math20C_W07_Practice_fin

math20C_W07_Practice_fin - g ( x, y, z ) = x 4 + y 4 + z 4...

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Math 20C, Practice Final Exam March 13, 2007 Name : PID : TA : Sec. No : Sec. Time : This exam consists of 11 pages including this front page. Ground Rules 1. No calculator is allowed. 2. Show your work for every problem. A correct answer without any justification will receive no credit. 3. You may use two 4-by-6 index cards, both sides. 4. You have two hours for this exam. Score 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 Total 100 1
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1. (a) Let f ( x, y ) = x y - ln | x + y 2 | . What is f x ? (b) Compute the double integral Z 1 0 Z x 0 e y dy dx 2
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2. (a) Let f ( x, y ) = xe y . Find the directional derivative of f at (1 , 0) in the direction of u = ( 3 5 , - 4 5 ). (b) Let a = 3 i - 1 j + 2 k and b = - i + 3 k . Compute a × b . 3
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3. Let S be the tangent plane to the surface defined by z = x 2 - 3 y 3 at (2 , 1 , 1). Find the symmetric equation of the line which is perpendicular to S and goes through the point (1 , 2 , 3). 4
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4. Compute the double integral Z 1 0 Z 2 2 x e y 2 dy dx. 5
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5. Use Lagrange multipliers to find the maximum and minimum values of the function f ( x, y, z ) = x 2 + y 2 + z 2 subject to the constraint
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Unformatted text preview: g ( x, y, z ) = x 4 + y 4 + z 4 = 1. 6 6. Let z = x 2 + xy 3 , x = uv 2 + w 3 , and y = u + ve w . Use the Chain Rule to nd z v , when u = 2 , v = 1 , w = 0. 7 7. Find all the points at which the direction of the fastest change of the function f ( x, y ) = x 2 + y 2-2 x-4 y is i + j . 8 8. Find the absolute maximum and minimum values of f ( x, y ) = x 2 + y 2 + x 2 y +4 on the region D = { ( x, y ) | | x | 1 , | y | 1 } . 9 9. Evaluate the triple integral R R R E z dV , where E is bounded by the cylinder y 2 + z 2 = 9 and the planes x = 0 , y = 3 x, and z = 0 in the rst octant. 10 10. Evaluate the triple integral R R R E x 2 dV , where E is the solid that lies within the cylinder x 2 + y 2 = 1, above the plane z = 0, and below the cone z 2 = 4 x 2 + 4 y 2 . 11...
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math20C_W07_Practice_fin - g ( x, y, z ) = x 4 + y 4 + z 4...

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