Part A.
[tex]\boxed{ARC_{A}=3} \\ \\ \boxed{ARC_{B}=36}[/tex]
Part B.
See in the explanation
Here we have the following function:
[tex]f(x) = 2(3)^x[/tex]
The average rate of change is the slope of the secant line between two points on the graph of a function. In this problem, we have two sections:
SECTION A:
[tex]\bullet \ From \ x = 0 \ to \ x = 1: \\ \\ ARC=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ \\ x_{1}=0 \\ \\ y_{1}=f(x_{1}})=2(3)^0=2 \\ \\ \\ x_{2}=1 \\ \\ y_{2}=f(x_{2}})=2(3)^1=6 \\ \\ ARC_{A}=\frac{6-1}{2-0} \\ \\ \boxed{ARC_{A}=3}[/tex]
SECTION B:
[tex]\bullet \ From \ x = 2 \ to \ x = 3: \\ \\ ARC=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ \\ x_{1}=2 \\ \\ y_{1}=f(x_{1}})=2(3)^2=18 \\ \\ \\ x_{2}=3 \\ \\ y_{2}=f(x_{2}})=2(3)^3=54 \\ \\ ARC_{B}=\frac{54-18}{3-2} \\ \\ \boxed{ARC_{B}=36}[/tex]
We can calculate how many times greater is the average rate of change of Section B than Section A as follows:
[tex]\frac{ARC_{B}}{ARC_{A}}=\frac{36}{3}=12 \ times[/tex]
So the average rate of change of Section B is 12 times greater than Section A. This is so because the function is exponential, so as you move from left to right you will get greater average rate of change whenever the change in x stays the same. In this case, the function is:
[tex]f(x) = 2(3)^x[/tex]
And since 3 > 0, then the exponential function grows as you move from left to right.
Exponential and linear functions: https://brainly.com/question/4119567
#LearnWithBrainly
