# tut4 - MATH 237 Tutorial 4 Wednesday May 31st 2006...

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Unformatted text preview: MATH 237 Tutorial 4 Wednesday, May 31st, 2006 Diﬀerentiability 1) Let g : R2 → R be deﬁned by g (x, y ) = |x|5/2 + |y |5/2 , (x, y ) = (0, 0) x2 + y 2 g (0, 0) = 0 . a) Determine whether g is continuous at (0, 0). b) Determine whether g is diﬀerentiable at (0, 0). 2) Let f (x, y ) = x(|y | − 2). a) Prove that f is diﬀerentiable at (0, 0). b) Using the result in (a), draw a conclusion about the continuity of f at (0, 0). c) Determine whether f is diﬀerentiable at (a, b), b < 0. d) Determine whether f is diﬀerentiable at (a, 0), a = 0. 3) Let f : R2 → R be deﬁned by f (x, y ) = x2 y 2 for (x, y ) = (0, 0) and f (0, 0) = 0 . x4 + y 2 Determine whether f is diﬀerentiable at (0, 0). 4) Determine whether the function f (x, y ) = is diﬀerentiable at (0, 0). 5) Consider the function f (x, y ) = |x|1/2 |y |p where p is a positive constant. 1 a) Show that f is diﬀerentiable at (0, 0) if p > . 2 b) Suppose that f is diﬀerentiable at (0, 0). What conclusion, if any, may be drawn about the continuity of fx and fy at (0, 0)? y 4 − x2 y for (x, y ) = (0, 0) and f (0, 0) = 0 x2 + y 2 ...
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