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It seems to me that Euler's method for solving ODEs is a topic that nicely illustrates the main strategy of calculus, which is to approximate a nonlinear function locally by a linear (or affine) function.

Suppose that a function $y:\mathbb R \to \mathbb R$ is differentiable and satisfies $$ y'(t) = f(y(t)) \,\text{ for all } t \in \mathbb R, \quad y(t_0) = y_0. $$ Here $f:\mathbb R \to \mathbb R$ is a continuous function. If $t_1$ is close to $t_0$, then from the definition of the derivative we have $$ y(t_1) \approx y(t_0) + y'(t_0)(t_1 - t_0). $$ Indeed, this approximation is the key idea of calculus. It follows that $$ y(t_1) \approx y(t_0) + f(y(t_0))(t_1 - t_0). $$ This gives us an estimate of the value of $y$ at $t_1$. We can now repeat this procedure to obtain estimates of the values of $y$ at points $t_2,t_3,\ldots,t_n$, where $t_{i+1} = t_i + \Delta t$ for all $i$ and $\Delta t$ is some small positive number. That's Euler's method.

I think this is accessible to calculus students, and is something they could implement with a computer program to obtain a cool visual result (a graph of $y$). Do you think this is a good topic to include in a first-semester calculus course?

(Maybe the answer is "obviously yes, just like we teach Newton's method." But I'm curious to see if anyone thinks this is a bad idea. Perhaps it would confuse students more than I realize.)

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    $\begingroup$ Reading this, I come to the conclusion that you have never taught a Calculus 1 course. (Except maybe at the Sorbonne or Moscow State.) $\endgroup$ – Gerald Edgar May 18 '18 at 13:08
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    $\begingroup$ I think that customarily this is delivered in the context of a differential equations course. As one example, it's in OpenStax Calculus II in Ch. 4, "Introduction to Differential Equations". $\endgroup$ – Daniel R. Collins May 18 '18 at 13:15
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    $\begingroup$ I teach in a polytechnic university in Spain and here it is routine to teach these things, in these terms, to first-year engineering students. Normally it is taught in a numerical methods course that follows the first semester calculus course, but it is entirely reasonable (to my mind desirable) to include Euler's method in calculus. In practice what is difficult for students to understand initially are two things: 1. the necessity of approximating solutions and 2. that the approximate solution consists of a finite collection of points that somehow approximate the graph of the solution. $\endgroup$ – Dan Fox May 18 '18 at 14:28
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    $\begingroup$ If you do so it might be helpful to employ physical analogies such as velocity, e.g. see V. Arnold's nice exposition of Euler's "broken line" method that I excerpted here. Though it is presented more generally there, it may lend ideas for exposition at lower levels. $\endgroup$ – Bill Dubuque May 18 '18 at 15:15
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    $\begingroup$ Euler's method is fairly standard (at least it used to be) in U.S. Calculus 2 courses (but not as much for Calculus 1 courses), when a brief introduction to separable ODE is covered. A quick google search seems to indicate this is still the case. Euler's method is also covered in AP-Calculus (the BC course), or at least it was just a few years ago. I haven't tutored AP-calculus since 2013 or so, and I haven't kept up with the topics, so the topic might have been dropped since then. $\endgroup$ – Dave L Renfro May 18 '18 at 18:09
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I think that it may be reasonable to spend some time on this, if you have the time for it. The content is, after all, essentially the definition of the tangent line and elementary ODEs are sometimes taught at the Calc 1/2 boundary. However, in a 1st semester calculus class, you should expect to heavily support the students.

For example, the explanation should include a visual component. The explanation you give above will probably not lead to much understanding.

Unless you know for certain that these students have CS skills, writing a program from scratch is probably too much to ask. Write it for them and explain in broad detail what it does. Visualization of the output is important here.

A good set of 1st order ODEs, where they can adjust the parameters, would be useful. (Ex: Exponential, Logistic, $y'=y^3+cy$ for positive and negative values of $c$) A comparison to the exact solutions would also be helpful. Make the step size adjustable so that they can see how convergence works.

Done right, it could be a good place for more qualitative questions and explorations, but for sure don't make them do it by paper and pencil on an exam.

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Personally, if I were not constrained by the heavy chains of course coordination, I would make Euler's method a major theme of my Calculus 1 course.

In fact, I would do Euler's method before defining the definite integral. Then applying Euler's method to $\frac{dy}{dx} = f(x)$ on the interval $[a,b]$ yields a Riemann sum. Defining the limit of this Riemann sum to be the "definite integral" of function $f$, the fundamental theorem of calculus is immediately obvious (assuming they already believe in the convergence of Euler's method). You can reinterpret the definite integral as a signed area later.

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