Answer: [tex]2(\dfrac{1-e}{\pi})[/tex] or StartFraction 2 (1 minus e) Over pi EndFraction
Step-by-step explanation:
The average rate of function f(x) over the interval [a,b] is given by:-
[tex]\dfrac{f(b)-f(a)}{b-a}[/tex]
Given : [tex]f(x)=e^{\sin x}[/tex]
The average rate of change over [tex][\dfrac{\pi}{2},\pi][/tex] will be:
[tex]\dfrac{f(\pi)-f(\dfrac{\pi}{2})}{\pi-\dfrac{\pi}{2}}=\dfrac{e^{\sin \pi}-e^{\sin \dfrac{\pi}{2}}}{\dfrac{\pi}{2}}\\\\=\dfrac{e^0-e^1}{ \dfrac{\pi}{2}}\\\\=2(\dfrac{1-e}{\pi})[/tex]
Hence, the average rate of change over [tex][\dfrac{\pi}{2},\pi][/tex] is [tex]2(\dfrac{1-e}{\pi})[/tex].
