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Unformatted text preview: hyun (hh7953) HW03 gogolev (57440) 1 This printout should have 20 questions. Multiplechoice 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 tan i 4 n without evaluating the limit. 1. braceleftBig ( x, y ) : 0 y tan x, x 4 bracerightBig correct 2. braceleftBig ( x, y ) : 0 y tan 3 x, x 8 bracerightBig 3. braceleftBig ( x, y ) : 0 y tan 2 x, x 8 bracerightBig 4. braceleftBig ( x, y ) : 0 y tan x, x 8 bracerightBig 5. braceleftBig ( x, y ) : 0 y tan 2 x, x 4 bracerightBig 6. braceleftBig ( x, y ) : 0 y tan 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 tan i 4 n , therefore, we see that f ( x i ) = tan i 4 n , x = 4 n . But in this case x i = i 4 n , f ( x ) = tan x, [ a, b ] = bracketleftBig , 4 bracketrightBig . Consequently, the area is that of the region under the graph of y = tan x on the interval [0 , / 4]. In setbuilder notation this is the region braceleftBig ( x, y ) : 0 y tan x, x 4 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 as sample points. 1. area 1 . 023 2. area 1 . 063 3. area 1 . 083 4. area 1 . 043 correct 5. area 1 . 103 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 ) = 3 sin x, b = 4 , n = 5 , hyun (hh7953) HW03 gogolev (57440) 2 and x 1 = 1 20 , x 2 = 1 10 , x 3 = 3 20 , x 4 = 1 5 , x 5 = 1 4 . Thus A 3 braceleftBig sin( 1 20 ) + . . . + sin( 1 4 ) bracerightBig 20 . After calculating these values we obtain the estimate area 1 . 043 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 01/19/2010 for the course M 57440 taught by Professor Gogolev during the Fall '09 term at University of Texas at Austin.
 Fall '09
 GOGOLEV
 Calculus

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