Question

Please help! please do not copy and paste i have seen the other ones

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.

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.

Answer 1

Given function:

`f(x)=2(3)^x`

Part A

The average rate of change of function f(x) over the interval a ≤ x ≤ b is given by:

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

`\begin{aligned}\textsf{Average rate of change - Section A} & =(f(1)-f(0))/(1-0)\\\\& =(2(3)^1-2(3)^0)/(1-0)\\\\& =(6-2)/(1)\\\\& =4 \end{aligned}`

`\begin{aligned}\textsf{Average rate of change - Section B} & =(f(3)-f(2))/(3-2)\\\\& =(2(3)^3-2(3)^2)/(3-2)\\\\& =(54-18)/(1)\\\\& =36 \end{aligned}`

Part B

`\frac{\textsf{Average rate of change of Section B}}{\textsf{Average rate of change of Section A}}=(36)/(4)=9`

Therefore, the average rate of change of Section B is 9 times greater than Section A.

The function is an exponential function (with 3 as the growth factor) so the rate of change increases over time.