Question

Place the three functions in order from the fastest decreasing average rate of change to the slowest decreasing average rate of change on the interval (0,3)

Answer 1

Answer: IN THIS ORDER!! g(x), f(x), h(x).

Answer 2

g(x), f(x) and h(x)

Explanation:

Given

Interval: (0,3)

See attachment for functions f(x), g(x) and h(x)

Required

Order from fastest to slowest decreasing average rate of change

The average rate of change is calculated as:

`Rate = (f(b) - f(a))/(b - a)`

In this case:

`(a,b) = (0,3)`

i.e.

`a = 0\\b=3`

For f(x)

`f(x) = 16((1)/(2))^x`

`Rate = (f(b) - f(a))/(b - a)`

`Rate = (f(3) - f(0))/(3 - 0)`

`Rate = (f(3) - f(0))/(3)`

Calculate f(3) and f(0)

`f(x) = 16((1)/(2))^x`

`f(3) = 16((1)/(2))^3 = 16 * (1)/(8) = 2`

`f(0) = 16((1)/(2))^0 = 16 * 1 = 16`

So:

`Rate = (f(3) - f(0))/(3)`

`Rate = (2 - 16)/(3)`

`Rate = -(14)/(3)`

For g(x)

`Rate = (g(b) - g(a))/(b - a)`

`Rate = (g(3) - g(0))/(3 - 0)`

`Rate = (g(3) - g(0))/(3)`

From the table of g(x)

`g(3) = 1`

`g(1) = 27`

So:

`Rate = (1 - 27)/(3)`

`Rate = -(26)/(3)`

For h(x)

`Rate = (h(b) - h(a))/(b - a)`

`Rate = (h(3) - h(0))/(3 - 0)`

`Rate = (h(3) - h(0))/(3)`

From the graph of h(x)

`h(3) = -3`

`h(0) = 4`

So:

`Rate = (-3 - 4)/(3)`

`Rate = -(7)/(3)`

So, the calculated rates of change are:

`f(x) = -(14)/(3)`
`= -4.67`

`g(x) = -(26)/(3)`
`=-8.67`

`h(x) = -(7)/(3)`
`=-2.33`

By comparison:

From the fastest decreasing to slowest, the order is: g(x), f(x) and h(x)