use the four - step process to find r(x) and then find r(1), r(2), and r(3).\nr(x)=6 - 4x^{2}\nr(x)=\n(type…

Explanation:

Step1: Recall the four - step process for finding the derivative

The four - step process for finding the derivative of a function $y = r(x)$ is based on the definition of the derivative $r^{\prime}(x)=\lim_{h\rightarrow0}\frac{r(x + h)-r(x)}{h}$. First, find $r(x + h)$: 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 $h\rightarrow0$

$r^{\prime}(x)=\lim_{h\rightarrow0}\frac{r(x + h)-r(x)}{h}=\lim_{h\rightarrow0}(-8x - 4h)=-8x$. Now, find $r^{\prime}(1),r^{\prime}(2),r^{\prime}(3)$:

• For $x = 1$, $r^{\prime}(1)=-8\times1=-8$.
• For $x = 2$, $r^{\prime}(2)=-8\times2=-16$.
• For $x = 3$, $r^{\prime}(3)=-8\times3=-24$.

Answer:

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