tag:blogger.com,1999:blog-5303307482158922565.post8984642025774302559..comments2024-03-22T13:39:55.941-07:00Comments on Math Mama Writes...: Derivatives of Sine and CosineSue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.comBlogger14125tag:blogger.com,1999:blog-5303307482158922565.post-6307814314569760792013-02-25T15:42:16.047-08:002013-02-25T15:42:16.047-08:00Alternatively you can compute the derivatives geom...Alternatively you can compute the derivatives geometrically. I've written a little diddy <a href="http://thephysicsvirtuosi.com/posts/trigonometric-derivatives.html" rel="nofollow">here</a>The Virtuosihttps://www.blogger.com/profile/03861940605943479600noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-36846875788567151202013-02-13T17:44:33.534-08:002013-02-13T17:44:33.534-08:00Learn latex! I swear, you'll be all "OMG,...Learn latex! I swear, you'll be all "OMG, why did I wait so long?!" (In my mind, that's in a very teenagey voice.)<br /><br />I 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.<br /><br />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.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-13227172993098108292013-02-11T18:46:58.381-08:002013-02-11T18:46:58.381-08:00lh, I saw it once at work, and now I can't get...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.<br />Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-43559560450535127572013-02-11T15:22:33.847-08:002013-02-11T15:22:33.847-08:00I'm not sure my own students will benefit from...I'm not sure my own students will benefit from a rigorous definition, but I want to try it out. <br /><br />I 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.<br />Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-53819822871928357532013-02-11T12:24:50.010-08:002013-02-11T12:24:50.010-08:00John, your post made me realize that I need to cha...John, 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.Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-3360090078111078332013-02-11T12:02:44.160-08:002013-02-11T12:02:44.160-08:00Very ambitious! Your handouts are very clear - es...Very ambitious! Your handouts are very clear - especially the squeeze theorem part.<br /><br />This is my attempt at a visual proof using geogebra:<br /><br />http://www.geogebratube.org/student/m29769<br /><br />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. <br /><br />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.<br /><br />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?l hodgenoreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-58075959064874730012013-02-11T07:07:49.844-08:002013-02-11T07:07:49.844-08:00I like the proof sheet that you've made for th...I 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.<br /><br />Whilst 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.<br /><br />http://www.youtube.com/watch?v=M0u9Qy3SERI&list=UUPhtaFX9pN6xEHez4YUFxZg&index=11Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-82330049605369957662013-02-11T06:31:53.274-08:002013-02-11T06:31:53.274-08:00Here's a blog post I wrote this morning in res...Here's a <a href="http://www.johndcook.com/blog/2013/02/11/differentiating-bananas-and-co-bananas/" rel="nofollow">blog post</a> I wrote this morning in response to your question. In a nutshell, emphasize that "co" means "complement" because that explains a lot.John Cookhttp://www.johndcook.com/noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-81899709470267029722013-02-10T14:03:42.327-08:002013-02-10T14:03:42.327-08:00Thanks, Rebecka. Next week I'll be able to let...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. <br /><br />I 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.<br /><br />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.Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-3233511848858684432013-02-10T13:58:34.496-08:002013-02-10T13:58:34.496-08:00I'm impressed! When I taught it, I just said,...I'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."<br /><br />But 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.<br /><br />I also really liked the first approximation exercise.Rebecka Petersonhttps://www.blogger.com/profile/12227797437296056645noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-50943906036044819012013-02-10T13:48:26.674-08:002013-02-10T13:48:26.674-08:00It would be nice, since this is so long and involv...It would be nice, since this is so long and involved, but I think this is the most straightforward way to get the results.Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-52639808314656865812013-02-10T13:21:39.109-08:002013-02-10T13:21:39.109-08:00Yep, you're right! I think I also dropped a fe...Yep, you're right! I think I also dropped a few minus signs here and there, too! ;)<br /><br />I do wonder if there's an easier way that doesn't involve taylor expansions or Euler's formula...Haohttps://www.blogger.com/profile/02348974241652264510noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-81173136253758454582013-02-10T10:56:41.054-08:002013-02-10T10:56:41.054-08:00I'm confused. It looks like you're using t...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.)Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-3312197588219580202013-02-10T10:48:21.208-08:002013-02-10T10:48:21.208-08:00If you use the trig identity cos(2x) = 1 - 2sin^2(...If you use the trig identity cos(2x) = 1 - 2sin^2(x) = 2cos^2(x) - 1, along with chain rule, you have:<br />2sin(2x) = -4sin(x) d/dx(sin(x)) = 4cos(x) d/dx(cos(x))<br /><br />Then, using sin(2x) = 2sin(x)cos(x), you have:<br />4sin(x)cos(x) = -4sin(x) d/dx(sin(x)) = 4cos(x) d/dx(sin(x))<br /><br />which gets you both derivatives.Haohttps://www.blogger.com/profile/02348974241652264510noreply@blogger.com