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Take Home Quiz 14, F09 due Nov. 3rd at the beginning of class Name ________________________ You can work with others and use references. You should write out your own final solutions. Show your work. 1. a) Use the chain rule to find z for z = (e xy ) tan y, x = s+4 t, y = t / s. t b) Use partial derivatives to find dy / dx for y6 + x3 y2 = 3 + x 2. (a) For f(x,y) = y e (b). For f f(x,y) = y e 5x 5x ey 2 find the gradient of f(x,y) at the point (1, -2). find the directional derivative of f(x,y) at the point (1, -2) in the direction v = <8, -6>. 3. Provide two geometric interpretations of the gradient – one interpretation should relate to level curves of a function and another should relate to the rate of increase of the function. 4. Find the following partial derivatives (a) x xy x y (OVER) (OVER) (OVER) (OVER) (b) 5 xy 2 2 x 3 y 3 xxy 5. Find the critical points and use second derivative tests to identify whether they are maximum, minimums or saddle points for f(x,y) = x y ( 4 – x – y ). 6. Find the absolute max and min of f(x,y) = 4 + xy – x – 2y over the closed triangle with vertices (1,0), (5,0) and (1,4). 7. Prove or disprove: if f ( x, y ) 5 xy then 4x 2 y 2 2 8. Find the tangent plane to the surface defined by lim ( x , y ) ( 0 , 0 ) f ( x, y ) = 0 z 2 x 3 y at the point (2,4,2)