Prelim1Solutions - Prelim 1 Math 1910 Calculus for...

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Unformatted text preview: Prelim 1 Math 1910 Calculus for Engineers February 23, 2010 You have 90 minutes to complete this test. The test has 100 points. Calculators are not allowed. You should show all your work, and explain where calculations are coming from. Write clearly and legibly. Please write your name and the time you meet for recitation. 1 . (a) [5 pts] Find Z p 2 t 3 + 2 t 2 + 1 (3 t 2 + 2 t )( t 3 + t 2 ) dt. (b) [5 pts] Calculate d dx Z cos( x ) 1 (2 u + 1) du. (c) [5 pts] Calculate the average of the function f ( x ) = sin( x ) sin(cos( x )) over the interval [0 , / 2]. (d) [5 pts] Suppose you are given numbers a 1 , a 2 , . . . , a 10 and you are given the following information: (i) the average of the first three numbers is 4; (ii) the average of the last four numbers is 2; (iii) 10 k =4 3 a k = 30. Compute 6 k =1 a k . Answer: (a) We use the substitution u = 2 t 3 + 2 t 2 + 1. Then du = (6 t 2 + 4 t ) dt , so Z p 2 t 3 + 2 t 2 + 1 (3 t 2 + 2 t )( t 3 + t 2 ) dt = Z u u 2- 1 2 1 2 du = 1 4 Z ( u 3 / 2- u 1 / 2 ) du = 1 4 2 5 u 5 / 2- 2 3 u 3 / 2 + C = 1 10 (2 t 3 + 2 t 2 + 1) 5 / 2- 1 6 (2 t 3 + 2 t 2 + 1) 3 / 2 + C. (b) By the fundamental theorem of calculus and the chain rule, d dx Z cos( x ) 1 (2 u + 1) du =- (2 cos( x ) + 1) sin( x ) ....
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Prelim1Solutions - Prelim 1 Math 1910 Calculus for...

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