numerical Integration.

Improper Integrals.

**9. Further Applications of Integration.**

Slope Fields and Separable Differential Equations.

First-Order Linear Differential Equations.

Euler's Method.

Graphical Solutions of Autonomous Equations.

Applications of First-Order Differential Equations.

**10. Conic Sections and Polar Coordinates.**

Conic Sections and Quadratic Equations .

Classifying Conic Sections by Eccentricity.

Quadratic Equations and Rotations.

Conics and Parametric Equations; The Cycloid.

Polar Coordinates .

Graphing in Polar Coordinates.

Area and Lengths in Polar Coordinates.

Conic Sections in Polar Coordinates.

**11. Infinite Sequences and Series.**

Sequences.

Infinite Series.

The Integral Test.

Comparison Tests.

The Ratio and Root Tests.

Alternating Series, Absolute and Conditional Convergence.

Power Series.

Taylor and Maclaurin Series.

Convergence of Taylor Series; Error Estimates.

Applications of Power Series.

Fourier Series.

**12. Vectors and the Geometry of Space.**

Three-Dimensional Coordinate Systems.

Vectors.

The Dot Product.

The Cross Product.

Lines and Planes in Space.

Cylinders and Quadric Surfaces .

**13. Vector-Valued Functions and Motion in Space.**

Vector Functions.

Modeling Projectile Motion.

Arc Length and the Unit Tangent Vector **T**.

Curvature and the Unit Normal Vector **N**.

Torsion and the Unit Binormal Vector **B**.

Planetary Motion and Satellites.

**14. Partial Derivatives.**

Functions of Several Variables.

Limits and Continuity in Higher Dimensions.

Partial Derivatives.

The Chain Rule.

Directional Derivatives and Gradient Vectors.

Tangent Planes and Differentials.

Extreme Values and Saddle Points.

Lagrange Multipliers.

*Partial Derivatives with Constrained Variables.

Taylor's Formula for Two Variables.

**15. Multiple Integrals.**

Double Integrals.

Areas, Moments and Centers of Mass*.

Double Integrals in Polar Form.

Triple Integrals in Rectangular Coordinates.

Masses and Moments in Three Dimensions.

Triple Integrals in Cylindrical and Spherical Coordinates.

Substitutions in Multiple Integrals.

**16. Integration in Vector Fields.**

Line Integrals.

Vector Fields, Work, Circulation, and Flux.

Path Independence, Potential Functions, and Conservative Fields.

Green's Theorem in the Plane.

Surface Area and Surface Integrals.

Parametrized Surfaces.

Stokes' Theorem.

The Divergence Theorem and a Unified Theory.

**Appendices.**

Mathematical Induction.

Proofs of Limit Theorems.

Commonly Occurring Limits .

Theory of the Real Numbers.

Complex Numbers.

The Distributive Law for Vector Cross Products.

Determinants and Cramer's Rule.

The Mixed Derivative Theorem and the Increment Theorem.

The Area of a Parallelogram's Projection on a Plane.