Question

follow the step by step process to determine the difference quotient, $\frac{f(x + h)-f(x)}{h}$. f(x)=6 - 8x^2 a. determine f(x + h). f(x + h)= (simplify your answer.) b. simplify the expression f(x + h)-f(x). f(x + h)-f(x)= (simplify your answer.) c. determine the difference quotient $\frac{f(x + h)-f(x)}{h}$. $\frac{f(x + h)-f(x)}{h}=$ (simplify your answer.)

Explanation:

Step1: Find f(x + h)

Substitute \(x+h\) into \(f(x)=6 - 8x^{2}\).
\[

$$\begin{align*} f(x + h)&=6-8(x + h)^{2}\\ &=6-8(x^{2}+2xh+h^{2})\\ &=6-8x^{2}-16xh - 8h^{2} \end{align*}$$

\]

Step2: Calculate f(x + h)-f(x)

\[

$$\begin{align*} f(x + h)-f(x)&=(6-8x^{2}-16xh - 8h^{2})-(6 - 8x^{2})\\ &=6-8x^{2}-16xh - 8h^{2}-6 + 8x^{2}\\ &=-16xh-8h^{2} \end{align*}$$

\]

Step3: Determine the difference quotient

\[

$$\begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{-16xh-8h^{2}}{h}\\ &=\frac{h(-16x - 8h)}{h}\\ &=-16x-8h \end{align*}$$

\]

Answer:

a. \(6-8x^{2}-16xh - 8h^{2}\)
b. \(-16xh-8h^{2}\)
c. \(-16x - 8h\)