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, ...
- 11.4k
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 ...
- 23k
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 ...
- 23k
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 ...
- 23k
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 ...
- 773
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 ...
- 23k
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 ...
- 11.4k
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 ...
- 21.6k
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 ...
- 296
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.)
...
quid♦
- 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:
$$
f(...
- 556
27
votes
Accepted
How does one tutor an A-level student past the derivative paradox?
There is no royal road to geometry. - Euclid
Nor calculus.
The essence of calculus thinking is really the limit concept. One needs to wrap one's mind around that. Formally: it's the core technique ...
- 21.6k
26
votes
How does one tutor an A-level student past the derivative paradox?
I teach calculus at a community college in the U.S. (2 year college, from which many students transfer to a university). I explain limits from about day two in an informal way ("h gets infinitely ...
- 18.9k
25
votes
Why do we teach that every line is a linear function?
The usage you object to is, in fact, the original meaning of "linear". "Linear" means "having to do with lines". The notion of "linear" in the sense of "linear transformation" is a more modern, ...
- 17k
25
votes
Why do no students know to change the limits of integration when doing substitutions?
Some students are instead taught to change the substitution variable back into the original variable before evaluating the antiderivative at the bounds.
$$\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 ...
- 1,944
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 ...
- 23k
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 ...
- 11.4k
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