Unformatted text preview: MAT 125B Homework 4
M.Fukuda
Please submit your answers at the discussion session on February 12 Tuesday. You can use theorems in the lecuture unless otherwise stated. When you do so write those statements clearly instead of quoting them by numbers. 1. Show, specifying the conditions you use, that
1 lim
y0 0 cos(x y + xy ) dx = 1, 2 2 and d dy 1 x2 y 2
1 + xy + y + 2 dx y=0 = 2 . 2 2. Let T L(Rn ; Rm ). Show that T is differentiable on Rn . 3. i) Prove that f : R2 R; f (x, y) = x4 +y 4 (x2 +y 2 ) (x, y) = (0, 0) (x, y) = (0, 0) is differentiable on R2 for < 3/2. ii) Prove that g : R2 R2 ; 0 g(x, y) = is differentiable on R2 for < 3/2. 4. Prove that f (x, y) = x3 xy2
x2 +y 2 f (x, y) 1 (x, y) = (0, 0) (x, y) = (0, 0) is continuous on R2 , has firstorder partial derivatives on R2 , but f is not differentiable at (0, 0). 0 1 ...
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This homework help was uploaded on 04/08/2008 for the course MATH 125B taught by Professor Fukuda during the Winter '07 term at UC Davis.
 Winter '07
 Fukuda

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