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Unformatted text preview: 20B Quiz Solutions Fall 2004 Bender Each quiz is worth 12 points. Q1. Each part is worth 6 points. (a) The first is an application of the Fundamental Theorem of Calculus and the rule for differentiating composition of functions. The answer is radicalbig ( x 2 ) 3 + 1 dx 2 dx = 2 x radicalbig x 6 + 1 . There are 3 points for each factor; that is, 3 points for 2 x and three points for radicalbig ( x 2 ) 3 + 1 (which can be written x 6 + 1). If you differentiate x 2 incorrectly, there are no points for that factor. If you get x 3 + 1 or some other incorrect factor, there are no points for that factor. (b) This is almost identical to a homework problem (5.5#63). Use the substitution u = t- 1 to obtain (remembering that du = dt ) integraldisplay t t- 1 dt = integraldisplay ( u + 1) u du = integraldisplay ( u 3 / 2 + u 1 / 2 ) du = 2 u 5 / 2 5 + 2 u 3 / 2 3 + C = 2( t- 1) 5 / 2 5 + 2( t- 1) 3 / 2 3 + C. Of course, you might not choose to call the new variable u . 1 point: giving correct substitution; 1 point: getting du = dt ; 1 point: doing the substitution correctly; 1 point: evaluating the resulting integral; 1 point: getting the final result in terms of t; 1 point: having + C . If you make an error somewhere along the line, you will lose the one point for that step. If the remainder of the work based on that error is done correctly, you will receive credit for those steps. Exception : If you write t- 1 = t- 1, or t- 1, or some similar misuse of square root, you will lose all subsequent points except possibly + C . Subsequent quizzes may not give such a detailed point break-down. Also, as we get more material, subsequent quizzes are likely to be more difficult. Q2. The region is bounded above by the parabola and below by the line. The curves intersect at (0,0) and (3,3). (a) A = integraldisplay 3 ( x (4- x ))- x ) dx = integraldisplay 3 (3 x- x 2 ) dx = parenleftbigg 3 x 2 2- x 3 3 bracketrightbigg 3 = 3 3 / 2- 3 2 . You need not carry out the arithmetic to get 9 / 2. 4 points for setting up the integral correctly; 2 points for the integration. 1 20B Quiz Solutions Fall 2004 Bender (b) We need to get x as a function of y , so we have x = y and x = 2- 4- y . The latter comes from y = 4 x- x 2 = 4- ( x- 2) 2 and the correct choice of the sign on the square root. Thus the answer is integraldisplay 3 parenleftBig y 2- (2- radicalbig 4- y ) 2 parenrightBig dy = integraldisplay...
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