Question

Compute the partial derivatives of the function at the given point, and determine what the total derivative would be if the function was differentiable there; then determine whether it is differentiable or not by definition; that is, determine whether or not 0. 22+y2 (x,y) # (0,0) fão + n) – $(20) - Df#, (h) lim ho hi 4. h(x, y) = 0 (2,y) = (0,0) at (0,0); 5. g(x, y, z) = x + 3y - 2 - 1 at (0,1,1); 6. y2+uz+1-1-2 (x, y, z) # (1,0,1) f(x, y, z) = (x-1)2+y2+(2-1)2 0 (x, y, z) = (1,0,1) at (x, y, z) = (1,0,1);

Answer 1

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we Can Ah = h (k, l) - h (0,0) = K +64 K+12 dh = Khx 10,0) + l hy 10,0) = 0. if show that sh-dh (k, l) → 0 then f will be differentiable. We ²+k2 here, sh-dh= Khul k² + 12 4h-dh K't (K48312 ah-dh 그 23 or Sin o) leb k=p Cos o l= r Sina. ph (Cost 0 + Sino) √ K ² + 1? (Cost of Sin 0 3/2 P (Cost Costot ot Singol (k² + 1² Er (1 Cost of + fin of < 20 sh-dh - r las 1xryl? lxl+ly) Csok Sinal 1, as 22 dk ² + 1² <2 E l < E² se if 1kk 2 / 2 Ilk این if ks 8² 2 sh-dh - O lim Kto 1 to (0,0) his differentiable at

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if we can show that, Ad- dd (h, k, l) (0,1,1) then & will be differentiable 6²+k²7 12 Ad- dd = o. here, there, ad-da О as (h, k, l) → (0,1,1). h²+k² ľ Therefore, & is differentiable at (0,1,1).

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