Question

use the four - step process to find r(x) and then find r(1), r(2), and r(3). r(x)=6 + 4x^2 r(x)=□

Explanation:

Step1: Recall the four - step process formula

The four - step process for finding the derivative of a function $y = r(x)$ is $r^{\prime}(x)=\lim_{h
ightarrow0}\frac{r(x + h)-r(x)}{h}$. Given $r(x)=6 + 4x^{2}$, then $r(x + h)=6+4(x + h)^{2}=6+4(x^{2}+2xh+h^{2})=6 + 4x^{2}+8xh+4h^{2}$.

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

$r(x + h)-r(x)=(6 + 4x^{2}+8xh+4h^{2})-(6 + 4x^{2})=8xh+4h^{2}$.

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

$\frac{r(x + h)-r(x)}{h}=\frac{8xh + 4h^{2}}{h}=8x+4h$.

Step4: Find the limit as $h

ightarrow0$
$r^{\prime}(x)=\lim_{h
ightarrow0}(8x + 4h)$. As $h
ightarrow0$, $r^{\prime}(x)=8x$.

To find $r^{\prime}(1)$, substitute $x = 1$ into $r^{\prime}(x)$: $r^{\prime}(1)=8\times1 = 8$.
To find $r^{\prime}(2)$, substitute $x = 2$ into $r^{\prime}(x)$: $r^{\prime}(2)=8\times2=16$.
To find $r^{\prime}(3)$, substitute $x = 3$ into $r^{\prime}(x)$: $r^{\prime}(3)=8\times3 = 24$.

Answer:

$r^{\prime}(x)=8x$, $r^{\prime}(1)=8$, $r^{\prime}(2)=16$, $r^{\prime}(3)=24$