P PREPARATION FOR CALCULUS. Graphs and Models. Linear Models and Rates of
Change. Functions and Their Graphs. Fitting Models to Data. Inverse Functions.
Exponential and Logarithmic Functions.
LIMITS AND THEIR PROPERTIES. A Preview of Calculus. Finding Limits Graphically
and Numerically. Evaluating Limits Analytically. Continuity and One-Sided
Limits. Infinite Limits. Section Project: Graphs and Limits of Trigonometric
DIFFERENTIATION. The Derivative and the Tangent Line Problem. Basic
Differentiation Rules and Rates of Change. Product and Quotient Rules and
Higher-Order Derivatives. The Chain Rule. Implicit Differentiation. Section
Project: Optical Illusions. Derivatives of Inverse Functions. Related Rates.
APPLICATIONS OF DIFFERENTIATION. Extrema on an Interval. Rolle's Theorem and the
Mean Value Theorem. Increasing and Decreasing Functions and the First Derivative
Test. Section Project: Rainbows. Concavity and the Second Derivative Test.
Limits at Infinity. A Summary of Curve Sketching. Optimization Problems. Section
Project: Connecticut River. Differentials.
INTEGRATION. Antiderivatives and Indefinite Integration. Area. Riemann Sums and
Definite Integrals. The Fundamental Theorem of Calculus. Section Project:
Demonstrating the Fundamental Theorem. Integration by Substitution. Numercal
Logarithmic Exponential Transcendtal The
Natural Logarithmic Function: Integration. Inverse Trigonometric Functions:
Integration. Hyperbolic Functions. Section Project: St. Louis Arch.
DIFFERENTIAL EQUATIONS. Slope Fields and Euler's Method. Differential Equations:
Growth and Decay. Differential Equations: Separation of Variables. The Logistic
Equation. First-Order Linear Differential Equations. Section Project: Weight
Loss. Predator-Prey Differential Equations.
APPLICATIONS OF INTEGRATION. Area of a Region Between Two Curves. Volume: The
Disk Method. Volume: The Shell Method. Section Project: Saturn. Arc Length and
Surfaces of Revolution. Work. Section Project: Tidal Energy. Moments, Centers of
Mass, and Centroids. Fluid Pressure and Fluid Force.
Integration Techniques, L'Hôpital's Rule, and Improper Integrals. Basic
Integration Rules. Integration by Parts. Trigonometric Integrals. Section
Project: Power Lines. Trigonometric Substitution. Partial Fractions. Integration
by Tables and Other Integration Techniques. Indeterminate Forms and L'Hôpital's
Rule. Improper Integrals.
INFINITE SERIES. Sequences. Series and Convergence. Section Project: Cantor's
Disappearing Table. The Integral Test and p-Series. Section Project: The
Harmonic Series. Comparisons of Series. Section Project: Solera Method.
Alternating Series. The Ratio and Root Tests. Taylor Polynomials and
Approximations. Power Series. Representation of Functions by Power Series.
Taylor and Maclaurin Series.
CONICS, PARAMETRIC EQUATIONS, AND POLAR COORDINATES. Conics and Calculus. Plane
Curves and Parametric Equations. Section Projects: Cycloids. Parametric
Equations and Calculus. Polar Coordinates and Polar Graphs. Section Project:
Anamorphic Art. Area and Arc Length in Polar Coordinates. Polar Equations of
Conics and Kepler's Laws.
VECTORS AND THE GEOMETRY OF SPACE. Vectors in the Plane. Space Coordinates and
Vectors in Space. The Dot Product of Two Vectors. The Cross Product of Two
Vectors in Space. Lines and Planes in Space. Section Project: Distances in
Space. Surfaces in Space. Cylindrical and Spherical Coordinates.
VECTOR-VALUED FUNCTIONS. Vector-Valued Functions. Section Project: Witch of
Agnesi. Differentiation and Integration of Vector-Valued Functions. Velocity and
Acceleration. Tangent Vectors and Normal Vectors. Arc Length and Curvature.
FUNCTIONS OF SEVERAL VARIABLES. Introduction to Functions of Several Variables.
Limits and Continuity. Partial Derivatives. Section Project: Moire Fringes.
Differentials. Chain Rules for Functions of Several Variables. Directional
Derivatives and Gradients. Tangent Planes and Normal Lines. Section Project:
Wildflowers. Extrema of Functions of Two Variables. Applications of Extrema of
Functions of Two Variables. Section Project: Building a Pipeline. Lagrange
MULTIPLE INTEGRATION. Iterated Integrals and Area in the Plane. Double Integrals
and Volume. Change of Variables: Polar Coordinates. Center of Mass and Moments
of Inertia. Section Project: Center of Pressure on a Sail. Surface Area. Section
Project: Capillary Action. Triple Integrals and Applications. Triple Integrals
in Cylindrical and Spherical Coordinates. Section Project: Wrinkled and Bumpy
Spheres. Change of Variables: Jacobians.
VECTOR ANALYSIS. Vector Fields. Line Integrals. Conservative Vector Fields and
Independence of Path. Green's Theorem. Section Project: Hyperbolic and
Trigonometric Functions. Parametric Surfaces. Surface Integrals. Section
Project: Hyperboloid of One Sheet. Divergence Theorem. Stoke's Theorem.