**Unformatted text preview: **f is continuous at-→ x . (d) Every limit point of a set E ⊆ R n is an interior point of E . 6. (10) Let f ( x, y, z ) = x 2 y 3 + (1 + zx ) 4 . (a) Calculate ∇ f ( x, y, z ). (b) Explain how you know that f is diﬀerentiable at (1 , ,-2). (c) For-→ v = (1 , 2 ,-3) calculate ∂f ∂-→ v (1 , ,-2). (d) Write down the expression for the ﬁrst degree Taylor approximation to f at (1 , ,-2). 7. (4) Let f ( x, y ) = xy ( x 2-y 2 ) ( x 2 + y 2 ) 2 . Show that lim ( x,y ) → (0 , 0) f ( x, y ) does not exist. This is what the problem should have been. On the sheet I originally had the typo where f ( x, y ) = xy ( x 2-y 2 ) x 2 + y 2 . In this case the limit exists and equals 0. 8. (5) Let f ( x, y ) = 3 + 2 x ( y 2-x 2 ) x 2 + y 2 if ( x, y ) 6 = (0 , 0) and f (0 , 0) = 3. Calculate ∂f ∂x (0 , 0). Typeset by A M S-T E X 1...

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- Spring '14
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- Derivative, #, Closure, General topology, Boundary