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# outline - University of Toronto at Scarborough Division of...

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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, examples—polynomials, 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 θ 0 sin θ θ . 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 Rolle’s Theorem; intermediate value theorem (no proof). Applications of intermediate value theorem and Rolle’s Theorem to roots of polynomials. Statement of the Mean Value Theorem (MVT); consequences of the theorem, including the Extended Mean Value Theorem (EMVT), and the fact that a positive derivative corresponds to an increasing function.

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