hw3 - padilla(tp5647 HW03 Gilbert(56650 This print-out...

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padilla (tp5647) – 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 π 2 n cos 2 n without evaluating the limit. 1. braceleftBig ( x, y ) : 0 y cos 2 x, 0 x π 4 bracerightBig 2. braceleftBig ( x, y ) : 0 y cos x, 0 x π 2 bracerightBig correct 3. braceleftBig ( x, y ) : 0 y cos 2 x, 0 x π 2 bracerightBig 4. braceleftBig ( x, y ) : 0 y cos 3 x, 0 x π 2 bracerightBig 5. braceleftBig ( x, y ) : 0 y cos x, 0 x π 4 bracerightBig 6. braceleftBig ( x, y ) : 0 y cos 3 x, 0 x π 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 ) Δ 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 Δ x = ( b - a ) /n . If A = lim n →∞ n summationdisplay i =1 π 2 n cos 2 n , therefore, we see that f ( x i ) = cos 2 n , Δ x = π 2 n . But in this case x i = 2 n , f ( x ) = cos x, [ a, b ] = bracketleftBig 0 , π 2 bracketrightBig . Consequently, the area is that of the region under the graph of y = cos x on the interval [0 , π/ 2]. In set-builder notation this is the region braceleftBig ( x, y ) : 0 y cos x, 0 x π 2 bracerightBig . 002 10.0 points Estimate the area under the graph of f ( x ) = sin x between x = 0 and x = π 2 using five approx- imating rectangles of equal widths and right endpoints. 1. area 1 . 149 correct 2. area 1 . 189 3. area 1 . 109 4. area 1 . 169 5. area 1 . 129 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 ) = sin x, b = π 2 , n = 5 , and x 1 = 1 10 π, x 2 = 1 5 π, x 3 = 3 10 π, x 4 = 2 5 π, x 5 = 1 2 π .
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padilla (tp5647) – HW03 – Gilbert – (56650) 2 Thus A braceleftBig sin parenleftBig 1 10 π parenrightBig + . . . + sin parenleftBig 1 2 π parenrightBigbracerightBig π 10 . After calculating these values we obtain the estimate area 1 . 149 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 + 3 n parenrightBig 2 + 5 n parenleftBig 6 + 6 n parenrightBig 2 + . . . + 5 n parenleftBig 6 + 3 n n parenrightBig 2 using sigma notation. 1. n summationdisplay i =1 5 i n parenleftBig 6 + 3 i n parenrightBig 2 2. n summationdisplay i =1 5 n parenleftBig 6 i + 3 i n parenrightBig 2 3. n summationdisplay i =1 3 i n parenleftBig 6 + 5 i n parenrightBig 2 4. n summationdisplay i =1 5 n parenleftBig 6 + 3 i n parenrightBig 2 correct 5.
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