homework 2 solutions - Mcbiles, Emily – Homework 2 –...

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Unformatted text preview: Mcbiles, Emily – Homework 2 – Due: Sep 12 2006, 3:00 am – Inst: David Fonken 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 n 4+ ‡ 1 9 · 2 o + n 8+ ‡ 2 9 · 2 o + ... + n 28+ ‡ 7 9 · 2 o using sigma notation. 1. 9 X i = 1 n 4 i + ‡ i 9 · 2 o 2. 7 X i = 1 4 n i + ‡ i 9 · 2 o 3. 9 X i = 1 4 n i + ‡ i 9 · 2 o 4. 7 X i = 1 n i + ‡ 4 i 9 · 2 o 5. 9 X i = 1 4 n i + ‡ 4 i 9 · 2 o 6. 7 X i = 1 n 4 i + ‡ i 9 · 2 o correct Explanation: The terms are of the form n 4 i + ‡ i 9 · 2 o , with i = 1 , 2 , ... , 7. Consequently, in sigma notation the sum becomes 7 X i = 1 n 4 i + ‡ i 9 · 2 o . keywords: Stewart5e, summation notation, 002 (part 1 of 1) 10 points Estimate the area, A , under the graph of f ( x ) = 4 x on [1 , 5] by dividing [1 , 5] into four equal subintervals and using right endpoints. Correct answer: 5 . 133 . 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 ≈ 5 . 133 . keywords: Stewart5e, area, rational function, Riemann sum, 003 (part 1 of 1) 10 points Decide which of the following regions has area = lim n → ∞ n X i = 1 π 5 n sin iπ 5 n without evaluating the limit. 1. n ( x, y ) : 0 ≤ y ≤ sin 3 x, ≤ x ≤ π 5 o 2. n ( x, y ) : 0 ≤ y ≤ sin x, ≤ x ≤ π 10 o 3. n ( x, y ) : 0 ≤ y ≤ sin x, ≤ x ≤ π 5 o correct 4. n ( x, y ) : 0 ≤ y ≤ sin 3 x, ≤ x ≤ π 10 o 5. n ( x, y ) : 0 ≤ y ≤ sin 2 x, ≤ x ≤ π 10 o 6. n ( x, y ) : 0 ≤ y ≤ sin 2 x, ≤ x ≤ π 5 o Explanation: Mcbiles, Emily – Homework 2 – Due: Sep 12 2006, 3:00 am – Inst: David Fonken 2 The area under the graph of y = f ( x ) on an interval [ a, b ] is given by the limit lim n → ∞ n X 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 , x n ] each of length Δ x = ( b- a ) /n . When the area is given by A = lim n → ∞ n X i = 1 π 5 n sin iπ 5 n , therefore, we see that f ( x i ) = sin iπ 5 n , Δ x = π 5 n , where in this case x i = iπ 5 n , f ( x ) = sin x, [ a, b ] = h , π 5 i . Consequently, the area is that of the region under the graph of y = sin x on the interval [0 , π/ 5]. In set-builder notation this is the region n ( x, y ) : 0 ≤ y ≤ sin x, ≤ x ≤ π 5 o . keywords: limit Riemann sum, area, trig func- tion 004 (part 1 of 1) 10 points Find an expression for the area of the region under the graph of f ( x ) = x 3 on the interval [3 , 10]....
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This note was uploaded on 03/19/2008 for the course M 408L taught by Professor Radin during the Fall '08 term at University of Texas at Austin.

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homework 2 solutions - Mcbiles, Emily – Homework 2 –...

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