Table of Contents
1. Functions, Graphs, and Models.
Functions and Mathematical Modeling. Graphs of Equations and Functions. Polynomials and Algebraic Functions. Transcendental Functions. Preview: What Is Calculus?
2. Prelude to Calculus.
Tangent Lines and Slope Predictors. The Limit Concept. More about Limits. The Concept of Continuity.
3. The Derivative.
The Derivative and Rates of Change. Basic Differentiation Rules. The Chain Rule. Derivatives of Algebraic Functions. Maxima and Minima of Functions on Closed Intervals. Applied Optimization Problems. Derivatives of Trigonometric Functions. Successive Approximations and Newton's Method.
4. Additional Applications of the Derivative.
Introduction. Implicit Differentiation and Related Rates. Increments, Differentials, and Linear Approximation. Increasing and Decreasing Functions and the Mean Value Theorem. The First Derivative Test and Applications. Simple Curve Sketching. Higher Derivatives and Concavity. Curve Sketching and Asymptotes.
5. The Integral.
Introduction. Antiderivatives and Initial Value Problems. Elementary Area Computations. Riemann Sums and the Integral. Evaluation of Integrals. The Fundamental Theorem of Calculus. Integration by Substitution. Areas of Plane Regions. Numerical Integration.
6. Applications of the Integral.
Riemann Sum Approximations. Volumes by the Method of Cross Sections. Volumes by the Method ofCylindrical Shells. Arc Length and Surface Area of Revolution. Force and Work. Centroids of Plane Regions and Curves.
7. Calculus of Transcendental Functions.
Exponential and Logarithmic Functions. Indeterminate Forms of L'Hôpital's Rule. More Indeterminate Forms. The Logarithm as an Integral. Inverse Trigonometric Functions. Hyperbolic Functions.
8. Techniques of Integration.
Introduction. Integral Tables and Simple Substitutions. Integration by Parts. Trigonometric Integrals. Rational Functions and Partial Fractions. Trigonometric Substitutions. Integrals Involving Quadratic Polynomials. Improper Integrals.
9. Differential Equations.
Simple Equations and Models. Slope Fields and Euler's Method. Separable Equations and Applications. Linear Equations and Applications. Population Models. Linear Second-Order Equations. Mechanical Vibrations.
10. Polar Coordinates and Parametric Curves.
Analytic Geometry and the Conic Sections. Polar Coordinates. Area Computations in Polar Coordinates. Parametric Curves. Integral Computations with Parametric Curves. Conic Sections and Applications.
11. Infinite Series.
Introduction. Infinite Sequences. Infinite Series and Convergence. Taylor Series and Taylor Polynomials. The Integral Test. Comparison Tests for Positive-Term Series. Alternating Series and Absolute Convergence. Power Series. Power Series Computations. Series Solutions of Differential Equations.
12. Vectors and Matrices.
Vectors in the Plane. Three-Dimensional Vectors. The Cross Product of Two Vectors. Lines and Planes in Space. Linear Systems and Matrices. Matrix Operations. Eigenvalues and Rotated Conics.
13. Curves and Surfaces in Space.
Curves and Motion in Space. Curvature and Acceleration. Cylinders and Quadric Surfaces. Cylindrical and Spherical Coordinates.
14. Partial Differentiation.
Introduction. Functions of Several Variables. Limits and Continuity. Partial Derivatives. Multivariable Optimizations Problems. Linear Approximation and Matrix Derivatives. The Multivariable Chain Rule. Directional Derivatives and Gradient Vectors. Lagrange Multipliers and Constrained Optimization. Critical Points of Multivariable Functions.
15. Multiple Integrals.
Double Integrals. Double Integrals over More General Regions. Area and Volume by Double Integration. Double Integrals in Polar Coordinates. Applications of Double Integrals. Triple Integrals. Integration in Cylindrical and Spherical Coordinates. Surface Area. Change of Variables in Multiple Integrals.
16. Vector Calculus.
Vector Fields. Line Integrals. The Fundamental Theorem and Independence of Path. Green's Theorem. Surface Integrals. The Divergence Theorem. Stokes' Theorem.