use the four - step process to find r(x) and then find r(1), r(2), and r(3).\nr(x)=3 + 3x^{2}\nr(x)=…

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 $r^{\prime}(x)=\lim_{h\rightarrow0}\frac{r(x + h)-r(x)}{h}$. 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 $h\rightarrow0$

[ r^{\prime}(x)=\lim_{h\rightarrow0}\frac{r(x + h)-r(x)}{h}=\lim_{h\rightarrow0}(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:

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