Question

given that f(x)=5x^2 + 8x - 13, determine an expression in terms of x and h that represents the average rate of change of f over any interval of length h. that is, over any interval (x,x + h). in other words, find the difference quotient for f(x). simplify your answer as much as possible.

Explanation:

Step1: Recall the difference - quotient formula

The difference quotient for a function $y = f(x)$ over the interval $(x,x + h)$ is $\frac{f(x + h)-f(x)}{h}$.

Step2: Find $f(x + h)$

Given $f(x)=5x^{2}+8x - 13$, then $f(x + h)=5(x + h)^{2}+8(x + h)-13$.
Expand $(x + h)^{2}=x^{2}+2xh+h^{2}$. So $f(x + h)=5(x^{2}+2xh + h^{2})+8(x + h)-13=5x^{2}+10xh+5h^{2}+8x + 8h-13$.

Step3: Calculate $f(x + h)-f(x)$

\[

$$\begin{align*} f(x + h)-f(x)&=(5x^{2}+10xh+5h^{2}+8x + 8h-13)-(5x^{2}+8x - 13)\\ &=5x^{2}+10xh+5h^{2}+8x + 8h-13 - 5x^{2}-8x + 13\\ &=10xh+5h^{2}+8h \end{align*}$$

\]

Step4: Calculate the difference - quotient

\[

$$\begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{10xh+5h^{2}+8h}{h}\\ &=\frac{h(10x + 5h+8)}{h}\\ &=10x+5h + 8 \end{align*}$$

\]

Answer:

$10x+5h + 8$