Question

Compare each of the functions shown below:

Answer 1

D. All three functions have the same rate of change.

Explanation:

1. For the function f(x):

• at
  `x=\pi,`
  `f(\pi)=0;`
• at
  `x=(3\pi )/(2),`
  `f\left((3\pi)/(2)\right)=-4.`

The rate of change is

`(f((3\pi)/(2))-f(\pi))/((3\pi)/(2)-\pi)=(-4-0)/((\pi)/(2))=-(8)/(\pi).`

2. For the function g(x):

• at
  `x=\pi,`
  `g(\pi)=0;`
• at
  `x=(3\pi )/(2),`
  `g\left((3\pi)/(2)\right)=-4.`

The rate of change is

`(g((3\pi)/(2))-g(\pi))/((3\pi)/(2)-\pi)=(-4-0)/((\pi)/(2))=-(8)/(\pi).`

3. For the function h(x):

• at
  `x=\pi,`
  `h(\pi)=4\cdot \sin \pi+2=2;`
• at
  `x=(3\pi )/(2),`
  `h\left((3\pi)/(2)\right)=4\cdot \sin (3\pi)/(2)+2=-4+2=-2.`

The rate of change is

`(h((3\pi)/(2))-h(\pi))/((3\pi)/(2)-\pi)=(-2-2)/((\pi)/(2))=-(8)/(\pi).`

All three functions have the same rate of change.