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HW2 Solution - Kim Jin Homework 2 Due 3:00 am Inst Diane...

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Kim, Jin – Homework 2 – Due: Sep 11 2007, 3:00 am – Inst: Diane Radin 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Rewrite the sum 3 n 2 + 4 n · 2 + 3 n 2 + 8 n · 2 + . . . + 3 n 2 + 4 n n · 2 using sigma notation. 1. n X i = 1 4 n 2 + 3 i n · 2 2. n X i = 1 3 n 2 + 4 i n · 2 correct 3. n X i = 1 3 n 2 i + 4 i n · 2 4. n X i = 1 4 i n 2 + 3 i n · 2 5. n X i = 1 3 i n 2 + 4 i n · 2 6. n X i = 1 4 n 2 i + 3 i n · 2 Explanation: The terms are of the form 3 n 2 + 4 i n · 2 , with i = 1 , 2 , . . . , n . Consequently in sigma notation the sum becomes n X i = 1 3 n 2 + 4 i n · 2 . keywords: Stewart5e, summation notation, Riemann sum form 002 (part 1 of 1) 10 points Estimate the area, A , under the graph of f ( x ) = 1 x on [1 , 5] by dividing [1 , 5] into four equal subintervals and using right endpoints. Correct answer: 1 . 283 . Explanation: With four equal subintervals and right end- points as sample points, A n f (2) + f (3) + f (4) + f (5) o 1 since x i = x * i = i + 1. Consequently, A 1 . 283 . keywords: Stewart5e, area, rational function, Riemann sum, 003 (part 1 of 1) 10 points The graph of a function f on the interval [0 , 10] is shown in -1 0 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 2 4 6 8 Estimate the area under the graph of f by dividing [0 , 10] into 10 equal subintervals and using right endpoints as sample points. 1. area 54
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Kim, Jin – Homework 2 – Due: Sep 11 2007, 3:00 am – Inst: Diane Radin 2 2. area 51 3. area 55 correct 4. area 52 5. area 53 Explanation: With 10 equal subintervals and right end- points as sample points, area n f (1) + f (2) + . . . f (10) o 1 , since x i = i . Consequently, area 55 , reading off the values of f (1) , f (2) , . . . , f (10) from the graph of f . keywords: Stewart5e, graph, estimate area, Riemannn sum 004 (part 1 of 1) 10 points Estimate the area under the graph of f ( x ) = 3 sin x between x = 0 and x = π 3 using five approx- imating rectangles of equal widths and right endpoints as sample points. 1. area 1 . 807 2. area 1 . 767 correct 3. area 1 . 847 4. area 1 . 787 5. area 1 . 827 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 ] = h ( i - 1) b n , ib n i and right endpoints x i as sample points is A n f ( x 1 ) + f ( x 2 ) + . . . + f ( x n ) o b n . For the given area, f ( x ) = 3 sin x, b = π 3 , n = 5 , and x 1 = 1 15 π, x 2 = 2 15 π, x 3 = 1 5 π, x 4 = 4 15 π, x 5 = 1 3 π . Thus A 3 n sin( 1 15 π ) + . . . + sin( 1 3 π ) o π 15 . After calculating these values we obtain the estimate area 1 . 767 for the area under the graph. keywords: estimate area, graph, Riemann sum 005 (part 1 of 1) 10 points Cyclist Joe accelerates as he rides away from 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
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Kim, Jin – Homework 2 – Due: Sep 11 2007, 3:00 am – Inst: Diane Radin 3 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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