Question

hw9 the derivative as a function (targets l6, d1, d2; §3.2) score: 4/9 answered: 4/9 question 5 for the function f(x)=3x² + 3, evaluate the following, enter your answers in simplest form: f(x + h)= f(x + h)-f(x)= (f(x + h)-f(x))/h= question help: message instructor

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$.
\[

$$\begin{align*} f(x + h)&=3(x + h)^2+3\\ &=3(x^{2}+2xh+h^{2})+3\\ &=3x^{2}+6xh + 3h^{2}+3 \end{align*}$$

\]

Step2: Find $f(x + h)-f(x)$

\[

$$\begin{align*} f(x + h)-f(x)&=(3x^{2}+6xh + 3h^{2}+3)-(3x^{2}+3)\\ &=3x^{2}+6xh + 3h^{2}+3 - 3x^{2}-3\\ &=6xh+3h^{2} \end{align*}$$

\]

Step3: Find $\frac{f(x + h)-f(x)}{h}$

\[

$$\begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{6xh + 3h^{2}}{h}\\ &=\frac{h(6x + 3h)}{h}\\ &=6x+3h \end{align*}$$

\]

Answer:

$f(x + h)=3x^{2}+6xh + 3h^{2}+3$
$f(x + h)-f(x)=6xh+3h^{2}$
$\frac{f(x + h)-f(x)}{h}=6x+3h$