To determine the average rate of change of the function [tex]\( h(t) = (t + 3)^2 + 5 \)[/tex] over the interval [tex]\([-5, -1]\)[/tex], we will follow these steps:
1. Identify the endpoints of the interval:
The interval given is [tex]\([-5, -1]\)[/tex], so we need to evaluate the function at [tex]\( t = -5 \)[/tex] and [tex]\( t = -1 \)[/tex].
2. Evaluate the function at the endpoints:
- Evaluate [tex]\( h(t) \)[/tex] at [tex]\( t = -5 \)[/tex]:
[tex]\[
h(-5) = ((-5) + 3)^2 + 5 = (-2)^2 + 5 = 4 + 5 = 9
\][/tex]
- Evaluate [tex]\( h(t) \)[/tex] at [tex]\( t = -1 \)[/tex]:
[tex]\[
h(-1) = ((-1) + 3)^2 + 5 = (2)^2 + 5 = 4 + 5 = 9
\][/tex]
3. Calculate the average rate of change:
The formula for the average rate of change of a function [tex]\( h(t) \)[/tex] over an interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[
\text{Average rate of change} = \frac{h(b) - h(a)}{b - a}
\][/tex]
In this case, [tex]\(a = -5\)[/tex] and [tex]\(b = -1\)[/tex].
- Substitute the values we determined:
[tex]\[
\frac{h(-1) - h(-5)}{-1 - (-5)} = \frac{9 - 9}{-1 + 5} = \frac{0}{4} = 0
\][/tex]
Therefore, the average rate of change of the function [tex]\( h(t) \)[/tex] over the interval [tex]\([-5, -1]\)[/tex] is [tex]\( 0 \)[/tex].
