One of the things I love about Desmos (besides the fact that it has better resolution than a graphing calculator and it is simple easy to use) is that it makes it easier for kids to see connections. Today in Calculus we considered at the function f(x) = (x - 2)2/3 + 1. Prior to actually graphing the function in Desmos, we calculated the derivative and determined that there would be a critical point at x = 1, because the derivative was undefined at that point. After we did a quick sketch of the graph by hand, we looked at both the function and its derivative in desmos. The two graphs are shown on the same axes below.
We could easily see when the derivative was negative and when it was positive and how that corresponded to the left and right sides of the graph. We could also see that the derivative had a vertical asymptote at x = 1 and that made sense since the derivative of the function was undefined at x = 1. What was more interesting however was that the derivative had large positive values immediately to the right of x = 1, but then the derivative had smaller positive values as x got larger. It was right around that moment that I could see the synapses in some students' brains firing as they were getting a better understanding - a visual understanding - of the derivative and its relationship to the graph of a function. Thank you, Desmos!