Question

Use a finite sum to estimate the average value of f on the given interval by partitioning the interval into four subintervals of equal length and evaluating f at the subinterval midpoints. f(x) 2/x on [1,17] The average value is (Simplify your answer.)​

Answer 1

`(122)/(385)`

Explanation:

To estimate the average value of f(x) = 2/x on the interval [1, 17] using a finite sum, begin by partitioning the interval into four subintervals of equal length.

The length of each subinterval Δx is given by:

`\Delta x = (17 - 1)/(4) = (16)/(4) = 4`

Therefore, the subintervals consist of:

`[1, 5], [5, 9], [9, 13], [13, 17]`

The midpoints of these subintervals are:

`\{3, 7, 11, 15\}`

Evaluate f(x) at each midpoint:

`f(3)=(2)/(3)`

`f(7)=(2)/(7)`

`f(11)=(2)/(11)`

`f(15)=(2)/(15)`

Sum these values and divide by the number of subintervals to find the average value:

`\begin{aligned}\text{Average Value} &= (1)/(4) \left[ f(3) + f(7) + f(11) + f(15) \right]\\\\&= (1)/(4) \left[(2)/(3)+(2)/(7)+(2)/(11)+(2)/(15) \right]\\\\&=(1)/(4)\left[(770)/(1155)+(330)/(1155)+(210)/(1155)+(154)/(1155)\right]\\\\&=(1)/(4)\left[(770+330+210+154)/(1155)\right]\\\\&=(1)/(4)\left[(1464)/(1155)\right]\\\\&=(1464)/(4620)\\\\&=(122)/(385)\end{aligned}`

Therefore, the average value is 122/385, which is approximately 0.317 (rounded to three significant figures).