HW03-solutions - griffin (ang684) HW03 Gilbert (56650) 1...

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Unformatted text preview: griffin (ang684) HW03 Gilbert (56650) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points Decide which of the following regions has area = lim n n summationdisplay i = 1 3 n sin i 3 n without evaluating the limit. 1. braceleftBig ( x, y ) : 0 y sin 2 x, x 6 bracerightBig 2. braceleftBig ( x, y ) : 0 y sin 3 x, x 3 bracerightBig 3. braceleftBig ( x, y ) : 0 y sin 3 x, x 6 bracerightBig 4. braceleftBig ( x, y ) : 0 y sin 2 x, x 3 bracerightBig 5. braceleftBig ( x, y ) : 0 y sin x, x 3 bracerightBig correct 6. braceleftBig ( x, y ) : 0 y sin x, x 6 bracerightBig Explanation: The area under the graph of y = f ( x ) on an interval [ a, b ] is given by the limit lim n n summationdisplay i = 1 f ( x i ) x when [ a, b ] is partitioned into n equal subin- tervals [ a, x 1 ] , [ x 1 , x 2 ] , . . . , [ x n 1 , b ] each of length x = ( b- a ) /n . If A = lim n n summationdisplay i = 1 3 n sin i 3 n , therefore, we see that f ( x i ) = sin i 3 n , x = 3 n . But in this case x i = i 3 n , f ( x ) = sin x, [ a, b ] = bracketleftBig , 3 bracketrightBig . Consequently, the area is that of the region under the graph of y = sin x on the interval [0 , / 3]. In set-builder notation this is the region braceleftBig ( x, y ) : 0 y sin x, x 3 bracerightBig . 002 10.0 points Estimate the area under the graph of f ( x ) = 3 sin x between x = 0 and x = 4 using five approx- imating rectangles of equal widths and right endpoints. 1. area . 983 2. area 1 . 023 3. area 1 . 003 4. area 1 . 063 5. area 1 . 043 correct Explanation: An estimate for the area under the graph of f on [0 , b ] with [0 , b ] partitioned in n equal subintervals [ x i 1 , x i ] = bracketleftBig ( i- 1) b n , ib n bracketrightBig and right endpoints x i as sample points is A braceleftBig f ( x 1 ) + f ( x 2 ) + . . . + f ( x n ) bracerightBig b n . For the given area, f ( x ) = 3 sin x, b = 4 , n = 5 , and x 1 = 1 20 , x 2 = 1 10 , x 3 = 3 20 , x 4 = 1 5 , x 5 = 1 4 . griffin (ang684) HW03 Gilbert (56650) 2 Thus A 3 braceleftBig sin parenleftBig 1 20 parenrightBig + . . . + sin parenleftBig 1 4 parenrightBigbracerightBig 20 . After calculating these values we obtain the estimate area 1 . 043 for the area under the graph. keywords: area, sin function, estimate area, numerical calculation, 003 10.0 points Rewrite the sum 5 n parenleftBig 6 + 2 n parenrightBig 2 + 5 n parenleftBig 6 + 4 n parenrightBig 2 + . . . + 5 n parenleftBig 6 + 2 n n parenrightBig 2 using sigma notation....
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This note was uploaded on 09/09/2009 for the course M 408L taught by Professor Gilbert during the Spring '09 term at University of Texas-Tyler.

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HW03-solutions - griffin (ang684) HW03 Gilbert (56650) 1...

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