PLEASE READ THIS EXPLANATION: It is an EXTREMELY foundational concept in algebra that must be understood. Ask questions in the comments if you do not understand anything because I love helping others learn! Do not be shy. :)
Alright, so let's just understand the problem. The interval [3,6] means that the smallest x-value is x=3 and the largest x-value is x=6, and they both INCLUDE x=3 and x=6 (It would exclude x=3 and x=6 if it was like interval (3,6) with parentheses). Since f(x) is an exponential function, the change per x is not consistent and its growth actually grows as it continues in the direction of infinity, so that is why the average rate of change is interesting and very much relies on what range there is. The average rate of change from x=3 to x=6 will be far smaller than the average rate of change farther down the road at x = 99999 and x = 100003.
So, how do we find the average rate of change? All you have to do is find the slope between two points. Simple as that.
So, how do we apply that here? Imagine a graph with two points, one point at x = 3 and one point at x = 6 and we do not know the y-value of either of these points. So, that's our first step!
We need to first find how much it grows at x=3 and x=6 separately to find the the y-value of these points. f(x) = 2^x for x = 3 means y = 8, and f(x) = 2^x for x = 6 means y = 64. We now have two points:
(3, 8) and (6, 64)
Now, all we have to do is find the slope between these lines. All we have to do is use the normal formula for finding slope:
(y₂ - y₁) / (x₂ - x₁) OR (y₁ - y₂) / (x₁ - x₂)
Most people use the first one, but they give you the exact same answer, so don't ever think it's impossible to use the second one. Let's just use the first formula anyway.
(3, 8) = (x₁, y₁)
(6, 64) = (x₂, y₂)
Now, let's use the formula (remember you can use either one because it won't affect your answer):
(y₂ - y₁) / (x₂ - x₁) = (64 - 8) / (6 - 3) = 56 / 3 = 18 and 2/3
So, the average rate of change is 18 and 2/3 or 18.666666667.
