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Calculus vs Analysis

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I can't say whether they would turn them off.  If you introduce the subject well, maybe it can be good.  Just understand that if you're writing to a student studying calculus, they may not have any understanding or exposure to proofs.  So it may be easy to say something about proofs that confuses them.
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Amir Aczel's *Infinitesimal* is about the early historical development of calculus and analysis and goes into some detail about the skepticism that naive calculus met with (some of it justifiied, some not).  Probably too much to fit in a blog post but worth knowing.
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I think some specific ideas from analysis, such as rigorous definitions of limits and continuity  etc are useful. But most of analysis is largely irrelevant to the average calculus student. Exceptional students will get a lot out of some supplementary analysis material though.
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You are being required to write a *blog post* for a course?

Anyway, it is a bad idea to target a post about proofs to an audience that has little to no experience with them. So do not blog about proofs. Instead, consider writing about the strange examples found in the 1800s whose unexpected properties led to the recognition of the need for more rigor in calculus: infinite series of continuous functions that are not continuous, sequences of functions where a limit and integral can’t be interchanged, continuous nowhere differentiable functions, and so on.
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I studied calculus before analysis. The proof of the Squeeze Theorem was a good introduction for me. Maybe pick that one or another well known concept to do it on?

Honestly I didn’t know you could do analysis before calculus. In my program, calculus was the first year and real analysis was in my last year.
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I don't think it is right to say that Calculus doesn't have proofs. That might be an insult to two great mathematicians, Newton and Leibniz. Why don't you just say that Analysis is the wider branch of mathematics that was developed and inspired by the rudiments of Calculus.
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Although, not directly related to your question, I want to know what EU students use to learn Analysis. I hear that they don't study Calculus (watered down) as a separate course like they do in the USA.
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Well, here are my two cents:

in this case, I think of Analysis as a rigorous study, including writing proofs to build up the theorems, lemmas, etc. of Calculus. When we think of "Calculus", I think of the volume of theorems built up through proof, to include the practical application of those theorems. When we take a single variable Calculus class, for example, we're learning about the big ideas proven through Analysis and how to use them.
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>I have studied only analysis and from what I know calculus is supposed to be analysis without proofs.

In my opinion, this is wrong.

"Calculus" ("differential and integral calculus", "calcul différentiel et intégral" or "Differential- und Integralrechnung") is an older name for the mathematical discipline that came to be known as analysis. To the extent there is a distinction between these two terms, it is that calculus deals more specifically with those aspects related to the processes of differentiation and integration.

In English-speaking countries, within the context of education, the use of the word "analysis" never caught on other than in advanced university courses, while in continental Europe, the reverse occurred, with the term "analysis" displacing the older usage of "differential and integral calculus".

The books "Calculus" by Apostol and Spivak and the book by Courant originally published in 1927 under the title "Vorlesungen über Differential- und Integralrechnung" would be described as nothing but rigorous, with full proofs. There are courses taught in American universities from these books, also under the title "Calculus." On the other hand, the subject "Analyse" is taught in school in many European countries without many proofs or in any case at a less rigorous level than the books above. No doubt first-year courses are far more rigorous on the whole in Germany than they are in the U.S., but this has nothing to do with an inherent distinction between the words "calculus" and "analysis."

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