Re: Wolfram, meet Stefan and Boltzmann

*To*: mathgroup at smc.vnet.net*Subject*: [mg117498] Re: Wolfram, meet Stefan and Boltzmann*From*: Helen Read <readhpr at gmail.com>*Date*: Mon, 21 Mar 2011 06:13:13 -0500 (EST)*References*: <im4isj$d2o$1@smc.vnet.net>

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

**Re: How to search mathgroup archives... or why does Export on linux need X?**

**Re: PopupWindow vs EventHandler**

**Re: Wolfram, meet Stefan and Boltzmann**

**Re: Wolfram, meet Stefan and Boltzmann**