Question

question
find the average rate of change of $f(x)=-x^{2}+\frac{9}{x}$ on the interval $1,6$.
provide your answer below:

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ on the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 1$, $b = 6$, and $f(x)=-x^{2}+\frac{9}{x}$.

Step2: Calculate $f(6)$

$f(6)=-6^{2}+\frac{9}{6}=-36+\frac{3}{2}=-\frac{72}{2}+\frac{3}{2}=-\frac{69}{2}$.

Step3: Calculate $f(1)$

$f(1)=-1^{2}+\frac{9}{1}=-1 + 9=8$.

Step4: Calculate the average rate of change

$\frac{f(6)-f(1)}{6 - 1}=\frac{-\frac{69}{2}-8}{5}=\frac{-\frac{69}{2}-\frac{16}{2}}{5}=\frac{-\frac{85}{2}}{5}=-\frac{85}{2}\times\frac{1}{5}=-\frac{17}{2}=- 8.5$. Rounding to the nearest whole - number gives - 8.

Answer:

-8