for the function f(x) = 5x² + 3x + 1, evaluat...
for the function f(x) = 5x² + 3x + 1, evaluate and fully simplify each of the following. f(x + h) = f(x + h) - f(x) / h =
Answer
# Explanation:
## Step1: Substitute $x + h$ into $f(x)$
\[
\begin{align*}
f(x + h)&=5(x + h)^2+3(x + h)+1\\
&=5(x^{2}+2xh+h^{2})+3x + 3h+1\\
&=5x^{2}+10xh+5h^{2}+3x + 3h+1
\end{align*}
\]
## Step2: Calculate $f(x + h)-f(x)$
\[
\begin{align*}
f(x + h)-f(x)&=(5x^{2}+10xh+5h^{2}+3x + 3h+1)-(5x^{2}+3x + 1)\\
&=5x^{2}+10xh+5h^{2}+3x + 3h+1 - 5x^{2}-3x - 1\\
&=10xh+5h^{2}+3h
\end{align*}
\]
## Step3: Calculate $\frac{f(x + h)-f(x)}{h}$
\[
\begin{align*}
\frac{f(x + h)-f(x)}{h}&=\frac{10xh+5h^{2}+3h}{h}\\
&=\frac{h(10x + 5h+3)}{h}\\
&=10x + 5h+3
\end{align*}
\]
# Answer:
$f(x + h)=5x^{2}+10xh+5h^{2}+3x + 3h+1$
$\frac{f(x + h)-f(x)}{h}=10x + 5h+3$