make content about derivatives of piecewise functions - Githubissues

lewischl / calculus_1

webwork and tools for teaching Calculus 1

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make content about derivatives of piecewise functions #37

Open weshuu opened 1 year ago

weshuu commented 1 year ago

In problem 4, the last question is:

"At which points must f be discontinuous? Use a comma-separated list of points."

It wants all x-values where f' is discontinuous to be listed. However:

One of them is a jump discontinuity for f'; it's possible f is continuous with a "corner" at that point.
One of them is an infinite discontinuity, with f' approaching -infinity from BOTH sides. It could be that f is shaped the way -x^(1/3) is shaped near the origin (with a vertical tangent line).
In problem 5, there's an infinite discontinuity with f' approaching infinity from one side and -infinity from the other. I think f can still be continuous here, by having a corner and a vertical "tangent" line (picture y=sqrt(|x|)).
Both problems also have removable discontinuities. I believe it's impossible for f to be continuous in these spots, but reasoning through this seems like a lot to ask of the students.

Did you intend to show a graph of f and ask where f' is continuous?

ofloveandhate commented 1 year ago

You're correct, this problem is flawed. Thank you so much for noticing and putting in the thought to show how it's flawed! Ultimately, I want a problem set that covers what you so beautifully described above.

In the meantime, I wonder: what we should do with this problem?

ofloveandhate commented 1 year ago

I kind of want to move it from this set into the proving grounds. I don't think the problem can be saved without a bunch of effort, which I cannot put in right now.

ofloveandhate commented 1 year ago

I think the right thing to do right now is just say the given graph is f, and ask where f' doesn't exist. I'm going to do that for these two problems, and then copy into the other instructors' courses.

ofloveandhate commented 1 year ago

I went ahead and changed these two problems to make them mathematically correct, for all instructors

ofloveandhate commented 1 year ago

i'm re-opening this because I really want to make a set exploring derivatives of piecewise functions and other discontinuous functions

weshuu commented 1 year ago

There is also at least one problem in the Challenge Set (or the corresponding group) with the same issue.