To find the average rate of change of the function [tex]\( f(t) = 2 + \cos t \)[/tex] over the interval [tex]\(\left[\frac{\pi}{2}, \pi\right]\)[/tex], we will follow these steps:
1. Evaluate the function at the endpoints of the interval:
- Calculate [tex]\( f\left(\frac{\pi}{2}\right) \)[/tex]:
[tex]\[
f\left(\frac{\pi}{2}\right) = 2 + \cos\left(\frac{\pi}{2}\right)
\][/tex]
Since [tex]\(\cos\left(\frac{\pi}{2}\right) = 0\)[/tex], we have:
[tex]\[
f\left(\frac{\pi}{2}\right) = 2 + 0 = 2
\][/tex]
- Calculate [tex]\( f(\pi) \)[/tex]:
[tex]\[
f(\pi) = 2 + \cos(\pi)
\][/tex]
Since [tex]\(\cos(\pi) = -1\)[/tex], we have:
[tex]\[
f(\pi) = 2 + (-1) = 1
\][/tex]
2. Determine the change in the function values:
[tex]\[
\Delta f = f(\pi) - f\left(\frac{\pi}{2}\right) = 1 - 2 = -1
\][/tex]
3. Determine the change in the [tex]\( t \)[/tex] values:
[tex]\[
\Delta t = \pi - \frac{\pi}{2} = \frac{\pi}{2}
\][/tex]
4. Calculate the average rate of change:
[tex]\[
\text{Average Rate of Change} = \frac{\Delta f}{\Delta t} = \frac{-1}{\frac{\pi}{2}} = -\frac{2}{\pi}
\][/tex]
Hence, the average rate of change of [tex]\( f(t) \)[/tex] over the interval [tex]\(\left[\frac{\pi}{2}, \pi\right]\)[/tex] is [tex]\(-\frac{2}{\pi}\)[/tex], which is approximately [tex]\(-0.6366\)[/tex].
