Part A:
The average rate of change refers to a function's slope. Thus, we are going to need to use the slope formula, which is:
[tex]m = \dfrac{y_2 - y_1}{x_2 - x_1}[/tex]
- [tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex] are points on the function
You can see that we are given the x-values for our interval, but we are not given the y-values, which means that we will need to find them ourselves. Remember that the y-values of functions refers to the outputs of the function, so to find the y-values simply use your given x-value in the function and observe the result:
[tex]h(0) = 3(5)^0 = 3 \cdot 1 = 3[/tex]
[tex]h(1) = 3(5)^1 = 3 \cdot 5 = 15[/tex]
[tex]h(2) = 3(5)^2 = 3 \cdot 25 = 75[/tex]
[tex]h(3) = 3(5)^3 = 3 \cdot 125 = 375[/tex]
Now, let's find the slopes for each of the sections of the function:
Section A
[tex]m = \dfrac{15 - 3}{1 - 0} = \boxed{12}[/tex]
Section B
[tex]m = \dfrac{375 - 75}{3 - 2} = \boxed{300}[/tex]
Part B:
In this case, we can find how many times greater the rate of change in Section B is by dividing the slopes together.
[tex]\dfrac{m_B}{m_A} = \dfrac{300}{12} = 25[/tex]
It is 25 times greater. This is because [tex]3(5)^x[/tex] is an exponential growth function, which grows faster and faster as the x-values get higher and higher. This is unlike a linear function which grows or declines at a constant rate.
