ef04 - df = f x ( x,y ) 4 x + f y ( x,y ) 4 y D u f ( x ) =...

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Final Exam Dec 13, 2004 MAC 2313 S Hudson Name Show all your work. Use the space provided, or leave a note. Don’t use a calculator or your own extra paper. The problems are 10pts each. 1) Use Stoke’s theorem to evaluate R R S (curl F ) · n dS where F = < 3 y, - 2 x,xyz > and S is the hemispherical surface z = p 4 - x 2 - y 2 with upper unit normal vector. 2) Find the moment of inertia R R S x 2 + y 2 dS of the surface S with respect to the z -axis. Assume δ = 1. S is the part of the plane z = x + y that lies inside the cylinder x 2 + y 2 = 9. 1
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3) Find a potential function for the vector field F ( x,y,z ) = yz i + xz j + xy k . Explain your reasoning. 4) Sketch the region of integration. Reverse the order of integration. Evaluate the resulting integral. Do not compute the original integral, unless you want to check your answer. Z 2 - 2 Z 4 x 2 x 2 y dy dx 5) Determine whether the line L and the plane P are parallel or intersect. If they intersect, find the point of intersection. L : x = 7 - 4 t , y = 3 + 6 t , z = 9 + 5 t P : 4 x + y + 2 z = 17 2
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6) Answer True or False. You do not have to explain. Assume f is differentiable. The differential of f ( x,y ) is
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Unformatted text preview: df = f x ( x,y ) 4 x + f y ( x,y ) 4 y D u f ( x ) = |∇ f ( x ) | u If f ( x,y ) ≥ 0 on a region R , then R R R f ( x,y ) dx dy ≥ 0. If a and b are parallel, then a × b = 0. (0,1) is an interior point of the ball B = { ( x,y ) : x 2 + y 2 ≤ 1. 7) Find the curvature κ for y = cos( x ) at (0,1). Use the formula for plane curves if you know it. If not, you may be able to work with the formula for space curves, κ = | r × r 00 | /v 3 . 8) Use integration in polar coordinates to find the volume of the solid bounded by z = 8-r 2 and z = r 2 . 3 9) Use LaGrange multipliers to find the max and min values for f ( x,y,z ) = xy +2 z subject to the constraint x 2 + y 2 + z 2 = 36. 10) Choose ONE. A) State and prove the Divergence Theorem. B) State and prove the Fundamental Theorem for Line Integrals. Bonus (5pts)) Compute the integral R + ∞ e-x 2 dx , as done in class, showing all work. 4...
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This note was uploaded on 12/27/2011 for the course MAC 2313 taught by Professor Grantcharov during the Fall '06 term at FIU.

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ef04 - df = f x ( x,y ) 4 x + f y ( x,y ) 4 y D u f ( x ) =...

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