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# Tag Info

## Hot answers tagged calculus

61

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 basics of groups, rings and fields? Why, for that matter, teach children to read first, instead of starting with the fundamentals of grammar and linguistics? ...

56

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 from instruction about what to do differently in the absence of a grade or numerical score. To repeat; individual scores discourage learning! (Explanations for ...

50

Here's the example I had which inspired me to post the question in the first place: The game League of Legends was the most-played PC game, in number of hours played, in North America and Europe in 2012. There is a good chance that League of Legends is a part of many of your students' daily life, especially if you are teaching engineering calculus. It doesn'...

48

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 into determinant ones is, generally speaking, the point. If the professor had some other solution in mind, then they made a mistake. They should have chosen a ...

42

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 undergrad, pharmacology major: absolutely no calculus used in biology courses. She actually laughed when I asked her. Grad student: Undergrad biophysics course used ...

42

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 justifies this by differentiating $\sin(x^2)$. In fact, this solution demonstrates deeper understanding of the meaning of these symbols than the variable ...

39

I have no evidence to back this up, nor do I know how I could obtain such a thing. But I strongly believe this is a vocabulary issue, and I would like to see the term "series" phased out of usage in this context. To many minds, "sequence" and "series" convey the same thing: a list of items. In modern parlance, we speak of a television series (a sequence of ...

38

This is a small tip based on the obvious idea that it needs to feel safe to answer questions. Suppose you need to take the derivative of x sin x, but you want students to speak up about it in the flow of lecture. Here are three ways to do it: "Now I need the derivative of x sin x. What should I do first?" "Now I need the derivative of x sin x. Which rule ...

38

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 quantifier on the +C without staring at the notation helps some of them separate the +C from the voodoo magic. I do a similar thing in college algebra classes ...

32

One important point to make is that you should ensure that interaction is part of the culture of your lectures. It isn't enough to pose a question now and again and expect them to suddenly leap in to action to answer it. So you need to be asking questions consistently through the course. The next point I'd like to make is that it will take time for the ...

32

The think-pair-share technique is an oldie but a goodie: Pose a question Give students 1 minute to quietly think of and write down their answer (if you have a computer/projector setup you can use an onscreen timer to enforce the "1 minute" frame) Give students 2 minutes to exchange / compare solutions with a neighbor Ask for volunteers to share results with ...

32

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 success rate seems to have lead to some streamlining and cutting corners in ways that mean that what gets taught doesn't really stick, and also doesn't set students ...

32

The specific identity $$\tag{A} \tfrac{1}{1 - \sin{x}} + \tfrac{1}{1 + \sin{x}} = 2\sec^{2}{x}$$ as such is probably not often encountered, but simplifications akin to $$\tag{B} \tfrac{1}{1-t} + \tfrac{1}{1 + t} = \tfrac{2}{1 - t^{2}}$$ occur frequently. For example, integration via partial fractions ...

31

I agree with vonbrand that it is important to stress that this is a convention that is used sometimes but not others. But I would add the emphasis that all conventions are local. There are places where it is helpful to adopt this convention but other settings in which it would be a disaster. My preference would be to make sure students understand that the ...

31

Bad Optimization Problems I thought that Jack M made an interesting comment about this question: There aren't any. There may be situations where it's possible to apply optimization to solve a problem you've encountered, but in none of these cases is it honestly worth the effort of solving the problem analytically. I optimize path lengths every day when I ...

30

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 not-so-great answers look like?

29

The most intuitive reason I know for $0^0 = 1$ comes from interpretation in terms of functions, namely $$\text{There are } |B|^{|A|} \text{ functions } A \to B \text{ for any finite A and B}.$$ Now, there are no functions $\{\spadesuit\} \to \varnothing$, so $0^1 = 0$, but there exists exactly one function $\varnothing \to \varnothing$ (its set of ...

29

For some reason there is a wide spread view that the $\epsilon-\delta$ definition of a limit is an obscure thing, relevant only to mathematicians, and that the only reason to care about them is to make limits rigorous''. This could not be further from the truth. $\epsilon-\delta$ analysis is all about learning how to control error in the outputs of a ...

29

No, it is a bad idea to avoid indefinite integrals, the reason being simply that your students will encounter them elsewhere, and therefore need to be familiar with them. Calculus is a service course. The purpose of the course is to make science and engineering majors fluent in the language of calculus as used in their fields. Rather than always using ...

28

The different notations are to a considerable extent outgrowth of slightly different ways of looking at "the derivative." To me these different ways is the important thing to discuss, the notation is a by product. And, I think it is quite useful to spend some time on discussing these different ways and to mention the different notations in that context. (...

28

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 distance in kilometers. You can use this to facilitate a discussion about, first of all, the convergence of these ratios $\frac{f_{n+1}}{f_n}\to\varphi$, and also ...

28

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, reducing fractions to lowest form fast, subtracting big numbers" By avoiding using a calculator you'll strengthen the basics.

28

I just start with constant rates of change, where it's pretty blazingly obvious that the chain rule works. E.g., Jane hikes 3 kilometers in an hour, and hiking burns 70 calories per kilometer. At what rate does she burn calories? Our class is generally much more comfortable with the $f'(x)$ notation, and as a result I stayed away from the $\frac{dy}{dx}=\... 27 Safety: When a wrong answer is given, if you can figure out what would have made it right, you help the student feel safer. Teacher: 2*3 is...? Student: 5 Teacher: oh, I bet you're thinking about 2+3. In calculus, teacher: integral of sinx? Student: cosx. Teacher: If I were asking the derivative, you'd be exactly right. What's the derivative of your answers?... 27 I'm no native English speaker, but you can tackle that question from the mathematical point of view as well. The best verb depends on how you view the nature of definite and indefinite integrals. Operators/Functionals Indefinite integrals are operators mapping functions to a set of functions or function (as representative of the equivalence class). ... 27 I would like to encourage consideration of a free textbook. The conventional textbooks are outrageously expensive. (Actually, if your department insists on one of the choices you mentioned, I'd want to go with the cheapest.) Students are suffering from massive amounts of debt these days, with no guarantees of good jobs with which to pay off the debt. ... 27 While all of your students at this point will have done (extensive) units on manipulating and simplifying expressions with exponents, this is the limits unit. When doing limits questions, most students are only searching their brains for limits techniques - they're extremely unlikely to come up with the$\frac{x^a}{x^b} = x^{a-b}$rule that had been driven ... 26 Problem of sloppy notation The notation is sloppy. Your students are justifiably confused. We've just gotten used to it. In order to untangle this, we need the notion of free variables and bound variables. These have somewhat confusing, perhaps even counter-intuitive names. So, I will use "local" as synonymous with "bound" and "non-local" as synonymous ... 25 What are some examples of math history that can be mentioned in calculus classes, either to liven things up or to provide additional perspective / insight on the material being learned? You mention two different goals here. Personally, I have used anecdotes about and discussion of historical events/people in calculus courses to... provide some context for ... 25 Some students are instead taught to change the substitution variable back into the original variable before evaluating the antiderivative at the bounds.$\$\int\limits_0^2 2x\cos(x^2) \;\mathrm{d}x = \int\limits_{x=0}^{x=2} \cos(u) \;\mathrm{d}u = \sin(u) \Big|_{x=0}^{x=2} = \sin(x^2) \Big|_{x=0}^{x=2} = \sin(4) \\\text{versus}\\ \int\limits_0^2 2x\cos(x^2)...

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