MATH 241 Midterm: MATH241H_HERB-R_FALL2003_0101_MID_EXAM_1

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turquoisegnu128

10 Jan 2019

School

University of Maryland

Department

Mathematics

Course

MATH 241

Professor

ALL

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Math 241h - exam 2 , october 27, 2003. No calculators - show your work. (21) 1. Find all critical points of the function f (x, y) = x3 + xy2 + 4xy 5x and determine whether each is a local maximum, local minimum, or saddle point. (20) 3. Integrate the function f (x, y) = x over the region bounded by y = x2 and y = 2x. (20) 5. Use the change of variables x = u + v, y = u v, to evaluate.

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1. Use Lagrange multipliers to find the least distance between the point P(?3, 0, 4) and the surface of the cone z 2 = x 2 + y 2 .

2. Evaluate ? 2 0 ? 4 y 2 y sin(x 2 )dxdy.

3. Show that the volume V of the solid in the first octant bounded by the surface z = e y?x , the plane x + y = 1, and the coordinate planes satisfy V < cosh(1)

4. Set up the integral in spherical coordinates ? 1 0 ? ? 1?x2 0 ? ? 2?x2?y 2 ? x2+y 2 xydzdydx

5. Set up in polar coordinates the following double integral ? ? R ? x 2 + y 2dA; (R) bounded by x 2 + y 2 = 2x, y ? x, y ? 0.

6. Suppose a particle moving in space has velocity v(t) = ?sin(2t), cos(t), e?t ? and initial position x(0) = ?2, 1, 0?. Find the position vector function x(t).

7. : Find the maximum value of y x over (x ? 2)2 + y 2 .

MATH 241 Midterm: MATH241H_HERB-R_FALL2003_0101_MID_EXAM_1

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