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18 February, 11:22

Function f (x) is positive, increasing and concave up on the closed interval [a, b]. The interval [a, b] is partitioned into 4 equal intervals and these are used to compute the upper sum, lower sum, and trapezoidal rule approximations for the value of Integral b a f (x) dx. Which one of the following statements is true?

Lower sum < Trapezoidal rule Value < Upper sum

Lower sum < Upper sum < Trapezoidal rule value

Trapezoidal rule < Lower sum < Upper sum

Cannot be determined without the x-values for the partitions

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Answers (2)
  1. 18 February, 11:45
    0
    The left sum would be f0+f1+f2+f3

    The right sum would be f1+f2+f3+f4

    The trapezoidal rule value is:

    (f0+f1) / 2 + (f1+f2) / 2 + (f2+f3) / 2 + (f3+f4) / 2

    This would put the trapezoidal rule in the middle, which makes the answer:

    Lower sum < Trapezoidal rule Value < Upper sum
  2. 18 February, 13:48
    0
    Step-by-step explanation:

    Given function f (x) is positive, increasing and concave up on the closed interval [a, b],

    it means f (x1) < f (x2) if x1 < x2

    So Lower sum < Upper sum

    As Trapezoidal is average of f (x1) and f (x2) = [f (x1) + f (x2) ] / 2

    it is average of the Lower and Upper sum.

    The answer is Lower sum < Trapezoidal rule Value < Upper sum
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