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.
>
> --