Answer:
The function that has the greatest rate of change is:
g(x)
Step-by-step explanation:
We know that the rate of change from x=a to x=b is determined as:
[tex]\dfrac{f(b)-f(a)}{b-a}[/tex]
We are asked to find the rate of change of each of the functions form x=0 to x=pi over 2.
f(x):
We are given that:
[tex]f(\dfrac{\pi}{2})=2[/tex]
and,
[tex]f(0)=0[/tex]
Hence,
[tex]rate\ of\ change=\dfrac{2-0}{\dfrac{\pi}{2}-0}\\\\\\rate\ of\ change=\dfrac{4}{\pi}[/tex]
g(x):
We have:
[tex]g(0)=0[/tex]
and
[tex]g(\dfrac{\pi}{2})=4[/tex]
Hence,
[tex]rate\ of\ change=\dfrac{4-0}{\dfrac{\pi}{2}-0}\\\\\\rate\ of\ change=\dfrac{8}{\pi}[/tex]
h(x):
[tex]h(x)=\sin (x-\pi)+5[/tex]
Now we have:
[tex]h(0)=\sin (-\pi)+5\\\\h(0)=0+5\\\\h(0)=5[/tex]
Also,
[tex]h(\dfrac{\pi}{2})=\sin (\dfrac{\pi}{2}-\pi)+5\\\\\\h(\dfrac{\pi}{2})=\sin (\dfrac{-\pi}{2})+5\\\\h(\dfrac{\pi}{2})=-\sin (\dfrac{\pi}{2})+5\\\\\\h(\dfrac{\pi}{2})=-1+5\\\\h(\dfrac{\pi}{2})=4[/tex]
Hence, the rate of change is calculated as:
[tex]rate\ of\ change=\dfrac{4-5}{\dfrac{\pi}{2}-0}\\\\\\rate\ of\ change=\dfrac{-2}{\pi}[/tex]
Hence, the greatest rate of change is:
g(x)
Since,
[tex]\dfrac{8}{\pi}>\dfrac{4}{\pi}>\dfrac{-2}{\pi}[/tex]
