To find the average rate of change of the function \( h(x) = -2x^3 + 5x^2 \) from \( x = -3 \) to \( x = 2 \), we need to follow these steps:
1. Calculate \( h(x) \) at \( x = -3 \):
[tex]\[
h(-3) = -2(-3)^3 + 5(-3)^2
\][/tex]
Let's break it down:
[tex]\[
(-3)^3 = -27
\][/tex]
[tex]\[
-2 \times -27 = 54
\][/tex]
[tex]\[
(-3)^2 = 9
\][/tex]
[tex]\[
5 \times 9 = 45
\][/tex]
Then,
[tex]\[
h(-3) = 54 + 45 = 99
\][/tex]
2. Calculate \( h(x) \) at \( x = 2 \):
[tex]\[
h(2) = -2(2)^3 + 5(2)^2
\][/tex]
Let's break it down:
[tex]\[
(2)^3 = 8
\][/tex]
[tex]\[
-2 \times 8 = -16
\][/tex]
[tex]\[
(2)^2 = 4
\][/tex]
[tex]\[
5 \times 4 = 20
\][/tex]
Then,
[tex]\[
h(2) = -16 + 20 = 4
\][/tex]
3. Calculate the average rate of change:
The average rate of change of a function \( h(x) \) over the interval \([x_1, x_2]\) is given by:
[tex]\[
\frac{h(x_2) - h(x_1)}{x_2 - x_1}
\][/tex]
Here, \( x_1 = -3 \) and \( x_2 = 2 \).
Substituting the values we calculated:
[tex]\[
\text{Average rate of change} = \frac{h(2) - h(-3)}{2 - (-3)}
\][/tex]
Let's substitute \( h(2) = 4 \) and \( h(-3) = 99 \):
[tex]\[
\frac{4 - 99}{2 - (-3)} = \frac{4 - 99}{2 + 3} = \frac{-95}{5} = -19
\][/tex]
Therefore, the average rate of change of [tex]\( h(x) = -2x^3 + 5x^2 \)[/tex] from [tex]\( x = -3 \)[/tex] to [tex]\( x = 2 \)[/tex] is [tex]\(\boxed{-19}\)[/tex].
