When using a Taylor polynomial to approximate a function value, wouldn't it be nice to have a way to see how far off your approximate actually was?  I mean, besides just putting the thing in a calculator.  You know, you're not always going to have a calculator in your back pocket!  Anyway, the Lagrange remainder will help us compute that error.  Duration: 12:41

Taylor's Theorem states that our function is exactly equal to its nth degree Taylor polynomial plus some remainder function.  Lagrange uses that remainder to help us bound the error generated with our Taylor polynomial approximation.  Think of it as the first neglected term, as in the (n+1)th term, with a maximized derivative.  Duration: 15:32

Example 9 demonstrates another instance of the Lagrange remainder, this time on our natural log approximation from Example 5.  Again, realize that the AP Exam will always give you a value for which to aim.  In other words, you will have to show that the remainder is less than some given value.  Duration: 8:12

In this video, Sal does an excellent job of demonstrating where the Lagrange remainder function--or, as he calls it, the error function--originates.  Continued in Part 2, which is embedded below.  Duration: 11:26

Conclusion of the previous Khan Academy video in which Sal does the integration necessary to derive the Lagrange remainder.  Duration: 15.07