```Date: Apr 17, 2013 5:32 PM
Author: Dave L. Renfro
Subject: Re: Research help needed

Daniel Sarkes wrote:http://mathforum.org/kb/message.jspa?messageID=8892992> Hi, I am conducting an undergrad research project involving L'Hopital's> rule and Excel. The idea is to use computation L'Hopital's rule works.> I have attached the lab and the teacher's guide. I also have a pre and> post test so I can make an attempt at getting some sort of quantifiable> data from this project. If you can help me and run the lab let me know> an I will email them.The problem I see with the activity is that no mathematical reasoningis employed. Indeed, there seems to be little more than substituting oneblack box (the statement that L'Hopital's rule works) for anotherblack box (graphs handed to students by calculator fiat).Let's say a student goes through your lab with the examples you gave.I come along and ask the student to go through the same process(except restrict to right hand limits at zero) with f(x)/g(x), wheref(x) = sqrt(x^3)*sin(1/x) and g(x) = x. [Or use sqrt(|x|^3)*sin(1/x)for f(x) if you want to continue using a 2-sided limit.] What doesthe student come away with? Probably not much more than this:L'Hopital's rule works when it works and it doesn't work whenit doesn't work.Below (between the parallel --- lines) is a May 2009 post ofmine in another discussion group that gives a calculatoractivity that I think will do a better job of developingcritical thinking while also reinforcing useful concepts.(Embedded in this post is another still earlier post, thisone between the parallel **** lines.)"Bibliographic information" for the post that follows:[ap-calculus] graphing calculator activities Posted: May 21, 2009 11:44 AMhttp://mathforum.org/kb/message.jspa?messageID=6720467- -------------------------------------------------------------- -------------------------------------------------------------Martha Martin wrote:http://mathforum.org/kb/message.jspa?messageID=6720150> Does anyone have any activities to help the students> better understand and learn how to use their graphing> calculators?There's a neat idea, which I first saw in this list(Richard Sisley, 22 November 2000), that you can doin either an AB class or a BC class. If it's in anAB class, you can get more mileage out of the idea byintroducing the "quadratic approximation of a functionat a point", which I've found to be fairly easilyunderstood and worked with by students who have alreadyworked with the idea of the linear approximation ofa function a point.While I was looking up my post about this topic(copied in full below), I came across another postof mine that might also be something to try, althoughin this case, if it's an AB class, you'll first wantto explain the idea of higher order polynomialapproximations to a function at a point. You don'tneed to explain how to get them -- although an easymechanical method is to assume sin(x) = a + bx + cx^2 + ...,plug in x = 0 to get a = 0, differentiate both sidesand then plug in x = 0 to get b = 1, differentiateboth sides again and then plug in x = 0 to get c = 0,differentiate both sides yet again and then plug inx = 0 to get d = -1/6, etc. -- and you don't haveto deal with convergence theory and the like) andhow to manipulate them algebraically.]Neat Taylor series application (19 September 2007)http://mathforum.org/kb/message.jspa?messageID=5917119************************************************************************************************************http://mathforum.org/kb/message.jspa?messageID=4487158Carol Hart wrote:http://mathforum.org/kb/thread.jspa?threadID=1338880> My BC students are just beginning Power Series .. and> I have to say how much fun is to teach the concept> with the Foerster text. I used to sort of dread this> relatively difficult concept .. I'm not sure I ever> really understood it completely before I started> using this text. I love the way it is approached.> We also use the "Explorations" text - Advanced> Placement Calculus with the TI-89 (Chapter 9 Infinite> Sequences and Series) from Texas Instruments for> very concise instructions on how to calculate and> graph the problems. I don't know if this is in Foerster's text, but here'ssomething neat that you can do with Taylor expansions.I first saw the idea on this list, back in Fall 2000.I tried it out the next day in a college calculus 2class (we were in the middle of Taylor series at thetime) where each student had a computer equipped withMathematica, and it went over very well, but even withgraphing calculators I've found it's nice (but not asneat as with CAS's).If you graph e^x - (1 + x), the behavior near x=0will look like a parabola.If you graph e^x - (1 + x + (x^2)/2), the behaviornear x=0 will look like a cubic.If you graph sin(x) - x, the behavior near x=0will look like a cubic.Moreover, these aren't just any parabolas or cubics,they appear to have the form (constant)*x^2 or(constant)*x^3.In general, the behavior of f(x) - P_n(x), whereP_n is the n'th order Taylor polynomial about x=0,will look like the graph of (constant)*x^m near x=0,where x^m is the first nonzero term in the Taylorexpansion of f(x) about x=0 that isn't in P_n.This provides a nice graphical illustration forthe remainder term of the Taylor series.You can also do this for expansions about otherpoints besides x=0. Moreover, you can do thisnear the beginning of a calculus 1 course (and Ihave done so many times) to illustrate that thetangent line is the *best* linear approximationin this way: Let L be the equation of a line thatisn't the tangent line and let TL be the equationof the tangent line. Then the graph of y = f(x) - Land the graph of y = f(x) - TL will differ nearthe tangency point in a way that is readilyvisible on a graphing calculator. When you zoomin, the former will look like a line through thex-axis (i.e. has a positive angle of intersectionwith the x-axis) and the latter will look likea parabola tangent to the x-axis (i.e. has a zeroangle of intersection with the x-axis). The ideais that BOTH L and TL can be used to approximatef(x) arbitrarily closely near the point in question(since both lines intersect the graph of y = f(x)at that point), but the error involved with TL is*qualitatively* better than the errors associatedwith any of the L's.This is such a neat idea that I did some searchingto see who brought it up. Here are the posts inthis list where I learned about this graphicalidea:Richard Sisley (November 22, 2000)http://mathforum.org/kb/thread.jspa?messageID=662138Mark Howell (November 26, 2000)http://mathforum.org/kb/thread.jspa?messageID=662142Incidentally, "angles zero measure" are calledhorn angles, and they were apparently even studiedby Euclid:http://aleph0.clarku.edu/~djoyce/java/elements/bookV/defV4.html"The result on horn angles in proposition III.16excludes ratios between horn angles and rectilinearangles. That proposition states that a horn angleis less than any rectilinear angle, hence nomultiple of a horn angle is greater than arectilinear angle."************************************************************************************************************- -------------------------------------------------------------- -------------------------------------------------------------Dave L. Renfro
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