Calculus

 

Functions & their graphs (Polynomial and rational functions, logarithmic and exponential functions, trigonometric functions and their inverses, hyperbolic functions and their inverses, combinations of such functions). Limit and continuity: Definitions and basic theorems on limit and continuity. limit at infinity and infinite limits Computation of limits.

 

Differentiation: Tangent lines and rates of change. Definition of derivative. One-sided derivatives. Rules of differentiation (proofs and applications). Successive differentiation. Leibnitz theorem (proofs and application). Related rates. Linear approximations and differentials.

 

Applications of Differentiation: Rolle’s theorm, mean value theorem. Maximum and minimum values of functions. Concavity and points of inflection. Oplimization problems, Curvature.

 

Function of several variables: Limit and continuty. Partial derivatives Defferentiability. linerarization and differentials. The chain rule. Partial derivatives with constrained variables Derictional variables. Lagrange multipliers, Taylor’s formula.

Integration: Antiderivatives and  indefinite integrals. Techniques of integration. Definite integration using antiderivatives. Definite integration using Riemann sums. Fundamental theorems of calculus (proofs and applications). Basic properties of integration. Integration reduction.

 

Applications of Integration: Arc length. Plane areas. Surfaces of revolution. Volumes of solids of revolution.

Graphing in polar coordinates. Tangents to polar curves. Areas in polar coordinates. Arc length in polar coordinates.

 

Multiple Integration: Double integrals and iterated integrals. Double integrals over nonrectangular regions. Double integrals in polar coordinates. Area by double integral. Triple integrals and iterated integrals. Volume as a triple integrals.

 

Improper integrals. Tests of convergence and their applications. Gamma and Beta functions.

 

Indeterminate forms, L’ Hospital’s rule.

 

Approximation and Series: Taylor polynomials and series. Convergence of series. Taylor’s sesies. Taylor’s theorem and remainders. Differentiation and integration of series. Validity of Tailor expansions and computations with series.