Question

Which function has the greatest rate of change on the interval from x = 3 pi over 2 to x = 2π?

A) f(x)
B) g(x)
C) h(x)
D)All three functions have the same rate of change.

Answer 1

The average rate of change for the function f(x) can be calculated from the following equation

`(f( x_(2))-f( x_(1) ))/( x_(2) - x_(1) )`

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

(2) the second function g(x)
from the graph g(3π/2) = -2 and g(2π) = 0
∴ The average rate of g =
`(g(2 \pi)-g( (3 \pi)/(2) ))/(2 \pi - (3 \pi)/(2) ) = (0-(-2))/( (\pi)/(2) )= (2)/( (\pi)/(2) ) = (4)/(\pi)`

(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 =
`(f(2 \pi)-f( (3 \pi)/(2) ))/(2 \pi - (3 \pi)/(2) ) = (1-(-5))/( (\pi)/(2) )= (6)/( (\pi)/(2) ) = (12)/(\pi)`

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)