given f(x)=x² + 7x, (a) find f(x + h) and sim...
given f(x)=x² + 7x, (a) find f(x + h) and simplify. (b) find (f(x + h)-f(x))/h and simplify. part: 0 / 2 part 1 of 2 (a) f(x + h) =
Answer
# Explanation:
## Step1: Sustituir \(x\) por \(x + h\) en \(f(x)\)
\[
\begin{align*}
f(x+h)&=(x + h)^2+7(x + h)\\
&=x^{2}+2xh+h^{2}+7x + 7h
\end{align*}
\]
## Step2: Encontrar \(f(x + h)-f(x)\)
\[
\begin{align*}
f(x + h)-f(x)&=(x^{2}+2xh+h^{2}+7x + 7h)-(x^{2}+7x)\\
&=x^{2}+2xh+h^{2}+7x + 7h - x^{2}-7x\\
&=2xh+h^{2}+7h
\end{align*}
\]
## Step3: Calcular \(\frac{f(x + h)-f(x)}{h}\)
\[
\begin{align*}
\frac{f(x + h)-f(x)}{h}&=\frac{2xh+h^{2}+7h}{h}\\
&=\frac{h(2x + h+7)}{h}\\
&=2x+h + 7
\end{align*}
\]
# Answer:
(a) \(f(x + h)=x^{2}+2xh+h^{2}+7x + 7h\)
(b) \(\frac{f(x + h)-f(x)}{h}=2x+h + 7\)