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How can I help a student who has a "wrong" kind of enthusiasm?

On a practical level, give extra credit assignments about things she is enthusiastic about. Give generous partial credit on these assignments for original ideas that don't necessarily work out. Let ...
Kostya_I's user avatar
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3 votes

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

One elementary property that fails already for functions of one variable with values in $\mathbb{R}^n$ is the chord-tangent property. For example, for the helix $$ h:t\mapsto (\sin t,\cos t, t),\quad ...
Kostya_I's user avatar
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10 votes

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

Existence of antiderivatives. Every continuous function $f: \mathbb{R} \to \mathbb{R}$ has an antiderivative. On the other hand, a continuous vector field $f: \mathbb{R}^d \to \mathbb{R}^d$ for $d \ge ...
Jochen Glueck's user avatar
2 votes

Advice and Remedial Algebra Resources for Students Committed to Calculus

I know this doesn't directly answer your question which is about helping one student. But I would be wanting to see my institution implement sections of calculus with support. This blog addresses that ...
Sue VanHattum's user avatar
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-2 votes

Advice and Remedial Algebra Resources for Students Committed to Calculus

My (paradoxical and politically incorrect) advice is to stick with the class they are accelerated into. but to do every single problem. Yes, even when the soft Millenials complain. Do every single ...
guest troll again's user avatar
24 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
17 votes

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

Critical points of smooth functions in one-dimension tend to be local extrema. This is because $f''(c)=0$ is an unlikely event. When $f'(c)=0$, we expect a local min or max, depending on the sign of ...
user52817's user avatar
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1 vote

What is the terminology for "self-referral" integrals in calculus?

From Does integration by parts with "deja vu" have a name?: Sheard ("Trick or Technique?", 2009) calls it the one-step algebra trick; OP says he saw it called integration by parts ...
user182601's user avatar
3 votes

Motivating a definition of "gap" in a line just barely more advanced than the one used in the typical first-year calculus course

A possibly motivating example might be the following: let $S$ be the set of all positive numbers whose square is not $3$. This description of $S$ seems operationally more straightforward than "...
Will Orrick's user avatar
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5 votes

Motivating a definition of "gap" in a line just barely more advanced than the one used in the typical first-year calculus course

How to explain to students why statements avoiding all mention of points not within the linearly ordered set being considered are better than statements that mention such points? You can't. Because ...
Daniel R. Collins's user avatar
1 vote

Why are calculators not allowed in post-secondary exams?

I administer individual math assessments to students. I think students need to learn to understand basic math concepts and apply them to every day situations. The best way to do that is to learn basic ...
Ron's user avatar
  • 11
6 votes

Motivating a definition of "gap" in a line just barely more advanced than the one used in the typical first-year calculus course

Your definition of a "gap" is reminiscent of a Dedekind cut. If we further assume $A$ and $B$ are sets of rational numbers and that $A$ and $B$ form a partition of ${\bf Q}$, then we have a ...
user52817's user avatar
  • 10.5k
12 votes

Motivating a definition of "gap" in a line just barely more advanced than the one used in the typical first-year calculus course

In my experience, the only way to successfully teach a more sophisticated technique is to present a problem where known simpler techniques fail. For instance, anyone who's taught algebra to kids will ...
Justin Skycak's user avatar

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