use the four - step process to find s(x) and then find s(1), s(2), and s(3). s(x)=8x - 4 s(x)= (simplify…

use the four - step process to find s(x) and then find s(1), s(2), and s(3). s(x)=8x - 4 s(x)= (simplify your answer. use integers or fractions for any numbers in the expression.) s(1)= (type an integer or a simplified fraction.) s(2)= (type an integer or a simplified fraction.) s(3)= (type an integer or a simplified fraction.)

use the four - step process to find s(x) and then find s(1), s(2), and s(3). s(x)=8x - 4 s(x)= (simplify your answer. use integers or fractions for any numbers in the expression.) s(1)= (type an integer or a simplified fraction.) s(2)= (type an integer or a simplified fraction.) s(3)= (type an integer or a simplified fraction.)

Answer

Explanation:

Step1: Recall the four - step process for derivative

The four - step process for finding the derivative of a function $y = s(x)$ is based on the limit definition $s^{\prime}(x)=\lim_{h\rightarrow0}\frac{s(x + h)-s(x)}{h}$. First, find $s(x + h)$: Given $s(x)=8x - 4$, then $s(x + h)=8(x + h)-4=8x+8h - 4$.

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

[ \begin{align*} s(x + h)-s(x)&=(8x + 8h-4)-(8x - 4)\ &=8x+8h - 4-8x + 4\ &=8h \end{align*} ]

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

[ \frac{s(x + h)-s(x)}{h}=\frac{8h}{h}=8 ]

Step4: Find the limit as $h\rightarrow0$

[ s^{\prime}(x)=\lim_{h\rightarrow0}\frac{s(x + h)-s(x)}{h}=\lim_{h\rightarrow0}8 = 8 ] Since $s^{\prime}(x)=8$ (a constant function), then:

  • For $x = 1$, $s^{\prime}(1)=8$.
  • For $x = 2$, $s^{\prime}(2)=8$.
  • For $x = 3$, $s^{\prime}(3)=8$.

Answer:

$s^{\prime}(x)=8$ $s^{\prime}(1)=8$ $s^{\prime}(2)=8$ $s^{\prime}(3)=8$