2011Solution - University of Toronto Scarborough Department...

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University of Toronto Scarborough Department of Computer & Mathematical Sciences MAT B41H 2011/2012 Term Test Solutions 1. (a) From the lecture notes we have Let f : U R n R k be a given function. We say that f is differentiable at a U if the partial derivatives of f exist at a and if lim x a f ( x ) - f ( a ) - Df ( a ) ( x - a ) x - a = 0 , where Df ( a ) is the k × n matrix ∂f i ∂x j evaluated at a . Df ( a ) is called the derivative of f at a . (b) From the lecture notes we have Chain Rule . Let f : U R n R m and g : V R m R k be given functions such that f [ U ] V so that g f is defined. Let a R n and b = f ( a ) R m .
MATB41H

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