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My calculus students diligently learn Newton's Method, the Trapezoidal Rule, Simpson's Rule, and Euler's Method throughout their calculus careers, but the topics always feel like strange digressions into a different field instead of coherent parts of the narrative of the calculus course.

I'm looking for any neat ways to apply these throughout calculus classes instead of following my normal approach, which is to cover them (or skip them when low on time) but fail to convince students that they really fit into our calculus courses.

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    $\begingroup$ I haven't tried this yet, so I cannot vouch, but ... I wonder whether this can be an opportunity to inject a little bit of coding experience into a calculus course. It's easy enough to write a little python script to do Newton's Method, for example. The instructor could have that code ready and show it to students, explaining how it encodes the process (after you teach the main idea visually). And then a homework assignment could have the students play around with specific examples. You could pick a situation that leads to divergence, for instance. $\endgroup$ – Brendan W. Sullivan Sep 30 '20 at 17:21
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    $\begingroup$ I am supposed to teach Newton's method in a couple of weeks. My intention is to take a day to review complex numbers, assert that Newton's method works for complex polynomials, and introduce them to Newton fractals (briefly). I'll try to remember to check back in an tell you how it goes. $\endgroup$ – Xander Henderson Oct 2 '20 at 3:37
  • $\begingroup$ (commenty comment) This is sort of a noble endeavor. But you might consider that coming up with a thematic treatment does not automatically make things better. May even derail from treatment of other topics. And/or the upside benefits from actually deliberately treating these topics as different, small oddities. (Life is prioritization, is time management.) Especially with weaker students, you may find that the extra elegance and theme-y ness benefit are outweighed by time/difficulty involved in expanding the treatment. Just something to weigh...in your uh calculus of benefits. $\endgroup$ – user14746 Oct 10 '20 at 16:22
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Applying Euler's method to the differential equation $F'(x) = f(x)$ over $[a,b]$ gives the approximation $$F(b) \approx F(a)+\sum_{i=1}^N f\left(a+\frac{i(b-a)}{N}\right) \frac{b-a}{N}$$

Taking the limit as $N \to \infty$ yields the Fundamental Theorem.

You can see this visually in the following Desmos app: https://www.desmos.com/calculator/l4sikc9mcd

The piecewise linear approximation to F is exactly what is obtained by following a slope-field given by $y' = f(x)$, aka applying Euler's method.

So Euler's method is a generalization of the Fundamental Theorem, and can actually be used to motivate the Fundamental Theorem.

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  • $\begingroup$ Mod note: This answer originally lived on another question. My following comments are copied from that answer. $\endgroup$ – Chris Cunningham Sep 30 '20 at 14:37
  • $\begingroup$ user1027: I guess I don’t get what others seem to, how Euler’s method helps us get to FTC rather than it being the other way around. I can’t get to equality as N→∞ with proving or applying the FTC. And Euler’s method seems a much bigger lift in Calc I than, say, starting from the fact that the net change in position s(B)−s(A) of a series of motions is equal to the sum of the average velocity times change in time for each motion, and taking the limit as N→∞ to “get” the FTC. $\endgroup$ – Chris Cunningham Sep 30 '20 at 14:38
  • $\begingroup$ Dave L Renfro: @user1027: I took this as a heuristic argument, probably intended for continuous functions. I don't have time now to dig into this (and probably won't anytime soon), but for subtleties about the convergence of various types of Riemann sums, see this 7 November 2007 sci.math post, and for a historical overview of what Riemann actually did, see my answer to Riemann's Contribution to Integration. $\endgroup$ – Chris Cunningham Sep 30 '20 at 14:39
  • $\begingroup$ The trapezoidal rule (or better, the mid-point rule) and Newton's method are practical numerical methods which are the basis for more powerful algorithms that are commonly used. Simpson's rule and the "classic" Euler forward-difference algorithm are not. If you want to prune bits of obsolete history from the syllabus, those two can certainly go IMO. $\endgroup$ – alephzero Sep 30 '20 at 21:27
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    $\begingroup$ @alephzero Simpson's rule is obsolete practically and also doesn't provide much insight. While Euler's forward difference algorithm might not be the best numerically, it can convey a lot of insight into what a differential equation is. It is the first thing you would naively try, and it is definitely worth exploring how well that naive idea works. $\endgroup$ – Steven Gubkin Sep 30 '20 at 22:03
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Calculus students usually see the theorem that a continuous function on a closed interval that takes on different signs at the intervals endpoints must have a zero in the interval. This can be proved rigorously via the bisection algorithm. In addition to being clearer than a more abstract proof, this can be used as a first introduction to numerical methods.

The geometric interpretation of Newton's method requires an understanding of the relation between the derivative of a funciton and the line tangent to the graph of the function. Consequently, Newton's method fits well in the early discussion of derivatives. On the other hand, it gives an application of the derivative (to root finding) that might help motivate interest in derivatives.

And so forth. Nearly every introductory topic can be related with numerical methods. That this is not customary everywhere reflects the inertial resistance to change in elementary curricula.

(However, coding these algorithms should be left for a subsequent class devoted to that task. Many students struggle with coding loops and conditionals and this requires its own pedagogy. But it is easier if the students already understand the algorithms they are asked to code.)

Calculus students usually acquire the mistaken notions that most integrals can be evaluated exactly and that exact expressions/formulas are useful. In fact most computation is numerical and most evaluation of exact expressions/formulas is also numerical. Particularly with students who will mostly use calculus on computers it is important to introduce them early to the issues related to practical computation of integrals and derivatives.

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    $\begingroup$ I had somehow never thought about using the bisection algorithm to prove the intermediate value theorem before, but it totally works! $\endgroup$ – Steven Gubkin Oct 2 '20 at 14:12
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    $\begingroup$ My experience is that it works in practice too. $\endgroup$ – Dan Fox Oct 2 '20 at 17:17
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Analysts would be horrified, but I'm a combinatorialist, so when I teach calculus, I take the viewpoint that a function is first and foremost a table of data. We have a bunch of $x$'s, and for those $x$'s we have values $f(x)$. (The $x$'s are not required to be evenly spaced.) Sometimes the table comes from plugging $x$'s into a symbolic formula - in which case particularly nice things happen, or they are read from a graph, but the model of function I want my students to have in mind first is that of a table of data.

Derivatives are first estimated with no limits (because we have discrete data) and integrals are first estimated numerically, both before any formulas are discussed.

Computations are done in an Excel spreadsheet, so no sigma notation is necessary (though you get the equivalent in Excel's SUM() constructions).

From this point of view, numerical methods are the most fundamental part of a calculus course.

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I've had a relatively good experience doing some of these via spreadsheets. (Google sheets works for everyone who has a web browser.) You can get ahold of some interesting data --- such as the melt rate of glaciers in Greenland --- and cook up the relevant column to apply the SUM() function to and that's your approximation to the integral. You can't let $h \to 0$, but that's what real data is like anyway.

I also did this sort of thing in a lower-than-calc level college class for calculating annuities, Fibonacci numbers, etc. If nothing else, students got a toe-hold on some spreadsheet skills.

Having taught a decent number of CS classes, I can see the temptation to try Python. I would avoid it at this level though; you want to teach math and not do tech support. Nor do you want to explain syntax, for and range, the different semantics of variables in math and cs, etc.

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  • $\begingroup$ Since I have to teach them summation notation anyway, I'm not sure I can avoid teaching them for and range :) $\endgroup$ – Chris Cunningham Oct 2 '20 at 7:21

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