Re: Wolfram, meet Stefan and Boltzmann
- To: mathgroup at smc.vnet.net
- Subject: [mg117544] Re: Wolfram, meet Stefan and Boltzmann
- From: DrMajorBob <btreat1 at austin.rr.com>
- Date: Tue, 22 Mar 2011 05:07:58 -0500 (EST)
Awesome work, Helen! Bobby On Mon, 21 Mar 2011 06:13:13 -0500, Helen Read <readhpr at gmail.com> wrote: > On 3/20/2011 5:54 AM, Andrzej Kozlowski wrote: >> >> One reason why the words "primitive" and "anti-derivative" are >> preferred to"indefinite integral" is that they emphasise the non-trivial >> nature of the fundamental theorem of calculus. To say that you can >> compute the definite integral by evaluating an indefinite integral at >> the limits of integration and subtracting sounds almost like a >> tautology; to say that you can compute the integral of a function by >> finding and evaluating its primitive, sounds like the profound result >> that it actually is. >> > > That's why I always use the word "anti-derivative" when first teaching > the concept in Calculus 1. I don't use the term "indefinite integral" at > all until *after* we get to the Fundamental Theorem of Calculus. Even in > Calculus 2, I continue to use the term "anti-derivative" much of the > time, especially when I want to emphasize what we are actually doing. > > In Calculus 1, I start off with approximating the area under a curve > with Riemann sums, and give the class a Mathematica worksheet where they > practice calculating Riemann sums numerically, and and a "lab" > assignment where they not only calculate Riemann sums, but also use some > functions I defined so they can plot the rectangles and get that visual > connection. Then they do a second lab assignment on what I call the > "area function" which is defined as the definite integral of f(x)dx from > (say) 0 to t, that is, the limit of the Riemann sums as \[CapitalDelta]x > -> 0. I show them how to enter a definite integral in Mathematica, > emphasizing that it is the limit of the Riemann sums (and not telling > them how Mathematica or anyone actually calculates that limit). The lab > has them plot f(x) and area(t), figure out how the graph of f(x) > predicts where area(t) is increasing/decreasing, concave up/down, etc., > and notice what happens to area(t) if you change the starting point from > 0 to something else. There's a big text box at the end of the lab where > they have to summarize all this, and also make note of "any other > observations they make" about the connection between the area function > and the original function. > > While they are working on the two lab assignments outside of class, we > finish up the applications of derivatives chapter and introduce > antiderivatives, always calling them that, and not using indefinite > integral notation. The students who did not have calculus in high school > really get it. In that second lab, they will have an aha moment, when > they realize that area(t) is an anti-derivative for the original > function, and how cool that is. I try to time it so that they finish the > second lab a day or so before we arrive in class at the Fundamental > Theorem of Calculus, and on that day I ask them a lot of questions about > the second lab, until they they essentially tell me the Fundamental > Theorem of Calculus. > > The students who had calculus in high school tend not to get it -- they > think "definite integral" is *defined* to be the indefinite integral > evaluated at the endpoints, and don't understand what I'm even asking > about. They miss out on the surprise of the Fundamental Theorem of > Calculus, because they think it is true by definition. > > -- > Helen Read > University of Vermont > -- DrMajorBob at yahoo.com