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Unformatted text preview: University of Toronto at Scarborough Division of Mathematical Sciences MAT A26Y (Calculus) 2002/2003 Course Outline Definition of a function; domain range; even and odd functions, examplespolynomials, rational, roots, trigonometric, absolute value, floor, ceiling. The number of roots of a non-zero polynomial is at most the degree. Rational functions: their domains, and the finiteness of the number of their zeroes and singularities. Definitions of sum, product, composition and inverse of functions. arcsin , arccos , arctan , and arcsec. Definition of limit at a point; examples. limit theorems (constant, sum, product, quotient, root, sandwich or squeeze law, substitu- tion). lim sin . Infinite limits (two kinds: lim x a f ( x ) = and lim x f ( x ) = L ); one-sided limits; properties; examples. Behaviour of rational functions as x ; order of a rational function with its properties. Left and right continuity of a function at a point; (two-sided) continuity at a point; examples of continuity; continuity theorems (sum product quotient, composition, inverse); one-sided continuity; extreme value theorem for continuous functions (no proof); examples. Differentiability at a point: f ( x ) = f ( a ) + F ( x )( x- a ), where F ( x ) is continuous at a ; examples. Sum, product, quotient, chain rule theorems; a selection of proofs; examples. Higher order derivatives; implicit differentiation (no proof), examples.Statement of Rolles Theorem; intermediate value theorem (no proof). Applications of intermediate value theorem and Rolles Theorem to roots of polynomials....
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