Calculus with Analytic Geometry

1. Preliminaries. The Real Number System. Decimals, Calculators, Estimation. Inequalities. Absolute Values, Square Roots, Squares. The Rectangular Coordinate System. The Straight Line. Graphs of Equations. 2. Functions and Limits. Functions and Their Graphs. Operations on Functions. The Trigonometric Functions. Introduction to Limits. Rigorous Study of Limits. Limit Theorems. Continuity of Functions. 3. The Derivative. Two Problems with One Theme. The Derivative. Rules for Finding Derivatives. Derivatives of Sines and Cosines. The Chain Rule. Leibniz Notation. Higher-Order Derivatives. Implicit Differentiation. Related Rates. Differentials and Approximations. 4. Applications of the Derivative. Maxima and Minima. Monotonicity and Concavity. Local Maxima and Minima. More Max-Min Problems. Economic Applications. Limits at Infinity, Infinite Limits. Sophisticated Graphing. The Mean Value Theorem. 5. The Integral. Antiderivatives (Indefinite Integrals). Introduction to Differential Equations. Sums and Sigma Notation. Introduction to Area. The Definite Integral. The Fundamental Theorem of Calculus. More Properties of the Definite Integral. Aids in Evaluating Definite Integrals. 6. Applications of the Integral. The Area of a Plane Region. Volumes of Solids: Slabs, Disks, Washers. Volumes of Solids of Revolution: Shells. Length of a Plane Curve. Work. Moments, Center of Mass. 7. Transcendental Functions. The Natural Logarithm Function. Inverse Functions and Their Derivatives. The Natural Exponential Function. General Exponential and Logarithmic Functions. Exponential Growth and Decay. The Inverse Trigonometric Functions. Derivatives of Trigonometric Functions. The Hyperbolic Functions and Their Inverses. 8. Techniques of Integration. Integration by Substitution. Some Trigonometric Integrals. Rationalizing Substitutions. Integration by Parts. Integration of Rational Functions. 9. Indeterminate Forms and Improper Integrals. Indeterminate Forms of Type O/O. Other Indeterminate Forms. Improper Integrals: Infinite Limits. Improper Integrals: Infinite Integrands. 10. Numerical Methods, Approximations. Taylors Approximation to Functions. Bounding the Errors. Numerical Integration. Solving Equations Numerically. Fixed-Point Methods. 11. Infinite Series. Infinite Sequences. Infinite Series. Positive Series: The Integral Test. Positive Series: Other Tests. Alternating Series, Absolute Convergence. Power Series. Operations on Power Series. Taylor and Maclaurin Series. 12. Conics and Polar Coordinates. The Parabola. Ellipses and Hyperbolas. More on Ellipses and Hyperbolas. Translation of Axes. Rotation of Axes. The Polar Coordinate System. Graphs of Polar Equations. Calculus in Polar Coordinates. 13. Geometry in the Plane, Vectors. Plane Curves: Parametric Representation. Vectors in the Plane: Geometric Approach. Vectors in the Plane: Algebraic Approach. Vector-Valued Functions and Curvilinear Motion. Curvature and Acceleration. 14. Geometry in Space, Vectors. Cartesian Coordinates in Three-Space. Vectors in Three-Space. The Cross Product. Lines and Curves in Three-Space. Velocity, Acceleration, and Curvature. Surfaces in Three-Space. Cylindrical and Spherical Coordinates. 15. The Derivative in n-Space. Functions of Two or More Variables. Partial Derivatives. Limits and Continuity. Differentiability. Directional Derivatives and Gradients. The Chain Rule. Tangent Planes, Approximations. Maxima and Minima. Lagranges Method. 16. The Integral in n-Space. Double Integrals Over Rectangles. Iterated Integrals. Double Integrals over Nonrectangular Regions. Double Integrals in Polar Coordinates. Applications of Double Integrals. Surface Area. Triple Integrals (Cartesian Coordinates). Triple Integrals (Cylindrical and Spherical Coordinates). 17. Vector Calculus. Vector Fields. Line Integrals. Independence of Path. Greens Theorem in the Plane. Surface Integrals. Gausss Divergence Theorem. Stokes Theorem. 18. Differential Equations. Linear First-Order Equations. Second-Order Homogeneous Equations. The Nonhomogeneous Equation. Applications of Second-Order Equations. Appendix. Mathematical Induction. Proofs of Several Theorems. A Backward Look. Numerical Tables. Answers to Odd-Numbered Problems. Index.