The average rate of change for the function f(x) can be calculated from the following equation
[tex] \frac{f( x_{2})-f( x_{1} )}{ x_{2} - x_{1} } [/tex]
By applying the last formula on the given equations
(1) the first function f
from the table f(3π/2) = -2 and f(2π) = 0
∴ The average rate of f = [tex] \frac{f(2 \pi)-f( \frac{3 \pi}{2} )}{2 \pi - \frac{3 \pi}{2} } = \frac{0-(-2)}{ \frac{\pi}{2} }= \frac{2}{ \frac{\pi}{2} } = \frac{4}{\pi} [/tex]
(2) the second function g(x)
from the graph g(3π/2) = -2 and g(2π) = 0
∴ The average rate of g = [tex] \frac{g(2 \pi)-g( \frac{3 \pi}{2} )}{2
\pi - \frac{3 \pi}{2} } = \frac{0-(-2)}{ \frac{\pi}{2} }= \frac{2}{
\frac{\pi}{2} } = \frac{4}{\pi} [/tex]
(3) the third function h(x) = 6 sin x +1
∴ h(3π/2) = 6 sin (3π/2) + 1 = 6 *(-1) + 1 = -5
h(2π) = 6 sin (2π) + 1 = 6 * 0 + 1 = 1
∴ The average rate of h = [tex] \frac{f(2 \pi)-f( \frac{3 \pi}{2} )}{2
\pi - \frac{3 \pi}{2} } = \frac{1-(-5)}{ \frac{\pi}{2} }= \frac{6}{
\frac{\pi}{2} } = \frac{12}{\pi} [/tex]
By comparing the results, The function which has the greatest rate of change is h(x)
So, the correct answer is option C) h(x)
