# We will follow the hint Away from 0,0, a little long...

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University of Toronto ScarboroughDepartment of Computer & Mathematical SciencesMAT B41H2010/2011Solutions #51. We will follow the hint. Away from (0,0), a little long division givesxy2-x2y+ 3x3-y3x2+y2=x-y+2x3x2+y2.Hencewecanrewritef(x, y)asf(x, y) =x-y+2x3x2+y2,(x, y) = (0,0)0,(x, y) = (0,0).4+ 6x2y23y(x2+y2)2.(a) For (x, y) = (0,0) we have∂f∂x= 1 +2x(x2+y2)2and∂f∂y=-1-4x(b) To calculatefx(0,0) andfy(0,0) we need to apply the definition.fx(0,0) = limh0f(h,0)-f(0,0)h= limh0h+2h3h2-0h= limh03h3h2h= limh03h3h3= 3.fy(0,0) = limh0f(0, h)-f(0,0)h= limh0(-h+0h2)-0h= limh0-hh=-1.2. Marsden & Tromba, page 140, #10.f(x, y) =x2+y2andg(x, y) =-x2-y2+xy3. At (x, y) = (0,0), we havefx(0,0) =2x(0,0)= 0,fy(0,0) = 2y(0,0)= 0,gx(0,0) =-2x+y3(0,0)= 0 andgy(0,0) =-2y+ 3xy2(0,0)= 0.Hence the graph ofz=f(x, y) at (0,0,0) and the graph ofz=g(x, y) at (0,0,0) have the same tangent plane; thexy–plane (z= 0). Since theyhave the same tangent plane we can think of them as tangent.3.f:R4R3is given byf(x, y, z, w) = (y z w, x2y, x z) soDf=0zwywyz2xyx200z0x0.g:R3R2is given byg(x, y, z) = (x y, y z) soD g=yx00zyandD g(f(x, y, z)) =x2yyzw00xzx2y.NowD(gf)(x, y, z, w) = [D g(f(x, y, z, w))] [D f(x, y, z, w)]
MATB41HSolutions # 5page2=x2yyzw00xz