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54 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 ...
Adam's user avatar
  • 5,733
54 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, ...
Henry Towsner's user avatar
47 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 ...
Steven Gubkin's user avatar
42 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 ...
Gerald Edgar's user avatar
  • 7,607
41 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, ...
BobaFret's user avatar
  • 836
41 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 ...
Steven Gubkin's user avatar
37 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 ...
Steven Gubkin's user avatar
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 ...
Peter Saveliev's user avatar
36 votes
Accepted

Why not think of derivatives as fractions?

If you have a function $f(x,y)$ where $x=x(t)$ and $y=y(t)$ are themselves functions of a parameter $t,$ and you blindly cancel out differentials, then you can get to incorrect statements like $$\...
Justin Skycak's user avatar
36 votes

What theorems from single-variable calculus break down in the multi-variable context?"

Path-dependency of limits can lead to very counterintuitive results. For instance, consider the following limit: \begin{align*}\lim\limits_{(x,y) \to (0,0)} \frac{x^2y}{x^4+y^2}\end{align*} An ...
Justin Skycak's user avatar
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 ...
Sue VanHattum's user avatar
  • 20.8k
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 ...
Henry Towsner's user avatar
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 \...
Dan Fox's user avatar
  • 5,869
32 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 ...
Daniel R. Collins's user avatar
32 votes

"Real life" examples of limits of functions at finite points

First thing that comes to my mind is the limit $$\lim\limits_{x \to 0} \dfrac{\sin x}{x} = 1.$$ This limit justifies the small-angle approximation $\sin \theta \approx \theta$ (for $\theta \approx 0$) ...
Justin Skycak's user avatar
31 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 ...
Sue VanHattum's user avatar
  • 20.8k
30 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(...
Kevin's user avatar
  • 586
29 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 ...
Daniel R. Collins's user avatar
29 votes

Students can't seem to grasp the intent of tangent lines and getting general trends of derivatives from graphs

Sadly, these students seem to think of math as a bunch of rules. You have done great work, and probably gotten them a little closer. But they are resisting the reasoning that can't easily be put into ...
Sue VanHattum's user avatar
  • 20.8k
27 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 ...
dfsjkhsldfkh's user avatar
27 votes

How to properly define volume for beginner calculus students?

It depends somewhat on the style of the course, but the majority of calculus students do not need a formal definition of volume or area, in my experience. They have studied geometry and (usually) ...
user22788's user avatar
  • 854
27 votes

A visualization for the quotient rule

Depending on how much algebra you allow, you could make the exact same rectangle picture but label the sides $g(x)$ and $q(x)$ with area $f(x)$. This geometrically enforces $g(x)q(x) = f(x)$, aka $q(...
Steven Gubkin's user avatar
26 votes

Why don’t we teach a topological view of continuity instead of epsilon-delta?

Yes. You are crazy to be spending time on another effort to bring real analysis style rigor into a calc 1 course. In this case, even off brand real analysis. Instead of a "different" way ...
guest troll's user avatar
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 =...
Mike Pierce's user avatar
  • 4,845
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 ...
JRN's user avatar
  • 10.8k
24 votes

Why are calculators not allowed in post-secondary exams?

In addition to @BobaFret's answer I'd like like to point out following: You said: Before you downvote this question, I actually want an answer to this. Is the calculator going to give me my ...
flawr's user avatar
  • 409
24 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 ...
Reese Johnston's user avatar
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 ...
Henry Towsner's user avatar
22 votes

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

This is a student who doesn't understand social cues. He only knows that if he is rude you keep trying. If acts disruptive, he will get the answer without working for it. As long as you reward him ...
Amy B's user avatar
  • 8,017

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