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Fall 2006 - Hall's Class - Practice Final Exam

Fall 2006 - Hall's Class - Practice Final Exam - Math 20E...

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Math 20E Practice Final Exma December 1, 2006 Name: ID#: Section Hour: TA: Guidelines for the test: No books, notes, or calculators are allowed. You may leave answers in symbolic form, like 42, unless they simplify further, like 9 = 3 , e 0 = 1, or cos(3 π/ 4) = - 2 / 2. Use the space provided. If necessary, write “see other side” and continue working on the back of the same sheet. Circle your final answers when relevant. Show all steps in your solutions and make your reasoning clear. Answers with no explanation will receive no credit, even if they are correct. You have 180 minutes. Question # Perfect Score Your Score 1 20 2 20 3 20 4 20 5 20 6 20 7 20 8 20 9 20 10 20 TOTAL 200
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1. Let D be the parallelogram with vertices (1 , 1) , (2 , 3) , (4 , 6), and (3 , 4). (a) (4 points) Let D * 1 be the unit square [0 , 1] × [0 , 1]. Explain why it is not possible to find a linear transformation T ( u, v ) = ( au + bv, cu + dv ) such that T ( D * 1 ) = D . (b) (8 points) Find a rectangular region D * 2 and a linear transformation T ( u, v ) = ( au + bv, cu + dv ) such that T ( D * 2 ) = D . (b) (8 points) Compute R R D x 2 dA . Page 1
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2. Let f ( u, v, w ) = ( e u - w , cos( u + v ) + sin ( u + v + w )) and g ( x, y ) = ( e x , cos( y - x ) , e - y ). (a) (7 points) Compute ( f g )( x, y ). (b) (6 points) Compute D ( f g )(0 , 0) directly, using your answer to part (a). (c) (7 points) Compute D ( f g )(0 , 0) using the Chain Rule. Page 2
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3. (a) (8 points) Find a parametrization for the surface S = { ( x, y ) : x 2 + y 2 + z 2 = R 2 , y 3 x } . Hint: S is a hemisphere. (b) (8 points) Find a parametrization for the boundary C of S . (c) (4 points) Is the orientation consistent between your two parametriza- tions? Explain. Page 3
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4. Let C be the curve parametrized by ~ c ( t ) = (cos( πt ) , t, t ) , 0 t 4 . (a) (7 points) Sketch a graph of C , labeling the endpoints. (b) (7 points) Find the equation of the tangent line to C at the point ( - 1 , 1 , 1).
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