Answer
$\frac{25}{96}\approx 0.2604$
Work Step by Step
Given-$ƒ(x) = \frac{1}{x}$ on the interval $[1, 9]$
Now, using a finite sum, we can partition the interval into four subintervals of equal length and evaluate ƒ at the subinterval midpoints.
The length of each subinterval can be calculated as follows:
Length of subinterval = $\frac{\text{upper bound} - \text{lower bound}}{\text{number of subintervals}}$
= $\frac{(9 - 1)}{4}$
= $2$
Now, we can calculate the subinterval midpoints:
Subinterval $1$: midpoint = $\frac{(1 + 3)}{2}= 2$
Subinterval $2$: midpoint = $\frac{(3 + 5) }{2}= 4$
Subinterval $3$: midpoint = $\frac{(5 + 7) }{2}= 6$
Subinterval $4$: midpoint = $\frac{(7 + 9) }{2}= 8$
Next, we evaluate ƒ at these subinterval midpoints:
$ƒ(2) = \frac{1}{2}$
$ƒ(4) = \frac{1}{4}$
$ƒ(6) = \frac{1}{6}$
$ƒ(8) = \frac{1}{8}$
Finally, we can calculate the average value of ƒ on the interval [1, 9] using the finite sum:
Average value of ƒ = $\frac{\text{sum of function values}}{\text{number of subintervals}}$
= $\frac{\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}}{4}$
≈ $\frac{25}{96}$
≈ $0.2604$
