Question

question
find the slope of the secant line between (x = - 2) and (x = 2) on the graph of the function (f(x)=5x^{3}-x^{2}-5x - 3).
provide your answer below:

Explanation:

Step1: Find \(f(-2)\)

\[

$$\begin{align*} f(-2)&=5(-2)^{3}-(-2)^{2}-5(-2)-3\\ &=5\times(-8)-4 + 10-3\\ &=-40-4 + 10-3\\ &=-37 \end{align*}$$

\]

Step2: Find \(f(2)\)

\[

$$\begin{align*} f(2)&=5(2)^{3}-(2)^{2}-5(2)-3\\ &=5\times8-4-10 - 3\\ &=40-4-10-3\\ &=23 \end{align*}$$

\]

Step3: Calculate the slope of the secant line

The slope \(m\) of the secant line between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Here \(x_1=-2,y_1 = f(-2)=-37,x_2 = 2,y_2=f(2)=23\).
\[

$$\begin{align*} m&=\frac{f(2)-f(-2)}{2-(-2)}\\ &=\frac{23-(-37)}{2 + 2}\\ &=\frac{23 + 37}{4}\\ &=\frac{60}{4}\\ &=17 \end{align*}$$

\]

Answer:

17