Fall 2006 - Takeda's Class - Practice Exam 1

Fall 2006 - Takeda's Class - Practice Exam 1 - = Z 1 (1-x 4...

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Math 20B, Sample Midterm 1 Solutions October 14, 2006 Name: Section: This exam consists of 7 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 one 4-by-6 index card, both sides. Score 1 10 2 10 3 10 4 10 5 10 6 10 Total 60 1
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1. (a) Evaluate Z 3 - 1 (6 x 2 - 2 x + 1) dx Z 3 - 1 (6 x 2 - 2 x + 1) dx = (2 x 3 - x 2 + x ) i 3 - 1 = (2 · 27 - 9 + 3) - ( - 2 - 1 - 1) = 52 (b) Find the indefinite integral Z x 2 x dx Z x 2 x dx = Z x 5 / 2 dx = 2 7 x 7 / 2 + c 2
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2. (a) Evaluate Z sin( t ) t dt Let u = t . Then du = 1 2 t dt . So Z sin( t ) t dt = Z 2 sin u du = - 2 cos u + c = - 2 cos( t ) + c (b) Evaluate Z 1 0 x (2 x 2 - 1) 2 dx Let u = 2 x 2 - 1. Then du = 4 xdx and x 0 1 u -1 1 Then Z 1 0 x (2 x 2 - 1) 2 dx = Z 1 - 1 1 4 u 2 du = 1 2 Z 1 0 u 2 du = 1 2 · 1 3 u 3 i 1 0 = 1 6 3
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3. Find the area of the region enclosed by y = | x | and y = x 2 - 2. Area = 2 Z 2 0 [ x - ( x 2 - 2)] dx = 2( 1 2 x 2 - 1 3 x 3 + 2 x ) i 2 0 = 20 3 4
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4. Find the volume of the solid by rotating about the x -axis the region in the first quadrant enclosed by x = 0 , y = 1 , and y = x 2 . Volume = Z 1 0 π (1 - ( x 2 ) 2 ) dx
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Unformatted text preview: = Z 1 (1-x 4 ) dx = ( x-1 5 x 5 ) i 1 = (1-1 5 ) = 4 5 5 5. Find the number b so that the average value of the function y = x 2 + bx + 2 on [0 , 1] is 1 3 . Average = 1 1-Z 1 ( x 2 + bx + 2) dx = ( 1 3 x 3 + 1 2 bx 2 + 2 x ) i 1 = 1 3 + 1 2 b + 2 = 1 2 b + 7 3 . Set 1 2 b + 7 3 = 1 3 . Then b =-4 6 6. (a) Sketch the curve with the polar equation r = . (b) Find the slope of the tangent line to the curve r = 2 sin at = / 6 dy dx = dy d dx d = d d (2 sin sin ) d d (2 sin cos ) = cos sin + sin cos cos cos -sin sin = sin(2 ) cos(2 ) Thus dy dx = / 6 = sin( / 3) cos( / 3) = 3 7...
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Fall 2006 - Takeda's Class - Practice Exam 1 - = Z 1 (1-x 4...

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