math 1050 written homework - section 1.4
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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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Question

Accepted Answer

# 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:
$-6x-3h + 1$

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QUESTION IMAGE

Question

Explanation:

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

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

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

Answer: