Question

use the four - step process to find r(x) and then find r(1), r(2), and r(3). r(x)=7 + 3x^2 r(x)= r(1)= (type an integer or a simplified fraction.) r(2)= (type an integer or a simplified fraction.) r(3)= (type an integer or a simplified fraction.)

Explanation:

Step1: Recall the 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 $r^{\prime}(x)=\lim_{h
ightarrow0}\frac{r(x + h)-r(x)}{h}$. First, find $r(x + h)$:
Given $r(x)=7 + 3x^{2}$, then $r(x + h)=7+3(x + h)^{2}=7+3(x^{2}+2xh+h^{2})=7 + 3x^{2}+6xh+3h^{2}$.

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

$r(x + h)-r(x)=(7 + 3x^{2}+6xh+3h^{2})-(7 + 3x^{2})=6xh+3h^{2}$.

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$
$r^{\prime}(x)=\lim_{h
ightarrow0}\frac{r(x + h)-r(x)}{h}=\lim_{h
ightarrow0}(6x + 3h)=6x$.

Now, find $r^{\prime}(1),r^{\prime}(2),r^{\prime}(3)$:

• When $x = 1$, $r^{\prime}(1)=6\times1 = 6$.
• When $x = 2$, $r^{\prime}(2)=6\times2=12$.
• When $x = 3$, $r^{\prime}(3)=6\times3 = 18$.

Answer:

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