234practice-2 - Math 234 Practice Test#2 Show your work in...

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Math 234, Practice Test #2 Show your work in all the problems. 1. In what directions is the derivative of f ( x, y ) = x 2 y 2 x 2 + y 2 at P = (1 , 1) equal to zero ? 2. Find an equation for the level surface of the function through the given point P . (a) f ( x, y, z ) = z x 2 y 2 , P = (3 , 1 , 1) (b) f ( x, y, z ) = integraldisplay y x dt 1 t 2 + integraldisplay z 2 dt t t 2 1 , P = ( 1 , 1 / 2 , 1) 3. Compute the limits of the following expressions if they exist. If you think they don’t, consider different paths of approach to show that they do not. (a) x y + 1 x y 1 , ( x, y ) (4 , 3) , x negationslash = y + 1 (b) x 2 + y 2 xy , ( x, y ) (0 , 0) , xy negationslash = 0 4. Compute all second order partial derivatives of the function f ( x, y ) = x sin y + y sin x + xy 5. Find dw dt if w = sin( xy + π ), x = e t and y = ln( t + 1). Then evaluate at t = 0. 1
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Solutions 1. We first compute the gradient vector of f : f = parenleftBigg 4 xy 2 ( x 2 + y 2 ) 2 , 4 x 2 y ( x 2 + y 2 ) 2 parenrightBigg Evaluating at (1 , 1) yields f (1 , 1) = (1 , 1) .
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