I've added two more animated GIF (click on it to see the animation) to illustrate how the secant approximation to the tangent line gets better as the two points get closer. This example involves the cube root of x, which is an important example of how a function which is continuous everywhere is not differentiable at x=0.

But $latex f(x)=x^{1/3}$ is one of those quirky Sage things you need to get used to. It doesn't graph it the way a calculator would:

Sage is complaining about raising a negative number to a rational power and even refuses to show the part from -3 to 0. So how do we get the entire graph? Take a look at the code for the animated GIF:

c = var('c')

p1 = plot(sign(x)*abs(x)^(1/3),-2,2)

a = animate([p1+point((0,0),rgbcolor='black',size=20)+ point((c,c^(1/3)),rgbcolor='black',size=20)+plot(c^(-2/3)*x,(x,-2,2),rgbcolor=(1,1/4,1/2)) for c in srange(1,0,-.05)],xmin=-2, xmax=2, ymin=-2,ymax=2)

a.show(delay=100)

The plot of $latex f(x)=x^{1/3}$ is created by: sign(x)*abs(x)^(1/3) and you've got an example why Sage is a tool for the teacher but not appropriate for the student. First, to get around x being negative (and then raised to rational power) we take the absolute value of it--but now that the answer is positive. The sign(x) is 1 if x is positive, 0 if x is 0, and -1 if x is negative. An annoying contortion to plot a common function. I've added this information to the Sage Essentials page.

A slow time of the year, but there have been a couple of events to report on:

- The TCEC Superfinal between reigning champion Stockfish and challenger Komodo is over and Komodo is the winner: 33.5-30.5. There is some information and links posted here.
- AlJazeera has an article on how a school was uninvited from a basketball tournament, "...because of concerns its players would wear T-shirts printed with the words "I Can't Breathe" during warm-ups.".
- The NY Times has an article on the "Rage Against the Common Core".
- A piece from the Las Vegas Review Journal tells us, "The cut scores established by the testing consortia and approved by the state superintendents have established a threshold that will leave only 33 percent of students declared proficient.I’m not sure how local superintendents, school principals or teachers explain to the communities they serve that their children, who attend 5-star schools and earn grades of A or B, are identified as nonproficient on a national test based on the Common Core standards.". Only 33 percent proficient? That will make for some interesting parent-teacher conferences--though I suspect as long as the A's and B's don't change most of the parents won't care.