Question

find the average rate of change of ( g(x) = -2x^3 + 5x^2 ) from ( x = 1 ) to ( x = 3 ). simplify your answer as much as possible.

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( g(x) \) from \( x = a \) to \( x = b \) is given by \( \frac{g(b)-g(a)}{b - a} \). Here, \( a = 1 \), \( b = 3 \), and \( g(x)=-2x^{3}+5x^{2} \).

Step2: Calculate \( g(1) \)

Substitute \( x = 1 \) into \( g(x) \):
\( g(1)=-2(1)^{3}+5(1)^{2}=-2 + 5=3 \)

Step3: Calculate \( g(3) \)

Substitute \( x = 3 \) into \( g(x) \):
\( g(3)=-2(3)^{3}+5(3)^{2}=-2\times27 + 5\times9=-54 + 45=-9 \)

Step4: Calculate the average rate of change

Using the formula \( \frac{g(3)-g(1)}{3 - 1} \), substitute the values of \( g(3) \), \( g(1) \):
\( \frac{-9 - 3}{3 - 1}=\frac{-12}{2}=-6 \)

Answer:

\(-6\)