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dis. Find the rate of change of f(x,y,z)=xyz in the direction normal to surface yx^(2)+xy^(2)+yz^(2)=3 at (1,1,1)Solution -:Rate of change of f at x_(0) in direction of unit veatos n is gules by grad f(x_(0))*nn is unit normal vectorSay, quad g(x,y,z)=yx^(2)+xy^(2)+yz^(2)then put value{:[Delta f* hat(r)=(3(1)(1)+4(1)(1)+2(1)(1))/(sqrt29)],[=(3+4+2)/(sqrt29)=(9)/(sqrt29)]:}rate of change (1,1,1)=(9)/(sqrt29) cinits]Qle: Find the planes tangent to the followiny surfaces. at indecated point.a) x^(2)+8y^(2)+9xz=36 at point (1,2,(1)/(3))Sol. f(x,y,z)=x^(2)+8y^(2)+9xz-36=0{:[grad f=(df)/(dx) hat(ı)+(df)/(dy) hat(ȷ)+(df)/(dz) hat(k)],[grad f=(2x+9z) hat(ı)+(16 y) hat(ȷ)+(9x) hat(k)],[grad f(1,2,(1)/(3))=5 hat(ı)+32 hat(ȷ)+3 hat(k)]:}Tangent plane grad f*[(x-x_(0))Omega+(y-y_(0))rho+(z-z_(0))k]=0{:[(5 hat(ı)+32 hat(ȷ)+3 hat(k))*[(x-1)( hat(ı))+(y-2)( hat(ı))+(z-(1)/(3))( hat(k))]=0],[5(x-1)+32(y-2)+3(z-(1)/(3))=0],[5x-5+32 y-6y+3z-1 ... See the full answer