Question

given that f(x) = 5/x, write an expression in terms of x and h that represents the average rate of change of f over any interval of length h. simplify your answer

Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function $y = f(x)$ over an interval $[x, x + h]$ is $\frac{f(x+h)-f(x)}{h}$.

Step2: Find $f(x + h)$

Given $f(x)=\frac{5}{x}$, then $f(x + h)=\frac{5}{x + h}$.

Step3: Substitute into formula

$\frac{f(x+h)-f(x)}{h}=\frac{\frac{5}{x + h}-\frac{5}{x}}{h}$.

Step4: Get a common - denominator for the numerator

$\frac{\frac{5x-5(x + h)}{x(x + h)}}{h}=\frac{5x-5x-5h}{hx(x + h)}$.

Step5: Simplify the numerator

$\frac{- 5h}{hx(x + h)}$.

Step6: Cancel out the common factor $h$

$\frac{-5}{x(x + h)}$.

Answer:

$\frac{-5}{x(x + h)}$