Next week I'll be working with my students to figure out these two derivatives. I don't want them taking notes on all the steps, so I made a handout that reflects what I think we'll do in class. I'm embarrassed to admit how many hours I put into this. The good thing is that I know the details of these proofs a lot better after having typed this out.
I used Word with MathType, which has usually looked fine to me. I'm starting to wish I knew LaTex. (It's definitely on the list of technology I want to learn soon.) I used Geogebra for diagrams, took snapshots using Jing, and pulled the .png files into my Word doc. There's probably a better way to do that too.
First is an estimation activity:
Estimating the Derivative From Graphs of Sine and Cosine by Sue VanHattum*
Then here is the complete proof I typed up:
Derivatives for Sine and Cosine by Sue VanHattum
What would you do differently?
[Note added on 2-21-14: I just taught this for the third time from my handouts. I loved how it went yesterday. When I came back to this post, I re-read the comment from Alemi at The Virtuosi, and studied the blog post referenced. It is exactly what I was hoping might exist: a nice short argument with the same result. Check it out!]
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*This is my first time using scribd in a very long time. I hope these are useful to someone. How do I get the documents to show up in my blog post?
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If you use the trig identity cos(2x) = 1 - 2sin^2(x) = 2cos^2(x) - 1, along with chain rule, you have:
ReplyDelete2sin(2x) = -4sin(x) d/dx(sin(x)) = 4cos(x) d/dx(cos(x))
Then, using sin(2x) = 2sin(x)cos(x), you have:
4sin(x)cos(x) = -4sin(x) d/dx(sin(x)) = 4cos(x) d/dx(sin(x))
which gets you both derivatives.
I'm confused. It looks like you're using the derivatives in your equations. If you're trying to show what the derivatives are, that seems circular. (I can't use chain rule yet, as I haven't taught it. I'm saving that for the next unit.)
ReplyDeleteYep, you're right! I think I also dropped a few minus signs here and there, too! ;)
DeleteI do wonder if there's an easier way that doesn't involve taylor expansions or Euler's formula...
It would be nice, since this is so long and involved, but I think this is the most straightforward way to get the results.
ReplyDeleteI'm impressed! When I taught it, I just said, "Ok, remember these identities from trig? No? Well, here they are again. We're going to need them today."
ReplyDeleteBut I think the way you've presented it here is much more valuable. It reveals all the knowledge that has to be in place to get to one simple derivative. Love it.
I also really liked the first approximation exercise.
Thanks, Rebecka. Next week I'll be able to let you know whether my students were able to tolerate this long a chain of reasoning.
ReplyDeleteI have a bad memory, and I wanted to get to where I could hold the sum of angles proof in my head. I think (after 25 years teaching these things) I've finally got it. The limit for sin h / h is just as unwieldy, and I think I might have that one in me now too.
None of the sites I looked at online pulled it all together the way I wanted, nor did any of the textbooks I have around me, so I had to do this if I want to get students to do more than take notes when we work on it in class.
Here's a blog post I wrote this morning in response to your question. In a nutshell, emphasize that "co" means "complement" because that explains a lot.
ReplyDeleteJohn, your post made me realize that I need to change the reference to a right angle on page 2, from 90 degrees to pi/2 radians. I will also add a link to your lovely post. It will become more relevant later in the week, as we use product and quotient properties (not learned yet) to get the derivatives of the other trig functions. Thank you.
DeleteI like the proof sheet that you've made for this Sue - it's rigorus, clear and well explained. In fact, I've never tried to prove or seen the proof of why the lim sin(h)/h tends to 1 - so I learnt something new from reading your post! I've never taught a course in which I felt students would benefit from seeing the proof so rigorously derived but I have presented students with a general feel for the derivatives of sine and cosine.
ReplyDeleteWhilst I'm not too happy with many of the videos I made last year, feel free to take a look at one I made when trialling a bit of flipped learning with an IB Standard Level group.
http://www.youtube.com/watch?v=M0u9Qy3SERI&list=UUPhtaFX9pN6xEHez4YUFxZg&index=11
I'm not sure my own students will benefit from a rigorous definition, but I want to try it out.
DeleteI loved your video. (Partly because I loved your accent.) I'll be having my students do that themselves (with the first handout) before I do a call-and-response style lecture that I hope matches the second handout.
Very ambitious! Your handouts are very clear - especially the squeeze theorem part.
ReplyDeleteThis is my attempt at a visual proof using geogebra:
http://www.geogebratube.org/student/m29769
I have not proved this in the past - just couldn't think of a way that I felt would give enough benefit to enough of my students. Some of that may just be my own bias/laziness when it comes to messy algebra, though.
The geogebra worksheet above provides a nice concrete and surprising visual for what you are trying to prove and a proof. It is all based on a fairly simple unit circle diagram with very little algebra. I think the argument is rigorous, though some may not consider it a proof.
With proofs, I sometimes wonder whether the definition of proof depend on the audience. What if a proof doesn't convince the audience of anything? Is it still a proof?
lh, I saw it once at work, and now I can't get it to load. The important parts to think about were small, and somehow that made it hard for me to think... But I'll try to look at it again later.
ReplyDeleteLearn latex! I swear, you'll be all "OMG, why did I wait so long?!" (In my mind, that's in a very teenagey voice.)
ReplyDeleteI went through the same thing, and it's seriously worth it. And graphics? TikZ all the way. It's so awesome. Also, tex.stackexchange.com is the most helpful place ever.
As far as the actual math content, I only ever teach Applied Calculus, so the level of rigor is not what you get in a math-major oriented calculus, and we just accept that the approximations from the tables are valid. I like the idea of doing a few more rigorous limits, but I think I would do those on one day, and then when it comes up in the next lesson say something like, "well, now we need a technique to find this particular limit, what have we got?" And then wait it out until someone looks in their notes from last time.
Alternatively you can compute the derivatives geometrically. I've written a little diddy here
ReplyDelete