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Some math:


My thesis:

Animated construction of a sliding derivative line; an animation I made using Python programming language and SAGE, an open-source math package. Step-daughter Katie tells me her calculus teacher used it. The derivative line is green where the deriaitive is positive (sloping up), red where the derivative is negative (sloping down), and black where the dervivitive equals zero (zero slope): of the right three, the first is a local max, the second a global min, and the third the global max. Not labeled, the are mins at (-1, 0.159) and (3, 2.24) (y-values are approximate); am using radians for angle measure. The point (0, 0) is not a max or a min, but is a saddle point. The second derivative equals 

f(x) = 2x(cos(x2) -2x2(sin(x2))

which is zero at (-0.994, 0), (0, 0), (0.994, 0), (1.882, 0), and (2.55, 0), (all except (0, 0) have the x-coordinate approximated) but in above graph the points of inflection are not noted. Below is f(x) graphed in black, with d2f/dx2 graphed in blue; as it has highs and lows, divided it by 12 to fit it on the graph; this does not affect the zeros, also graphed it from a lower limit of -1.05 to demonstrate the sign change at (-0.994, 0):

It is obvious by how the second derivative (blue) switches signs at its zeros that at (-0.994, 0), which is a meaningless endpoint, but at (0, 0), (0.994, 0), (1.882, 0), and (2.55, 0) on the graph of f(x) there are points of inflection, at (0, 1), (0.994, 1.830), (1.882, 0.267) and (2.55, 1.57).

Function graph

The image links to the file, done in Mac Grapher.

© dino 2018