Answer:
g(x), f(x) and h(x)
Step-by-step 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:
[tex]Rate = \frac{f(b) - f(a)}{b - a}[/tex]
In this case:
[tex](a,b) = (0,3)[/tex]
i.e.
[tex]a = 0\\b=3[/tex]
For f(x)
[tex]f(x) = 16(\frac{1}{2})^x[/tex]
[tex]Rate = \frac{f(b) - f(a)}{b - a}[/tex]
[tex]Rate = \frac{f(3) - f(0)}{3 - 0}[/tex]
[tex]Rate = \frac{f(3) - f(0)}{3}[/tex]
Calculate f(3) and f(0)
[tex]f(x) = 16(\frac{1}{2})^x[/tex]
[tex]f(3) = 16(\frac{1}{2})^3 = 16 * \frac{1}{8} = 2[/tex]
[tex]f(0) = 16(\frac{1}{2})^0 = 16 * 1 = 16[/tex]
So:
[tex]Rate = \frac{f(3) - f(0)}{3}[/tex]
[tex]Rate = \frac{2 - 16}{3}[/tex]
[tex]Rate = -\frac{14}{3}[/tex]
For g(x)
[tex]Rate = \frac{g(b) - g(a)}{b - a}[/tex]
[tex]Rate = \frac{g(3) - g(0)}{3 - 0}[/tex]
[tex]Rate = \frac{g(3) - g(0)}{3}[/tex]
From the table of g(x)
[tex]g(3) = 1[/tex]
[tex]g(1) = 27[/tex]
So:
[tex]Rate = \frac{1 - 27}{3}[/tex]
[tex]Rate = -\frac{26}{3}[/tex]
For h(x)
[tex]Rate = \frac{h(b) - h(a)}{b - a}[/tex]
[tex]Rate = \frac{h(3) - h(0)}{3 - 0}[/tex]
[tex]Rate = \frac{h(3) - h(0)}{3}[/tex]
From the graph of h(x)
[tex]h(3) = -3[/tex]
[tex]h(0) = 4[/tex]
So:
[tex]Rate = \frac{-3 - 4}{3}[/tex]
[tex]Rate = -\frac{7}{3}[/tex]
So, the calculated rates of change are:
[tex]f(x) = -\frac{14}{3}[/tex] [tex]= -4.67[/tex]
[tex]g(x) = -\frac{26}{3}[/tex] [tex]=-8.67[/tex]
[tex]h(x) = -\frac{7}{3}[/tex] [tex]=-2.33[/tex]
By comparison:
From the fastest decreasing to slowest, the order is: g(x), f(x) and h(x)
