fx265s2009

fx265s2009 - ( x,y ) = y occupying the triangle with...

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Math 265 Final Exam S2009 Instructor: Answer each question completely. Show all work. No credit for mere answers with no work shown. Show the steps of calculations and give exact answers. State the reasons that justify conclusions. 1. A particle moves in space with position vector r ( t ) = cos(2 t ) i + 3 t j + sin(2 t ) k . a) Show that the velocity vector v ( t ) and the acceleration vector a ( t ) have constant length. b) Show that v ( t ) and a ( t ) are orthogonal for each t . c) Find the distance the particle moves between t = 0 and t = 2 π . 2. Find an equation of the plane containing the point (2 , 1 , 3) and the line x = 1 + 3 t,y = - 2 - t,z = 3 t .
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3. Find an equation of the tangent plane to the surface x 2 + xy + y 2 + z 2 = 16 at the point (1 , 2 , 3). 4. Find all of the critical points of the function f ( x,y ) = x 3 - 9 xy + y 3 , and classify each one as local maximum, local minimum, or saddle point.
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5. Find the mass and the center of mass of a thin plate with density function
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Unformatted text preview: ( x,y ) = y occupying the triangle with vertices (0 , 0), (1 , 1), and (-1 , 1). 6. Convert the integral below from cylindrical coordinates to an equivalent integral in a) Cartesian, b) spherical coordinates. DO NOT EVALUATE. I = Z Z 1 Z 3 r r 2 sin d z d r d . 7. Let S be the solid sphere centered at the origin with radius 2, let F = (3 xz 2 + 2 yz 2 ) i + (3 x 2 z 3-yz 2 ) j + (3 x 2 y 2 + z 3 ) k , and let n be the outward pointing unit normal vector on the boundary of S . Calculate RR S F n dS . 8. Consider the vector eld F ( x,y ) = h 3 x-y, 2 z, 4 x-z i . a) Calculate curl F and div F . b) Use Stokes Theorem to calculate R C F T d s , where C is the triangular path from (2 , , 0) to (0 , 3 , 0) to (0 , , 2) and back to (2 , , 0)....
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fx265s2009 - ( x,y ) = y occupying the triangle with...

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