Question

Please help !

Answer 1

The second option has the greatest rate of change

Answer 2

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:

$\dfrac{f(b)-f(a)}{b-a}$

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:

$f(\dfrac{\pi}{2})=2$

and,

$f(0)=0$

Hence,

$rate\ of\ change=\dfrac{2-0}{\dfrac{\pi}{2}-0}\\\\\\rate\ of\ change=\dfrac{4}{\pi}$

g(x):

We have:

$g(0)=0$

and

$g(\dfrac{\pi}{2})=4$

Hence,

$rate\ of\ change=\dfrac{4-0}{\dfrac{\pi}{2}-0}\\\\\\rate\ of\ change=\dfrac{8}{\pi}$

h(x):

$h(x)=\sin (x-\pi)+5$

Now we have:

$h(0)=\sin (-\pi)+5\\\\h(0)=0+5\\\\h(0)=5$

Also,

$h(\dfrac{\pi}{2})=\sin (\dfrac{\pi}{2}-\pi)+5\\\\\\h(\dfrac{\pi}{2})=\sin (\dfrac{-\pi}{2})+5\\\\h(\dfrac{\pi}{2})=-\sin (\dfrac{\pi}{2})+5\\\\\\h(\dfrac{\pi}{2})=-1+5\\\\h(\dfrac{\pi}{2})=4$

Hence, the rate of change is calculated as:

$rate\ of\ change=\dfrac{4-5}{\dfrac{\pi}{2}-0}\\\\\\rate\ of\ change=\dfrac{-2}{\pi}$

               Hence, the greatest rate of change is:

                                g(x)

Since,

$\dfrac{8}{\pi}>\dfrac{4}{\pi}>\dfrac{-2}{\pi}$