Test2Old

# Test2Old - (25 points Find the directions in which the...

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MATH 231, Calculus III, Section 1, Fall 2007 Test 2 Date: Oct. 24, Wednesday Time: 10:00 am - 10:50 am Please write clearly, reduce answers to their simplest form, and box your answers. To receive full credit you must show ALL your work. Student's Name (Please print): __________________________________________ Pledge: On my honor as a student at the University of Virginia I have neither given nor received aid on this test. Signature: _______________________________ Problem Points Score 1 25 2 25 3 25 4 25 5 (Bonus) 10 Total 110/100

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Problem 1 (25 points) (a) (8 points) Evaluate the double integral R (4 - x ) dA, where R = { ( x, y ) R 2 | 0 x 4 , 0 y 3 } , by first identifying it as the volume of a solid. (b) (8 points) If z = z ( x, y ) is given by sin( xyz ) = x + 2 y + 3 z , find ∂z ∂x . (c) (9 points) Let z = z ( u 2 - v 2 , v 2 - u 2 ). Show that u ∂z ∂v + v ∂z ∂u = 0 . 1
Problem 2

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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 y-axis 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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