M1_2009_Math124 - MATH124 CALCULUS II for Engineers Midterm...

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MATH124 CALCULUS II for Engineers Course Section: Midterm 1 (50 pts) February 11, 2009 Please print names and IDs in ink : NSID: Family Name: First Name: Student ID: INSTRUCTIONS: 1 . Time Limit: 80 minutes 4 . Closed book. Closed notes. 2 . No cheating. 5 . Write clearly & legibly. 3 . Simplify all answers unless otherwise instructed. 6 . No calculators. SECTION I
14 pts
1. (2 pts) Evaluate the sum 100
i =1 2 i + 1 100 100
i =1 2 i + 1 100 = 100
i =1 2 i 100 + 100
i =1 1 100 = 1 50 100
i =1 i + 1 100 100
i =1 1 = 1 50 · 100(100 + 1) 2 + 1 100 · 100 = 101 + 1 = 102 2. (3 pts) Express the integral
1 0 3(1 + x ) 2 3 dx as a limit of Riemann sums. DO NOT EVALUATE THE LIMIT. Observe that x = 1 n and x i = 0 + i · 1 n = i n . Therefore,
1 0 3(1 + x ) 2 / 3 dx = lim n →∞ n
i =1 3
1 + i n
2 / 3 · 1 n
3. (3 pts) Compute the average of the function f ( x ) = 2 x on the interval [1 , 4]. f ave = 1 3
4 1 2 x dx = 2 3
4 1 x 1 / 2 dx = 2 3 · x 1 / 2 · 2
4 1 = 4 3 x
4 1 = 4 3 (2 1) = 4 3 4. (3 pts) Use the Fundamental Theorem of Calculus to find F
( x ) where F ( x ) =
x 0 sin( t 2 ) dt . F
( x ) = d dx
x 0 sin( t 2 ) dt Let x = u . Then d dx
x 0 sin( t 2 ) dt = d dx
u 0 sin( t 2 ) dt = d du
u 0 sin( t 2 ) du dx = sin( u 2 ) · du dx = sin x · 1 2 x = sin x 2 x

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