Questions
MATH_1225_64262_202506 5.1 Areas and Distances
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Suppose is increasing on the interval and we want to estimate the area under the curve on this interval. If is concave down on this interval, using left endpoints would give an underestimation. . If is concave down on this interval, using right endpoints would give an overestimation . If is concave up on this interval, using left endpoints would give an underestimation . If is concave up on this interval, using right endpoints would give an overestimation . If is not known to be increasing or decreasing on , which of the above statements must be true? None of them.
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Rewriting the problem in my own words helps set the stage: we’re estimating the area under f on [a,b] using Riemann sums, and we have four qualitative statements about under- or over-estimation depending on concavity and endpoint choice. The final question asks, if we do not know whether f is increasing or decreasing, which statements must be true?
Option 1: 'an underestimation.' (Left endpoints when f is concave down on the interval and increasing) In this scenario, the left-endpoint rectangle height is f(x_{i-1}). Since f is increasing, f(x_{i-1}) ≤ f(x) for x in [x_{i-1}, x_i], so the left-rectangle area is less than or equal to the actual area over that subinterval. Howeve......Login to view full explanationLog in for full answers
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