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

Question

Mattie Austin

Mathematics

17 June, 07:38

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

+3

Answers (2)

Know the Answer?

Dangelo · Answer 1

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

Collin Tucker · Answer 2

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