given the function defined in the table below, find the average of the function over the interval 25 ≤ x ≤…

Explanation:

Step1: Recall average - value formula

The average value of a function $y = f(x)$ over the interval $[a,b]$ is given by $\bar{y}=\frac{1}{b - a}\int_{a}^{b}f(x)dx$. For a discrete - valued function, we use the formula $\bar{f}=\frac{\sum_{i = 1}^{n}f(x_{i})\Delta x_{i}}{\sum_{i = 1}^{n}\Delta x_{i}}$. In the case of equally - spaced $x$ values, if $x_{i+1}-x_{i}=h$ (constant), and we want to find the average value over $[a,b]$, we can also use $\bar{f}=\frac{\sum_{x\in[a,b]}f(x)}{n}$, where $n$ is the number of data points in the interval $[a,b]$. Here, $a = 25$, $b = 85$, and the $x$ values are equally spaced with a step of $15$. The data points in the interval $25\leq x\leq85$ are $x = 25,40,55,70,85$.

Step2: Calculate the sum of $f(x)$ values

The corresponding $f(x)$ values for $x = 25,40,55,70,85$ are $f(25)=9$, $f(40)=12$, $f(55)=15$, $f(70)=18$, $f(85)=21$. The sum $S=\sum_{i = 1}^{5}f(x_{i})=9 + 12+15+18+21$. $S=75$.

Step3: Calculate the number of data points

The number of data points $n = 5$.

Step4: Calculate the average value

The average value of the function over the interval $25\leq x\leq85$ is $\bar{f}=\frac{S}{n}$. $\bar{f}=\frac{75}{5}=15$.

Answer:

$15$