Example of derivative as limit of average rate of change
Stacy wants to find the derivative of f of x = x² + 1 at the point x = 2. Her table below shows the average rate of change of f over the intervals from x to 2 or from 2 to x, and these are closed intervals for x values. They get increasingly closer to two, so we're talking about the average rate of change of f over these closed intervals for x values that get increasingly close to two.
It looks like we're going to be dealing with some type of a limit, or we're trying to calculate some type of a limit or approximate some type of a limit. So let's read this data here. So these are the x values, and she's trying to find the average rate of change between each of these x values and 2, or the average rate of change of the function when x is one of these x values and 2.
Then she has the average rate of change that she pre-calculated, so we don't have to get a calculator out or anything like that. And just as a reminder, how did she calculate this 3.9? Well, they tell us she took f of 1.9. What is the function equal when x is 1.9? From that, she subtracted what is the value of the function when f is equal to 2. So that's really our change in f, and she divided it by the x, which is 1.9 minus 2.
So change in f over change in x, what is the average rate of change of our function over that interval? So she did it between 1.9 and 2, she got 3.9. Then she gets closer to two, so now she's doing it between 1.99 and 2, and it becomes 3.99. It looks like it's getting closer to four.
She gets even closer to two and the average rate of change gets even closer to four. Then she goes on the other side of two. You could view it as this is approaching. This is approaching x approaching two from the left-hand side, and this is x approaching two from the right-hand side. So when it's 2.1, the average rate of change is 4.1. When it's 2.01, once again we're getting closer to two; we're getting closer to two, the average rate of change is getting closer to four.
The closer we get to two, the closer the average rate of change gets to four. So what this data is really helping us approximate, it's really saying, "Okay, the average rate of change we know is f of x minus f of 2 over x - 2," but what we're really thinking about is, "Well, what is the limit as x approaches two?" Right over here, that's what this data is helping us to get at, and it looks like this limit is equal to four.
They give us the data here and says, “Look, the closer that x gets to two from either the left-hand side or the right-hand side, the closer that this expression right over here, which is this number, gets to four." You might recognize this as one of the definitions of a derivative. This is one of the definitions of a derivative. This right over here would be f prime of 2.
The derivative at x = 2 is equal to the limit as x approaches 2 of all of this business. There's other ways to express a derivative as a limit, but this is one of them. And so there you go from the table, what does the derivative of f of x equals x² + 1 at x = 2 appear to be? Well, the derivative at x = 2 appears to be equal to 4, and we're done.