The table gives the values of a function obtained from an experiment. Use them to estimate 9f(x) dx3 using three equal subintervals with right endpoints, left endpoints, and midpoints.x 3 4 5 6 7 8 9f(x) −3.3 −2.2 −0.6 0.2 0.8 1.3 1.9(a) Estimate 9f(x) dx3 using three equal subintervals with right endpoints.R3 = If the function is known to be an increasing function, can you say whether your estimate is less than or greater than the exact value of the integral?less thangreater than one cannot say(b) Estimate 9f(x) dx3 using three equal subintervals with left endpoints.L3 = If the function is known to be an increasing function, can you say whether your estimate is less than or greater than the exact value of the integral?less thangreater than one cannot say(c) Estimate 9f(x) dx3 using three equal subintervals with midpoints.M3 = If the function is known to be an increasing function, can you say whether your estimate is less than or greater than the exact value of the integral?less thangreater than one cannot say
Question
The table gives the values of a function obtained from an experiment. Use them to estimate 9f(x) dx3 using three equal subintervals with right endpoints, left endpoints, and midpoints.x 3 4 5 6 7 8 9f(x) −3.3 −2.2 −0.6 0.2 0.8 1.3 1.9(a) Estimate 9f(x) dx3 using three equal subintervals with right endpoints.R3 = If the function is known to be an increasing function, can you say whether your estimate is less than or greater than the exact value of the integral?less thangreater than one cannot say(b) Estimate 9f(x) dx3 using three equal subintervals with left endpoints.L3 = If the function is known to be an increasing function, can you say whether your estimate is less than or greater than the exact value of the integral?less thangreater than one cannot say(c) Estimate 9f(x) dx3 using three equal subintervals with midpoints.M3 = If the function is known to be an increasing function, can you say whether your estimate is less than or greater than the exact value of the integral?less thangreater than one cannot say
Solution
The given table represents the function f(x) at different x values. We are asked to estimate the integral of f(x) from 3 to 9 using three equal subintervals. The subintervals are [3,5], [5,7], and [7,9].
(a) Using right endpoints, we have: R3 = (2)(f(5) + f(7) + f(9)) = 2(-0.6 + 0.8 + 1.9) = 2.2 If the function is increasing, the right endpoint estimate will be greater than the exact value of the integral.
(b) Using left endpoints, we have: L3 = (2)(f(3) + f(5) + f(7)) = 2(-3.3 - 0.6 + 0.8) = -4.2 If the function is increasing, the left endpoint estimate will be less than the exact value of the integral.
(c) Using midpoints, we need to find the values of f at the midpoints of the subintervals. However, we don't have these values in the table. If we had these values, we would calculate the midpoint estimate as: M3 = (2)(f(midpoint of [3,5]) + f(midpoint of [5,7]) + f(midpoint of [7,9])) If the function is increasing, the midpoint estimate will be less than the right endpoint estimate and greater than the left endpoint estimate.
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