> 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 reasoning is employed. Indeed, there seems to be little more than substituting one black box (the statement that L'Hopital's rule works) for another black 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), where f(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 does the student come away with? Probably not much more than this: L'Hopital's rule works when it works and it doesn't work when it doesn't work.
Below (between the parallel --- lines) is a May 2009 post of mine in another discussion group that gives a calculator activity that I think will do a better job of developing critical thinking while also reinforcing useful concepts. (Embedded in this post is another still earlier post, this one between the parallel **** lines.)
"Bibliographic information" for the post that follows:
> 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 do in either an AB class or a BC class. If it's in an AB class, you can get more mileage out of the idea by introducing the "quadratic approximation of a function at a point", which I've found to be fairly easily understood and worked with by students who have already worked with the idea of the linear approximation of a function a point.
While I was looking up my post about this topic (copied in full below), I came across another post of mine that might also be something to try, although in this case, if it's an AB class, you'll first want to explain the idea of higher order polynomial approximations to a function at a point. You don't need to explain how to get them -- although an easy mechanical method is to assume sin(x) = a + bx + cx^2 + ..., plug in x = 0 to get a = 0, differentiate both sides and then plug in x = 0 to get b = 1, differentiate both sides again and then plug in x = 0 to get c = 0, differentiate both sides yet again and then plug in x = 0 to get d = -1/6, etc. -- and you don't have to deal with convergence theory and the like) and how to manipulate them algebraically.]
> 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's something 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 2 class (we were in the middle of Taylor series at the time) where each student had a computer equipped with Mathematica, and it went over very well, but even with graphing calculators I've found it's nice (but not as neat as with CAS's).
If you graph e^x - (1 + x), the behavior near x=0 will look like a parabola.
If you graph e^x - (1 + x + (x^2)/2), the behavior near x=0 will look like a cubic.
If you graph sin(x) - x, the behavior near x=0 will 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), where P_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 Taylor expansion of f(x) about x=0 that isn't in P_n.
This provides a nice graphical illustration for the remainder term of the Taylor series.
You can also do this for expansions about other points besides x=0. Moreover, you can do this near the beginning of a calculus 1 course (and I have done so many times) to illustrate that the tangent line is the *best* linear approximation in this way: Let L be the equation of a line that isn't the tangent line and let TL be the equation of the tangent line. Then the graph of y = f(x) - L and the graph of y = f(x) - TL will differ near the tangency point in a way that is readily visible on a graphing calculator. When you zoom in, the former will look like a line through the x-axis (i.e. has a positive angle of intersection with the x-axis) and the latter will look like a parabola tangent to the x-axis (i.e. has a zero angle of intersection with the x-axis). The idea is that BOTH L and TL can be used to approximate f(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 associated with any of the L's.
This is such a neat idea that I did some searching to see who brought it up. Here are the posts in this list where I learned about this graphical idea:
"The result on horn angles in proposition III.16 excludes ratios between horn angles and rectilinear angles. That proposition states that a horn angle is less than any rectilinear angle, hence no multiple of a horn angle is greater than a rectilinear angle."