Question

find the average rate of change of g(x) = x³ - 16x + 20 over the interval -6, -1. write your answer as an integer, fraction, or decimal rounded to the nearest tenth. simplify any fractions.

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( g(x) \) over the interval \([a, b]\) is given by \(\frac{g(b) - g(a)}{b - a}\). Here, \( a=-6 \) and \( b = - 1 \).

Step2: Calculate \( g(-6) \)

Substitute \( x=-6 \) into \( g(x)=x^{3}-16x + 20 \):
\[

$$\begin{align*} g(-6)&=(-6)^{3}-16\times(-6)+20\\ &=-216 + 96+20\\ &=-216+116\\ &=-100 \end{align*}$$

\]

Step3: Calculate \( g(-1) \)

Substitute \( x = - 1 \) into \( g(x)=x^{3}-16x + 20 \):
\[

$$\begin{align*} g(-1)&=(-1)^{3}-16\times(-1)+20\\ &=-1 + 16+20\\ &=35 \end{align*}$$

\]

Step4: Calculate the average rate of change

Using the formula \(\frac{g(b)-g(a)}{b - a}\) with \( a=-6 \), \( b=-1 \), \( g(-6)=-100 \) and \( g(-1) = 35 \):
\[

$$\begin{align*} \frac{g(-1)-g(-6)}{-1-(-6)}&=\frac{35-(-100)}{-1 + 6}\\ &=\frac{35 + 100}{5}\\ &=\frac{135}{5}\\ &=27 \end{align*}$$

\]

Answer:

\( 27 \)