evaluate the difference quotient for the given function. simplify your answer.$ f(x) = 5 + 3x - x^2, \\ \frac{f(4 + h) - f(4)}{h} $

Question

Accepted Answer

# Explanation:
## Step1: Find $ f(4 + h) $
Substitute $ x = 4 + h $ into $ f(x) = 5 + 3x - x^2 $:
$$
\begin{align*}
f(4 + h)&= 5 + 3(4 + h) - (4 + h)^2\
&= 5 + 12 + 3h - (16 + 8h + h^2)\
&= 17 + 3h - 16 - 8h - h^2\
&= 1 - 5h - h^2
\end{align*}
$$
## Step2: Find $ f(4) $
Substitute $ x = 4 $ into $ f(x) = 5 + 3x - x^2 $:
$$
\begin{align*}
f(4)&= 5 + 3(4) - 4^2\
&= 5 + 12 - 16\
&= 1
\end{align*}
$$
## Step3: Compute $ f(4 + h) - f(4) $
Subtract $ f(4) $ from $ f(4 + h) $:
$$
\begin{align*}
f(4 + h) - f(4)&= (1 - 5h - h^2) - 1\
&= -5h - h^2
\end{align*}
$$
## Step4: Divide by $ h $ ( $ h
eq 0 $)
Divide the result by $ h $:
$$
\frac{f(4 + h) - f(4)}{h} = \frac{-5h - h^2}{h} = -5 - h
$$
# Answer:
$ -5 - h $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Find \( f(4 + h) \)

Step2: Find \( f(4) \)

Step3: Compute \( f(4 + h) - f(4) \)

Step4: Divide by \( h \) ( \( h

Answer: