(6) Let f(r, y)-a2 +3y2 +1. Prove directly from the definition that thus functio

Question

(6) Let f(r, y)-a2 +3y2 +1. Prove directly from the definition that thus function is differentiable at (xo, Jo)- (1,2) (7) Use the chain rule to compute w'-f(x(t,y(t)) for the following function: (b) w-f(a, y)2y + 4y2 and a-3 cos t, y sin 4t and cos 8) Find Du./(ro, yo) for the following functions: (a) f(x,y)=y In x+xy2,(zo, yo)= (1,2) and u= (cosn/6. Sinn/6) (b) f(z. )-x2y2+3ry, (xo, yo)-(-1,3) and u-(1,-2) (c) f(x, y, z)zn zy2z, (xo, Vo, zo) - (1,-1,2) and u - (9) Compute Vf(x, y) for the following functions at (xo, yo)- and de- termine the direction of maximum and minimum increase. (b) f(x,y) = sin xy+ x tany at (x0,Yo) = (-/4, /4) (c) f(x, y, z) = x2 + + at (240 , yo, a) = (-1, 1.2) (10) Find the equation of the tangent plane and the normal line for the following functions: (a) f (x, y)-2+ y4+3 (ro, yo(1,-2) (b) f(x, y, z) x2 + y2 + z2 (zo, yo.zo)-(-1,2. 2) (c) f(z, y, z)-zex2+92 (zo. , 20-(0.0.1)

Explanation / Answer

solution(6)-

Since f(x,y) is a polynomial function in R^2, hence it is differentiable everywhere.

Hence it is also differentiable at the point (x0,y0) = (1,2).

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