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After such amazing answers I got here for a related question (link at the end if someone still wants to share with me their views)...

Here is the concept:

If you were to create the ULTIMATE Calculus syllabus - a tree of topics - from pre-requisites to fundamentals to basic to intermediate to advanced to very advanced... and beyond...

What would this tree of topics look like? When I say a "tree" I mean a hierarchy of subjects, not just a list of topics.


UPDATE: Reading some comments (thank you @schremmer) - I would definitely appreciate also a narrower question, as follows:

What MIGHT a tree of topics look like for a "Calculus I" course that's required for most science and engineering undergrads?


(Here is my previous question - regarding teaching only by problem-solving: Can mathematics be learned by ONLY solving problems?)

Thank you :)

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    $\begingroup$ In fact, I don't think that there is a true/correct poset/tree of such topics. Yes, one can contrive it. But mathematics, as reality, does not have any natural order to it. "Crazy!" $\endgroup$ – paul garrett Jun 29 '17 at 23:55
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    $\begingroup$ @paul garrett I agree, "mathematics, as reality, does not have any natural order to it." But when we take a walk you and I probably go different ways. Now rephrase the question as, for instance, "What might a tree of topics look like?". Ok, ok. Now specify a kind of students. Incoming freshmen at Harvard? Incoming students at a Community College. Ok, ok. I am probably missing other specifications. But I for one have long taken the question seriously: if you don't specify where you are starting and where you want to go ... Anyway, for better or worse, see my (freemathtexts.org). $\endgroup$ – schremmer Jul 1 '17 at 16:48
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    $\begingroup$ The ultimate calculus syllabus would be at least two different syllabi. At least one should be like the current standard, and at least one should be a one-semester terminal course for biology majors. It's absurd that we're now requiring require future pharmacists and podiatrists to learn integration by parts. $\endgroup$ – Ben Crowell Jul 1 '17 at 21:29
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    $\begingroup$ @schremmer Great point. I'll go ahead and be more specific in my question! $\endgroup$ – Amir Hardoof Jul 3 '17 at 8:47
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    $\begingroup$ @AmirHardoof Harvard undergrads and Community Colleges undergrads are vastly different. Yet, I have known CC students ending up with an MS in engineering. Just because you have a bad leg does not mean you can't run but you have to acknowledge the bad leg. $\endgroup$ – schremmer Jul 4 '17 at 14:00
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Hopefully the short and sweet is best. Here's what I want out of a first semester class regardless of background. I think depending on the individual student they will be ready to engage with these ideas at different levels of rigor and complexity.

  • Calculus I: Concept of Differentiation and Integration, connection through simple ODE's. Students should be able to use these ideas to solve real problems, and constantly be moving back and forth between approximate and exact solutions, discussing accuracy of models and solutions.

Next, depending on the context the first is a more traditional approach(service class) or an alternative more focused on modeling with calculus.

  • Calculus II: Representing functions with series and doing calculus on these. Basically, a complexifying ideas from first semester through tasks that necessitate representing functions with Taylor series. Light connections to work in ODE and PDE problems to motivate what would likely be a class or two in Diff Eq's. Finally, students should experience some work with vectors and higher dimensional problems.

  • Calculus II(alternative): Introduce Taylor polynomials and series through solving differential equations problems, spend half the course here focusing on both solving problems and dealing with the rigor of series conversion. Spend the second half focused on vector calculus and solving some good problems in 3D. I prefer this, and would love if students had an introductory understanding of solving physical problems so important to the emergence of calculus, particularly something of Newton's laws and their use in modeling things like a wave.

In any of these I would want a student to use a computer well to help solve the problems. Python is the best from my perspective in terms of doing calculus with students.

This means there are only two concepts I suppose: Integration and Differentiation. These concepts are related through the FTC, which describes their inverse character. Students need to understand this first.

Second, we can't differentiate or integrate most functions that we would want to. As such, we need approximation techniques which come in the form of Taylor series.

Third, all of these problems above can occur in dimensions higher than 2, and students should understand some of the operational complexity that comes when moving to a higher dimension with the derivative and integral, but also recognize their utility in situations much like those from first semester.

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A contrarian answer: I believe the standard curriculum is optimal. Think Granville. Syllabus should emphasize drill and problems that help a general technical track. This gives best options for most students who don't know their future major. I think incorporating baby real analysis or the like is not optimal. Many students don't need that ever. And those who do can better get it later.

LATER EDIT: Still not the soup to nuts spread you were asking for (as I think real analysis is better done as a recursion into calculus instead of same time), but one guilty pleasure I would have loved as a student would be to work all the way through Edwards (especially the mammoth Tripos tricks filled Integral volumes). You would be a badass at doing integrals if you did all those problems!

https://archive.org/details/treatiseonintegr01edwauoft

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As already pointed out, there is no ULTIMATE calculus syllabus but, considering that a large majority of students in the Precalculus-Calculus sequence are not prospective mathematicians or even prospective physicists or prospective engineers, the following might be worth giving it some thought before reaching for one's gun:

Arithmetic-Algebra. A major emphasis on decimal numbers. In particular, look at fractions and roots as just code for algorithms generating decimals. Then, show how the four operations on Laurent polynomials are a mere continuation, sometimes simpler (No carryover, borrowing, etc), of the four operations on (finite) decimal numbers. Forego just about everything else.

Precalculus-Differential Calculus. As a continuation of the above and in place of a rehash of Algebra and a bad introduction to trigonometry followed by avoidance maneuvers around limits, bring in the differential calculus as the calculus of approximations rather than as the "calculus of change".

So, study algebraic functions by looking at the Laurent Polynomial Approximations of $f(x_{0}+h)$. Define $f^{(n)}$ as the function which outputs the coefficient of $h^{n}$ (divided by $n!$ to permit recursion.). Study similarly the transcendental functions defined from the differential equations.

Integral Calculus. Replace by Dynamical Systems.

Notes.

  1. A treatment such as the above has a much better chance of being "convincing"---as in Stephen Toulmin's The uses of Argument with non---mathematics majors---and for the late mathematical bloomer offers a serious basis on which to ask serious questions about rigor. For more details, see How Content Matters and To Calculate In Calculus.

  2. Also, concerning the use of decimals, see Timothy Gowers: ``physical measurements are not real numbers. That is, a measurement of a physical quantity will not be an exactly accurate infinite decimal. Rather, it will usually be given in the form of a finite decimal together with some error estimate: x=3.14 +/- 0.02 or something like that.'' One way to arrive at the definition of continuous functions and What is so wrong with thinking of real numbers as infinite decimals?

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