Lagrange Remainder For Taylor Series
The full proof of the Lagrange formula for the Taylor approximation of a function.
There is an exact error formula which one can verify by doing integration by parts.
How much rigor is needed and desirable in a calculus course for a rather general
audience with mostly non-math majors? There are many opinions here. I myself believe that if an argument can be told in less than one minute and is at the central core of calculus like the Lagrange error formula, it is desirable that a student knows why the statement is true without having just to takes it for granted on a leap of faith. There is enough "because I told you so" in science these days.
There is a nice article of Thomas Tucker "Rethinking Rigor in Calculus: The Role of the Mean Value Theorem" which discusses the Taylor error bound and how it explains also error bounds for integration methods like the Trapezoid or Simpson method:
https://maa.org/sites/default/files/0...
I believe that the simple proof given in this video using integration by parts can be absorbed quickly. It provides a corner stone of mathematics, where a non-specialist can say with confidence: yes, I understand the argument and I can accept the formula also without having to believe an authority.