08wHW4 - MAT 125B Homework 4 M.Fukuda Please submit your...

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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 first-order partial derivatives on R2 , but f is not differentiable at (0, 0). 0 1 ...
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