65 votes
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Is it worth grading calculus homework?

The evidence says no What research I'm aware of is all about how giving any overall data about their own performance is actively harmful in promoting further learning. They learn considerably more ...
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  • 1,912
62 votes

Why would you teach Calculus before teaching Real Analysis?

You may as well ask: Why teach elementary school children how to perform whole-number arithmetic without teaching them the Peano axioms first? Why teach high-school Algebra without starting with the ...
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  • 16.3k
52 votes
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Grading a limit problem

The student changed something which was indeterminate ($\infty-\infty$) into something which was not ($\infty\cdot \infty$). How does that not merit a perfect score? Changing indeterminate expressions ...
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  • 4,784
52 votes
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How rigorous should high school calculus be?

Not very rigorously at all, but that doesn't (and shouldn't) mean just memorizing calculations. (I should add that I'm basing this on my experience teaching calculus to non-major college students, ...
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48 votes

Should we avoid indefinite integrals?

On quizzes, homeworks, and tests, I repeatedly ask questions like this: Find three different functions that have derivative equal to $x^2 + x$. Forcing them to do antiderivatives and deal with the ...
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  • 19.1k
43 votes
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Intergration by differentiating will get you $0$ marks - but how to explain why?

Your student should get full marks. In fact, I would say that even a more complicated example, like $$\int 2x\cos(x^2) dx = \sin(x^2) + C$$ should be awarded full points as long as the student ...
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42 votes
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How is calculus helpful for biology majors?

I'm an old-school biologist (animal physiology) who works with mostly cell biologists. I sent out an email to a bunch of grad students and postdocs I work with. Here is the data so far: Senior ...
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  • 1,494
38 votes

Why are calculators not allowed in post-secondary exams?

As a professor/teacher I have some insight. You just answered your own question: "my level of math right now is not basic. So, like many, I tend to forget the basics of math, like adding fractions, ...
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  • 786
36 votes

What is the best way to intuitively explain the relationship between the derivative and the integral?

You start by noticing that the Riemann sums (multiplication followed by addition) and the difference quotients (subtraction followed by division) undo each other. Their limits -- the integral and the ...
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35 votes

Why is differential calculus often presented before integral calculus?

While there are no theoretical difficulties with developing integration first ("from scratch" measure theory books demonstrate this), it presents some pedagogical challenges. Most problems ...
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35 votes

Do I really need to cover solids of revolution in my Calculus I class?

An operation is born when we recognize the regularity in repeated reasoning. Take multiplication for example. If we are living lives which involve even a modest amount of arithmetic thinking, we will ...
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33 votes

Grading a limit problem

If this is calc I, that deserves a 5/5. If this is analysis, it depends on what you taught them. Don't you set up a grading rubric ahead of time? What do the 5 point answers look like? What do other ...
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  • 17.3k
33 votes

Difference between high school and college calculus courses

Assuming we're talking about mostly US students, most American high schools teach calculus in a way that's very focused on the AP test. The pressure to get students through that with an adequate ...
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32 votes
Accepted

How important is knowledge of trig identities for use in Calculus

The specific identity \begin{equation}\tag{A} \tfrac{1}{1 - \sin{x}} + \tfrac{1}{1 + \sin{x}} = 2\sec^{2}{x} \end{equation} as such is probably not often encountered, but simplifications akin to \...
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  • 5,083
29 votes
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Near-universal student mistake on $\lim_{x\rightarrow\infty}e^{x+1}/e^x$

While all of your students at this point will have done (extensive) units on manipulating and simplifying expressions with exponents, this is the limits unit. When doing limits questions, most ...
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  • 4,850
28 votes
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What interesting properties of the Fibonacci sequence can I share when introducing sequences?

Since $\varphi$ is rather close to the conversion rate between miles and kilometers, one can use the Fibonacci numbers to convert: if $f_n$ is the distance in miles, then $f_{n+1}$ is (roughly) the ...
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28 votes
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What is a good reason to change calculus texts?

I would like to encourage consideration of a free textbook. The conventional textbooks are outrageously expensive. (Actually, if your department insists on one of the choices you mentioned, I'd want ...
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  • 17.3k
28 votes

Tutoring a recalcitrant/awkward/exasperating student---special needs?

How do I reach [this] kid? Let me be blunt: You probably don't. This is a person who is so intransigent that you effectively need to black-tag them. A hard lesson is that you can't save everyone. At ...
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27 votes
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What is the proper verb for "doing" an integral?

I'm no native English speaker, but you can tackle that question from the mathematical point of view as well. The best verb depends on how you view the nature of definite and indefinite integrals. ...
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  • 3,362
27 votes

How to convince students of the integral identity $\int_0^af(x)dx=\int_0^af(a-x)dx$?

Problem of sloppy notation The notation is sloppy. Your students are justifiably confused. We've just gotten used to it. In order to untangle this, we need the notion of free variables and bound ...
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  • 421
27 votes
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Using $dx$ for $h$ in the definition of derivative

To use $$f'(x)=\lim_{dx\to0}\frac{f(x+dx)-f(x)}{dx}$$ is mathematically correct if $dx$ is the name for a real variable. (If it should be something else it needs to be made clear what it should be.) ...
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  • 7,582
27 votes

How to explain what's wrong with this application of the chain rule?

The root of the difficulty is that $x$ appears free in $f(z)$, but we are trying to "capture" it with $g(x)$, which is illegal. When we substitute $g(x)$ into $f(g(x))$, we have a variable clash: $$ f(...
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  • 556
27 votes
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How does one tutor an A-level student past the derivative paradox?

There is no royal road to geometry. - Euclid Nor calculus. The essence of calculus thinking is really the limit concept. One needs to wrap one's mind around that. Formally: it's the core technique ...
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26 votes
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Historical tidbits to liven up calculus classes

What are some examples of math history that can be mentioned in calculus classes, either to liven things up or to provide additional perspective / insight on the material being learned? You mention ...
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26 votes

How does one tutor an A-level student past the derivative paradox?

I teach calculus at a community college in the U.S. (2 year college, from which many students transfer to a university). I explain limits from about day two in an informal way ("h gets infinitely ...
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  • 17.3k
25 votes

Why do we teach that every line is a linear function?

The usage you object to is, in fact, the original meaning of "linear". "Linear" means "having to do with lines". The notion of "linear" in the sense of "linear transformation" is a more modern, ...
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  • 16.3k
25 votes

Why do no students know to change the limits of integration when doing substitutions?

Some students are instead taught to change the substitution variable back into the original variable before evaluating the antiderivative at the bounds. $$\int\limits_0^2 2x\cos(x^2) \;\mathrm{d}x =...
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  • 4,412

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