Question

math 1050 written homework - section 1.4
name:

1. compute and simplify the difference quotient, $\frac{f(x + h)-f(x)}{h}$, for the function $f(x)=-3x^{2}+x - 8$.

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Explanation:

Step1: Find \(f(x + h)\)

Substitute \(x+h\) into \(f(x)=- 3x^{2}+x - 8\).
\[

$$\begin{align*} f(x + h)&=-3(x + h)^{2}+(x + h)-8\\ &=-3(x^{2}+2xh+h^{2})+x + h-8\\ &=-3x^{2}-6xh-3h^{2}+x + h-8 \end{align*}$$

\]

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

\[

$$\begin{align*} f(x + h)-f(x)&=(-3x^{2}-6xh-3h^{2}+x + h-8)-(-3x^{2}+x - 8)\\ &=-3x^{2}-6xh-3h^{2}+x + h-8 + 3x^{2}-x + 8\\ &=-6xh-3h^{2}+h \end{align*}$$

\]

Step3: Compute \(\frac{f(x + h)-f(x)}{h}\)

\[

$$\begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{-6xh-3h^{2}+h}{h}\\ &=\frac{h(-6x-3h + 1)}{h}\\ &=-6x-3h + 1 \end{align*}$$

\]

Answer:

\(-6x-3h + 1\)