Given the function f(x) = 2(3)x, 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:

To find the average rate of change, let us first write out the equation to find it.

Δy/Δx = average rate of change. 

Finding average rate of change for Section A

Δy = f(1) - f(0) = 2(3)^1 - 2(3)^0 = 6 - 1 = 5

Δx = 1- 0 = 1

Plug the numbers in: Δy/Δx = 5/1 = 5

Therefore, the average rate of change for Section A is 5. 

Finding average rate of change for Section B

Δy = f(3) - f(2) = 2(3)^3 - 2(3)^2 = 2(27) - 2(9) = 54 - 18 = 36

Δx = 3 - 2 = 1

Plug the numbers in: Δy/Δx = 36/1 = 36

Therefore, the average rate of change for Section B is 36. 

Part B:

(a) How many times greater is the average rate of change of Section B than Section A?

If Section B is on the interval [2,3] and Section A is on the interval [0,1]. 

For the function f(x) = 2(3)^x, the average rate of change of Section B is 7.2 times greater than the average rate of change of Section A. 

(b) Explain why one rate of change is greater than the other. 

Since f(x) = 2(3)^x is an exponential function the y values do not increase linearly, instead increase exponentially. In an interval with smaller x values the rate of change is lower than an interval with larger x values.