Calc 1, Lec 21: Linear Approximations, Error, Intro Taylor Polynomials, & a Life-Changing Animation

Math can sometimes change your life.

(0:00) Lecture plan.
(1:06) Standard linear approximations, typically near x = 0 (so "a" = 0). Demonstrate how to find these with the tangent line (local linearization) formula. Show how to write them in terms of infinitesimals as well.
(14:35) Approximate the cube root of 10 with a linear approximation and compute its error.
(19:47) A theoretical result about the error function and show its graph.
(23:58) Introduction to Taylor polynomials. Find a quartic (fourth degree) approximation to the cosine function cos(x) near x = 0 (so "a" = 0) by matching up both the function values and the derivative values.
(28:30) Give a theoretical upper bound for the error function and show the graphs.
(30:21) A life-changing animation (for an old student of mine) showing how Taylor polynomial approximations to the cosine function approximate it better and better as the degree gets higher and higher. This is related to how calculators (and their circuits) compute quantities like the cosine function's values.
(33:33) The increasing function theorem, the decreasing function theorem, and the constant function theorem. Statements and reasonableness. Mention that the converse of the increasing function theorem is not true.