Question

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.)

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

ightarrow0$
\[
s^{\prime}(x)=\lim_{h
ightarrow0}\frac{s(x + h)-s(x)}{h}=\lim_{h
ightarrow0}8 = 8
\]
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:

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