Question

find the average rate of change of ( g(x)=-3 x^{3}-2 ) from ( x=-4 ) to ( x = 3 )

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = g(x)\) from \(x=a\) to \(x = b\) is \(\frac{g(b)-g(a)}{b - a}\). Here, \(a=-4\), \(b = 3\), and \(g(x)=-3x^{3}-2\).

Step2: Calculate \(g(a)\) and \(g(b)\)

• Calculate \(g(-4)\):

Substitute \(x=-4\) into \(g(x)\):
\(g(-4)=-3(-4)^{3}-2=-3(-64)-2 = 192-2=190\).

• Calculate \(g(3)\):

Substitute \(x = 3\) into \(g(x)\):
\(g(3)=-3(3)^{3}-2=-3\times27-2=-81 - 2=-83\).

Step3: Substitute into the average - rate - of - change formula

\(\frac{g(3)-g(-4)}{3-(-4)}=\frac{-83 - 190}{3 + 4}=\frac{-273}{7}=-39\).

Answer:

\(-39\)