Question

use the four - step process to find r(x) and then find r(1), r(2), and r(3).
r(x)=3 + 3x^{2}
r(x)= square
r(1)= square (type an integer or a simplified fraction.)
r(2)= square (type an integer or a simplified fraction.)
r(3)= square (type an integer or a simplified fraction.)

Explanation:

Step1: Recall four - step process for derivative

The four - step process for finding the derivative of a function $y = r(x)$ is based on the limit definition of the derivative $r^{\prime}(x)=\lim_{h
ightarrow0}\frac{r(x + h)-r(x)}{h}$. First, find $r(x + h)$:
\[

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

\]

Step2: Calculate $r(x + h)-r(x)$

\[

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

\]

Step3: Calculate $\frac{r(x + h)-r(x)}{h}$

\[
\frac{r(x + h)-r(x)}{h}=\frac{6xh + 3h^{2}}{h}=6x+3h
\]

Step4: Find the limit as $h

ightarrow0$
\[
r^{\prime}(x)=\lim_{h
ightarrow0}\frac{r(x + h)-r(x)}{h}=\lim_{h
ightarrow0}(6x + 3h)=6x
\]
To find $r^{\prime}(1)$:
Substitute $x = 1$ into $r^{\prime}(x)$, so $r^{\prime}(1)=6\times1 = 6$.
To find $r^{\prime}(2)$:
Substitute $x = 2$ into $r^{\prime}(x)$, so $r^{\prime}(2)=6\times2=12$.
To find $r^{\prime}(3)$:
Substitute $x = 3$ into $r^{\prime}(x)$, so $r^{\prime}(3)=6\times3 = 18$.

Answer:

$r^{\prime}(x)=6x$
$r^{\prime}(1)=6$
$r^{\prime}(2)=12$
$r^{\prime}(3)=18$