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) = 5/x = on [2,18]
The average value is
(Simplify your answer.)

Answer 1

`\sf (125)/(192)`

Explanation:

To estimate the average value of
`\sf f(x) = (5)/(x)` on the interval
`\sf [2,18]` by partitioning the interval into four subintervals of equal length and evaluating
`\sf f` at the subinterval midpoints, we can use the following steps:

Find the
`\triangle x`.

`\sf \triangle x = (b - a)/(n) = (18 - 2)/(4) = 4`

Create a list of subintervals.

`\sf \textsf{subintervals} = [[2, 6], [6, 10], [10, 14], [14, 18]]` t

Find the midpoints of each subinterval.

`\sf \text{midpoints} = \left[(2 + 6)/(2), (6 + 10)/(2), (10 + 14)/(2), (14 + 18)/(2)\right]\\\\ = [4, 8, 12, 16]`

Evaluate
`\sf f(x)` at each midpoint.

`\begin{aligned} f(4) & = (5)/(4) \\\\ f(8) = & = (5)/(8) \\\\\ f(12) =& = (5)/(12) \\\\ f(16) = & = (5)/(16) \\\\ \end{aligned}`

Compute the average of the four function values.

`\begin{aligned} \text{average value} & = ((5)/(4) (5)/(8) + (5)/(12) + (5)/(16))/(4) \\\\ & =(( 125)/(48))/(4) \\\\ & =(125)/(48)\cdot (1)/(4)\\\\ & = (125)/(192)\end{aligned}`

Therefore, the estimated average value of
`\sf f(x) = (5)/(x)` on the interval
`\sf [2,18]` is:

`\sf (125)/(192)`