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Unformatted text preview: Version 010 – Homework 2 – Odell – (58340) 1 This printout should have 23 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. 001 (part 1 of 3) 10.0 points Below is the graph of a function f . 1 2 3 1 2 3 2 4 6 8 2 4 6 (i) Estimate the definite integral I = integraldisplay 3 − 3 f ( x ) dx with six equal subintervals using right end points. 1. I ≈ 10 correct 2. I ≈ 8 3. I ≈ 6 4. I ≈ 7 5. I ≈ 9 Explanation: Since [ 3 , 3] is subdivided into six equal subintervals, each of these will have length 1 and the six corresponding rectangles are shown as the shaded areas in 1 2 3 1 2 3 2 4 6 8 2 4 6 The heights of the rectangles are right end point sample values of f that can be read off from the graph. Thus, with right endpoints, I ≈ 2 4 1 + 2 + 4 + 7 = 10 . 002 (part 2 of 3) 10.0 points (ii) Estimate the definite integral I = integraldisplay 3 − 3 f ( x ) dx with six equal subintervals using left end points. 1. I ≈ 6 2. I ≈ 4 3. I ≈ 5 4. I ≈ 7 5. I ≈ 8 correct Explanation: Since [ 3 , 3] is subdivided into six equal subintervals, each of these will have length 1 and the six corresponding rectangles are shown as the shaded areas in Version 010 – Homework 2 – Odell – (58340) 2 1 2 3 1 2 3 2 4 6 8 2 4 6 The heights of the rectangles are left endpoint sample values of f that can be read off from the graph. Thus, with left endpoints, I ≈ 5 + 2 4 1 + 2 + 4 = 8 . 003 (part 3 of 3) 10.0 points (iii) Estimate the definite integral I = integraldisplay 3 − 3 f ( x ) dx with six equal subintervals using midpoints. 1. I ≈ 9 correct 2. I ≈ 8 3. I ≈ 11 4. I ≈ 12 5. I ≈ 10 Explanation: Since [ 3 , 3] is subdivided into six equal subintervals, each of these will have length 1 and the six corresponding rectangles are shown as the shaded areas in 1 2 3 1 2 3 2 4 6 8 2 4 6 The heights of the rectangles are midpoint sample values of f that can be read off from the graph. Thus, with midpoints, I ≈ 4 2 4 + 1 + 4 + 6 = 9 . 004 10.0 points Express the limit lim n →∞ n summationdisplay i =1 3 x i sin x i Δ x as a definite integral on the interval [2 , 7]. 1. limit = integraldisplay 2 7 3 sin x dx 2. limit = integraldisplay 7 2 3 x sin x dx correct 3. limit = integraldisplay 7 2 3 x dx 4. limit = integraldisplay 2 7 3 x dx 5. limit = integraldisplay 7 2 3 sin x dx 6. limit = integraldisplay 2 7 3 x sin x dx Explanation: By definition, the definite integral I = integraldisplay b a f ( x ) dx Version 010 – Homework 2 – Odell – (58340) 3 of a continuous function f on an interval [ a, b ] is the limit I = lim n →∞ n summationdisplay i = 1 f ( x ∗ i ) Δ x of the Riemann sum n summationdisplay i = 1 f ( x ∗ i ) Δ x formed when the interval [ a, b ] is divided into n subintervals of equal width Δ x and x ∗ i is any sample point in the i th subinterval [ x i − 1 , x i ]....
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 Spring '10
 ZHENG
 Calculus, Derivative, lim, dx

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