Origin of the “Product Rule”, and Visualizing Taylor polynomial approximations

This blog entry provides a helpful follow-up for a couple of calculus-related topics that we covered during today’s Mathematics Tutorial.

  1. See page 12 of the above-referenced lecture note.  There, the equation for a parabola (y = {x^2}) appears, and the claim that \frac{{dy}}{{dx}} = 2x is corroborated by solving the following expression:
    In the 11-minute Khan Academy video at https://youtu.be/HEH_oKNLgUU, Sal Kahn takes on the solution of this problem in a very succinct and easy-to-comprehend fashion.
  2. In his video lesson entitled “Visualizing Taylor polynomial approximations”, Sal Kahn replicates the tail end of today’s Finance 4366 class meeting in which we approximated y = ex with a Taylor polynomial centered at x=0 (as also shown in pp. 18-23 of the Mathematics Tutorial lecture note).  Sal approximates y = ex with a Taylor polynomial centered at x=3 instead of x=0, but the same insight obtains in both cases, which is that the accuracy of Taylor polynomial approximations increases as the order of the polynomial increases.

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