Answer:
D. All three functions have the same rate of change.
Step-by-step explanation:
1. For the function f(x):
- at [tex]x=\pi,[/tex] [tex]f(\pi)=0;[/tex]
- at [tex]x=\dfrac{3\pi }{2},[/tex] [tex]f\left(\dfrac{3\pi}{2}\right)=-4.[/tex]
The rate of change is
[tex]\dfrac{f(\frac{3\pi}{2})-f(\pi)}{\frac{3\pi}{2}-\pi}=\dfrac{-4-0}{\frac{\pi}{2}}=-\dfrac{8}{\pi}.[/tex]
2. For the function g(x):
- at [tex]x=\pi,[/tex] [tex]g(\pi)=0;[/tex]
- at [tex]x=\dfrac{3\pi }{2},[/tex] [tex]g\left(\dfrac{3\pi}{2}\right)=-4.[/tex]
The rate of change is
[tex]\dfrac{g(\frac{3\pi}{2})-g(\pi)}{\frac{3\pi}{2}-\pi}=\dfrac{-4-0}{\frac{\pi}{2}}=-\dfrac{8}{\pi}.[/tex]
3. For the function h(x):
- at [tex]x=\pi,[/tex] [tex]h(\pi)=4\cdot \sin \pi+2=2;[/tex]
- at [tex]x=\dfrac{3\pi }{2},[/tex] [tex]h\left(\dfrac{3\pi}{2}\right)=4\cdot \sin \frac{3\pi}{2}+2=-4+2=-2.[/tex]
The rate of change is
[tex]\dfrac{h(\frac{3\pi}{2})-h(\pi)}{\frac{3\pi}{2}-\pi}=\dfrac{-2-2}{\frac{\pi}{2}}=-\dfrac{8}{\pi}.[/tex]
All three functions have the same rate of change.
