Question

`Given the function f(x) = 5^(x) , Section A is from x = 0 to x = 1 and Section B is from x = 2 to x = 3 .\\How many times greater is the average rate of change of Section B than Section A?\\ Explain why one rate of change is greater than the other. \\`

Answer 1

The average rate of change in section 2 is 25 times larger than the average rate of change in section 1.

Explanation:

Recall that the average rate of change of a function
`f(x)` in an interval [a,b] (section of the number line) is defined as:

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

Therefore:

1) First interval: evaluating the rate of change from x=0, to x=1 (interval [0, 1]) it becomes

Rate of change =
`(f(b)-f(a))/(b-a)=(f(1)-f(0))/(1-0)=(5^1-5^0)/(1)=5-1=4`

2) Second interval: we now evaluate the rate of change from x=2 to x=3 (interval [2, 3]), so it becomes

`(f(b)-f(a))/(b-a)=(f(3)-f(2))/(3-2)=(5^3-5^2)/(1)=125-25=100`

Therefore, the rate of change in the second interval is much larger than the rate of change in the first one. The second rate of change is in fact 100/4 = 25 times larger than the first rate of change. this is due to the fact that the function is an exponential function and not a linear function (where the rate of change is constant)