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Unformatted text preview: (25 points) Find the directions in which the directional derivative of the function f ( x, y ) = x 2 + sin( xy ) at the point (1 , 0) has the value 1. 2 Problem 3 (25 points) Evaluate the following double integral: I = ZZ D sin( xy ) dA, where D is bounded by y = 1, y = 2, the yaxis and x = π y . 3 Problem 4 (25 points) Evaluate the following double integral: I = ZZ D y dA p x 2 + y 2( x 2 + y 2 ) 2 , where D = { ( x, y ) ∈ R 2  x ≥ , y ≥ , 1 2 ≤ x 2 + y 2 ≤ 1 } . 4 Bonus Problem 5 (10 points) Suppose that f is a diﬀerentiable function of three variables. Show that the maximum value of the directional derivative of f in the direction of u is ∇ f ( x, y, z )  and it occurs when u has the same direction as the gradient vector of f . 5...
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 Spring '09
 Stojanovic
 Calculus, Derivative

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