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…

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)