# 2008 Solution - University of Toronto Scarborough...

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University of Toronto ScarboroughDepartment of Computer & Mathematical SciencesMAT B41H2008/2009Term Test Solutions1.(a) From the lecture notes we haveLetf:URnRkbe a given function.We say thatfisdifferentiable ataUif the partial derivatives offexist ataand iflimxabardblf(x)f(a)Df(a) (xa)bardblbardblxabardbl= 0,whereDf(a) is thek×nmatrixparenleftbigg∂fi∂xjparenrightbiggevaluated ata.Df(a) is called thederivative offata.(b) From the lecture notes we haveExtreme Value Theorem.LetDbe a compact set inRnand letf:DRnRbe continuous.Thenfassumes both a (global)maximum and a (global) minimum onD.2.(a)lim(x,y)(1,-1)f(x, y) = 3 does not say anything aboutf(1,1).The limitsays something about what happens as a point is approached, not whathappens at that point.(x+y)23
MATB41HTerm Test Solutionspage23.f(x, y) =xx2+y2.Domain is{(x, y)R2|(x, y)negationslash= (0,0)}.Puttingf(x, y) =cwe havexx2+y2=c. Forc= 0, the level curve isx= 0.Forcnegationslash= 0,we havex=c(x2+y2)⇐⇒x2xc+y2=0⇐⇒parenleftbiggx12cparenrightbigg2+y2=parenleftbigg12

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