Origin of the “Power 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 the 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 dy/dx = 2x is corroborated by solving the following expression: \displaystyle\frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \left[ {\frac{{{{(x + \Delta x)}^2} - {x^2}}}{{\Delta x}}} \right].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. On pp. 18-23 of the Mathematics Tutorial, I showed how y = ex can be approximated with a Taylor polynomial centered at x=0 for \delta x values ranging from -2 to +2.  In his video lesson entitled “Visualizing Taylor polynomial approximations”, Sal Kahn essentially replicates the work I did; the main difference between my example and Sal’s example is that Sal approximates y = ex with a Taylor polynomial centered at x=3 instead of x=0.  The important insight provided in both cases is that the accuracy of Taylor polynomial approximations increases as the order of the polynomial increases.

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