61 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 ...
  • 17k
53 votes
Accepted

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 ...
  • 4,868
53 votes
Accepted

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, ...
46 votes

What do you do in order to drag out lectures?

What you are describing is so far outside of my, and I suspect most educators, experience that it appears to be literally incredible. The very strongest Universities in the country, with some of the ...
43 votes
Accepted

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 ...
41 votes
Accepted

Is there a canonical name for a polynomial-like expression allowing for negative powers?

There is also the term "Laurent series", where we allow an infinite number of terms... $$ \dots +5x^{-3} + 2x^{-2} + 3x^{-1} + 2 + 4x - 7x^2+\dots $$ So perhaps yours is a "finite ...
  • 7,006
40 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 ...
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, ...
  • 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 ...
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 ...
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 ...
  • 18.9k
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 ...
33 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 \...
  • 5,660
29 votes
Accepted

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 ...
  • 4,940
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 ...
28 votes
Accepted

What are some rationales to teach Computer Science students Sequences and Series?

Big O and related notations relate closely to these notions. They are often defined using limits. Although it is also possible to define them without using limits, the style of those definitions is ...
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 ...
  • 421
27 votes
Accepted

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.) ...
  • 7,612
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(...
  • 556
27 votes
Accepted

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 ...
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 ...
  • 18.9k
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, ...
  • 17k
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 =...
  • 4,506
25 votes
Accepted

An introductory example for Taylor series (12th grade)

One practical reason for choosing a Taylor Series approximation of a function over the function itself is if you are able to compute using only the four arithmetic operations. For example, if you are ...
  • 10.6k
23 votes
Accepted

How can students self-check derivatives?

As a student currently taking calculus, something I do is select a couple of arbitrary points, and compare my derivative at those points with a small secant: $f'(a) \approx \frac{f(a+h) - f(a)}{h}$ ...
  • 346
23 votes
Accepted

Should we tell students to never replace parts of an expression by their limits when taking a limit?

The important thing is whether students' reasoning is logically valid — and in particular, that they only use the conclusion of a theorem after they've checked that all its hypotheses hold — not ...
  • 4,823
23 votes

How shall we teach math online?

Instructor at University of Washington here - we were one of the early closures, so I feel like we're starting to get the hang of it. Here's what I'm using: Zoom: Zoom is similar to Skype, with ...
22 votes
Accepted

Students strictly follow the steps and notations in sample problems without understanding them

The 25% of students did not think of the problem as easier. It was harder because it did not exactly follow the form of the example problems. As such, it required some (little tiny bit) of creative ...
22 votes

I'm worried that my struggles with calc 2 mean I won't be able to become a professor later

A lot of students seem to make it through high school and well into college with the idea that school is supposed to be easy, and that having to work hard, or being confused at times, or struggling ...

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