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Calculus

Functions, continuity

The derivative

Differentiability, differentiation rules and formulas

Differentiation, implicit differentiation, derivatives of higher order, critical points, inflection points, maxima and minima

Rules of exponents and logarithms

The exponential and logarithmic functions, logarithmic differentiation

Tangents, normals, parametric equations, vectors, curvilinear motion, velocity, acceleration, related time-rates problems

Differentials, differential of arc length, curvature, radius of curvature, circle of curvature, center of curvature, evolute

Position vector, vector representation of plane curves, unit tangent, principal normal, curvature, curvilinear motion, velocity, acceleration, tangential and normal components of acceleration

Polar coordinates

Mean value theorems for derivatives; Rolle's Theorem, Mean Value Theorem, Cauchy's Generalized Mean Value Theorem, Extended Law of the Mean (Taylor's Theorem)

Indeterminate forms, L'hopital's rule

Primitive, Definite Integral, Fundamental Theorem of Integral Calculus, Properties of Definite Integrals

Integration, indefinite integral, fundamental formulas and rules

Trigonometry: formulas, identities, solution of triangles

Integration by parts

Integration by substitution

Evaluation of trigonometric integrals

Rational expression, Rational algebraic fraction, Partial fractions, Expressing a proper algebraic fraction as a sum of partial fractions

Elementary functions, General methods of integration

Improper integrals

Applications: areas, volumes of solids of revolution, length of a curve, areas of surfaces of revolution, work, fluid pressure

Approximate integration, Trapezoidal rule, Prismoidal formula, Simpson's rule

Partial derivative, Total differential, Total derivative, Chain rule, Directional derivative, Differentiation under the integral sign, Leibnitz’s rule

Implicit functions, Derivatives of implicit functions, Jacobian

Maxima and minima of functions of several variables, Stationary point, Lagrange's Method of Multipliers

Double integral, triple integral, iterated integral

Moments, center of gravity, centroid, moment of inertia, radius of gyration

Calculus of variations. Functionals. Euler’s equation. Beltrami Identity. Brachistochone problem. Minimal surface of revolution. Isoperimetric problems.

Space curves and surfaces