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.
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