This sketch follows an earlier experiment with piecewise linear tactile functions. These functions are defined using a small number of user-positioned independent points which use a simple convention to define a function. The linear one offered what I thought to be an interesting look at the antiderivative. This sketch shows a piecewise cubic function, meaning that consecutive independent points are joined with a segment of a cubic function.
Let me know what you think.
There is something called a Bezier curve that involves cubics and addresses some similar applications, but I am really not sure if my construction is different.
I decided that I wanted a manipulable function that was smooth on its domain. I could not do that with lines, and quadratics offer only the most unwieldy options if we insist on differentiability. I did make one with sinusoidal curves, but the first construction I tried demanded in horizontal tangents at each of the independent points, which was a bit limiting. In the end, I realized it needed to be cubic and I was pretty sure that a unique curve existed where endpoints and slopes (at the endpoints) were defined.
I'm not sure if it is good for anything.
In this sketch, the derivative (red) and antiderivative (blue) can be shown and hidden easily. The curve in the construction is really seven distinct objects, but it would be possible to create it as a single function object.
1 comments:
Brilliant and fun. I tried manipulating and creating a similar analogy on GSP a few years ago and had a nightmare with the Antiderivative. My kids will have a good time trying out problems based off the file. Thanks!
Post a Comment