Exam_exam_2_(3)

Exam_exam_2_(3) - x y =(2 5 we have ∂z/∂x ∂z/∂y...

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MATH 241 – Exam 2 – Boyle Friday the 13th, March 1998 Show your work. Put a box around the result of a computation. 1. (15 points) Let f ( x, y ) = ye 3 x . For which unit vector u is the directional derivative D u f (0 , 1) greatest? 2. (20 points) Let f ( x, y ) = xy + 8 / ( x 2 ) + 8 / ( y 2 ). The critical points of f are (2 , 2) and ( - 2 , - 2). Determine what information the second partial derivatives test gives you about f at these critical points. 3. (25 points) Let f ( x, y ) = x 3 y . (a) (10 points) What is the derivative approximation to f (2+Δ x, - 1+Δ y ) - f (2 , - 1)? (b) (15 points) Suppose z is a function of x and y , and at the point (
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Unformatted text preview: x, y ) = (2 , 5) we have ( ∂z/∂x, ∂z/∂y ) = (10 ,-1). Also suppose x = u 2 v and y = 3 u + uv . What is ∂z/∂u at the point ( u, v ) = (1 , 2)? 4. (20 points) Below, determine the limit or determine it does not exist (DNE). You must show some work to justify your conclusion. lim ( x,y ) → (0 , 0) ( x + y ) 2 x 2 + y 2 lim ( x,y ) → (0 , 0) sin( xy ) y 5. (25 points) Find the extreme values of the function f ( x, y ) = 3 x 2 + 2 y 2-4 y subject to the constraint x 2 + y 2 ≤ 9....
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