Mean value theorem for divided differences
Okay kiddo, let me explain the mean value theorem for divided differences to you.
Imagine you have a bunch of numbers, let's say 5, 9, and 12. You want to find the average (or mean) of these numbers. So you add them up and divide by how many there are, which in this case is 3. So the mean is (5+9+12)/3 = 8.6.
Now let's imagine we have more numbers, maybe 5, 9, 12, and 15. And let's say we want to find the mean of just the first three numbers (5, 9, and 12). We could add them up and divide by 3 again, but what if there was an easier way?
This is where the mean value theorem for divided differences comes in. It says that if we have a function (which is like a rule that tells us how to get an output from an input) that is continuous (which means it doesn't have any holes or jumps) on an interval (which is just a fancy word for a range of numbers) that contains all of our numbers (in this case 5, 9, 12, and 15), then there is a number between the first and third numbers (5 and 12) where the divided difference equals the mean of those numbers, just like we found earlier.
But wait, what's a divided difference? Well, it's just a fancy way of saying we take the difference between two numbers and divide it by the difference between their inputs (or x values). So for our example, we would calculate the divided difference of the first three numbers like this:
[(9-5)/(1-0)] and [(12-9)/(2-1)]
And then we would take the average of those two divided differences:
[(9-5)/(1-0)] and [(12-9)/(2-1)] = (4/1) and (3/1)
Mean of the divided differences = (4/1 + 3/1)/2 = 3.5
So the mean value theorem tells us that there must be a number, somewhere between 5 and 12, where the divided difference equals 3.5. And this is true for any set of numbers we choose, as long as the function that we're working with is continuous on the interval that contains our numbers.
Does that make sense?
Imagine you have a bunch of numbers, let's say 5, 9, and 12. You want to find the average (or mean) of these numbers. So you add them up and divide by how many there are, which in this case is 3. So the mean is (5+9+12)/3 = 8.6.
Now let's imagine we have more numbers, maybe 5, 9, 12, and 15. And let's say we want to find the mean of just the first three numbers (5, 9, and 12). We could add them up and divide by 3 again, but what if there was an easier way?
This is where the mean value theorem for divided differences comes in. It says that if we have a function (which is like a rule that tells us how to get an output from an input) that is continuous (which means it doesn't have any holes or jumps) on an interval (which is just a fancy word for a range of numbers) that contains all of our numbers (in this case 5, 9, 12, and 15), then there is a number between the first and third numbers (5 and 12) where the divided difference equals the mean of those numbers, just like we found earlier.
But wait, what's a divided difference? Well, it's just a fancy way of saying we take the difference between two numbers and divide it by the difference between their inputs (or x values). So for our example, we would calculate the divided difference of the first three numbers like this:
[(9-5)/(1-0)] and [(12-9)/(2-1)]
And then we would take the average of those two divided differences:
[(9-5)/(1-0)] and [(12-9)/(2-1)] = (4/1) and (3/1)
Mean of the divided differences = (4/1 + 3/1)/2 = 3.5
So the mean value theorem tells us that there must be a number, somewhere between 5 and 12, where the divided difference equals 3.5. And this is true for any set of numbers we choose, as long as the function that we're working with is continuous on the interval that contains our numbers.
Does that make sense?