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Unformatted text preview: pokharel (yp624) HW03 Radin (57410) 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 4 n cos i 4 n without evaluating the limit. 1. braceleftBig ( x, y ) : 0 y cos 3 x, x 8 bracerightBig 2. braceleftBig ( x, y ) : 0 y cos 2 x, x 8 bracerightBig 3. braceleftBig ( x, y ) : 0 y cos 2 x, x 4 bracerightBig 4. braceleftBig ( x, y ) : 0 y cos x, x 4 bracerightBig correct 5. braceleftBig ( x, y ) : 0 y cos x, x 8 bracerightBig 6. braceleftBig ( x, y ) : 0 y cos 3 x, x 4 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 4 n cos i 4 n , therefore, we see that f ( x i ) = cos i 4 n , x = 4 n . But in this case x i = i 4 n , f ( x ) = cos x, [ a, b ] = bracketleftBig , 4 bracketrightBig . Consequently, the area is that of the region under the graph of y = cos x on the interval [0 , / 4]. In set-builder notation this is the region braceleftBig ( x, y ) : 0 y cos x, x 4 bracerightBig . 002 10.0 points Estimate the area under the graph of f ( x ) = sin x between x = 0 and x = 4 using five approx- imating rectangles of equal widths and right endpoints as sample points. 1. area . 288 2. area . 308 3. area . 268 4. area . 328 5. area . 348 correct Explanation: An estimate for the area, A , 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 ) = 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 . pokharel (yp624) HW03 Radin (57410) 2 Thus A braceleftBig sin( 1 20 ) + . . . + sin( 1 4 ) bracerightBig 20 . After calculating these values we obtain the estimate area . 348 for the area under the graph. 003 10.0 points Cyclist Joe brakes as he approaches a stop sign. His velocity graph over a 5 second period (in units of feet/sec) is shown in 1 2 3 4 5 4 8 12 16 20 Compute best possible upper and lower es- timates for the distance he travels over this period by dividing [0 , 5] into 5 equal subinter- vals and using endpoint sample points....
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This note was uploaded on 12/23/2009 for the course M 408L taught by Professor Radin during the Fall '08 term at University of Texas at Austin.
- Fall '08