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Homework 2

# Homework 2 - Version 070 Homework 02 Gompf(58370 This...

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Version 070 – Homework 02 – Gompf – (58370) 1 This print-out should have 23 questions. Multiple-choice 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 8 2. I 9 3. I 11 4. I 12 5. I 10 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 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 - 5 - 1 + 3 + 5 + 6 = 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 12 2. I 10 3. I 9 4. I 8 correct 5. I 11 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

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Version 070 – Homework 02 – Gompf – (58370) 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 4 + 2 - 5 - 1 + 3 + 5 = 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 2. I 11 3. I 8 4. I 10 correct 5. I 12 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 3 - 2 - 3 + 2 + 4 + 6 = 10 . 004 10.0 points Express the limit lim n → ∞ n summationdisplay i =1 4 x i sin x i Δ x as a definite integral on the interval [1 , 9]. 1. limit = integraldisplay 9 1 4 x dx 2. limit = integraldisplay 1 9 4 x sin x dx 3. limit = integraldisplay 1 9 4 sin x dx 4. limit = integraldisplay 9 1 4 x sin x dx correct 5. limit = integraldisplay 1 9 4 x dx 6. limit = integraldisplay 9 1 4 sin x dx Explanation: By definition, the definite integral I = integraldisplay b a f ( x ) dx
Version 070 – Homework 02 – Gompf – (58370) 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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Homework 2 - Version 070 Homework 02 Gompf(58370 This...

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