Question

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

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:

\( -5 - h \)