Given the function f(x)=5^2, section a is from x = 0 to x = 1 and section b is from x = 2 to x = 3. part a: find the average rate of change of each section. (4 points) part b: 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. (6 points)

Part A.
For an average rate of change you can just find Δf(x)/Δx using the points (x, f(x)).

x going from 0 --> 1

f(0) = 5^0
= 1
(0,1)

f(1) = 5^1
= 5
(1,5)

average rate of change Sect. A:
Δy/Δx = (5-1)/(1-0)
= 4

x going from 2-->3

f(2) = 5^2
= 25
(2, 25)

f(3) = 5^3
= 125
(3, 125)

average rate of change Sect. B:
Δy/Δx = (125-25)/(3-2)
= 100

Part B.
How many times greater is Sect B rate of change than Sect. A ?
100/4 = 25 times greater

Also, it is an exponential function so the rate of change is not constant. That is why the rates of change over the two sections are different.

Part a) section A      4(2)2−4(2)12−1=8 section B  4(2)4−4(2)34−3=32Part b) The rate of change is greater between x = 3 and x = 4 than between x = 2 and x =1  because an exponential function's rate of change is increasing, unlike a linear function which has a  a constant rate of change.