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Why do some (pre-) calculus text allow $r<0$ in polar coordinates?

There is still a simple geometric meaning for polar coordinates when we allow for negative values of $r$. (It's how I was taught polar coordinates.) Given an ordered pair $(r, \theta)$, Start at the ...
Justin Hancock's user avatar
9 votes

Intuition for order of operations in compound transformations

For teaching this concept, I always come back to tables of values. In Calculus Reform, there was the "Rule of Three" which emphasized that learning about functions should involve tables, ...
user52817's user avatar
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8 votes

Why do some (pre-) calculus text allow $r<0$ in polar coordinates?

It would be nice if we distinguished between "polar coordinates" and "polar parameterizations". Let $\Omega \subset \mathbb{R}^2$ be an open set which does not contain any loop ...
Steven Gubkin's user avatar
8 votes

Sources on inequity in precalculus sequence

Here's a hair-raising article I sometimes use as a touchstone: Kenschaft, Patricia Clark. "Racial equity requires teaching elementary school teachers more mathematics." Notices of the AMS 52....
Daniel R. Collins's user avatar
5 votes

Educational resources commonly address slant asymptotes. Why not general polynomial asymptotes?

This is kind of a joke answer, but in my favorite math story ever we have the following exchange: Eric pondered a moment. "But... but if it's that simple, why don't my textbooks talk about it?&...
Steven Gubkin's user avatar
4 votes

Chinese and Japanese most important high school textbooks

The only full-length English translations of Chinese or Japanese secondary math textbooks that I'm aware of are the ones listed here, which were produced by the University of Chicago School ...
Justin Hancock's user avatar
4 votes

Intuition for order of operations in compound transformations

One way to build intuition around this is to think about the effects on the intercepts. Vertical Shifts/Stretches: Effect on $y$-Intercept Suppose you want to graph $y=2\sqrt{x}-6$ using ...
Justin Skycak's user avatar
4 votes

Generating function example

Why not give the generating function for the Catalan numbers as an example? For a quick overview: The Catalan numbers ($C_n$, A000108) is defined as the number of ways that parenthesis can be ...
CrSb0001's user avatar
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4 votes

Why do some (pre-) calculus text allow $r<0$ in polar coordinates?

Allowing $r<0$ certainly does not de-emphasize the geometric picture. The image below is the graph of $r=\sin(3\theta).$ As $\theta$ goes from $0$ to $\pi/3,$ $3\theta$ goes from $0$ to $1$ and ...
Michael Hardy's user avatar
4 votes

Generating function example

I suggest using generating functions to find two dice whose sum has the same probability distribution as a pair of d6's. You can find this written up in the Sicherman Dice wikipage (where the section ...
Benjamin Dickman's user avatar
3 votes
Accepted

Generating function example

Chapter 6 of "Applied Combinatorics" by Alan Tucker contains several elementary but powerful examples and exercises of generating functions. Example 4 asks for the generating function of the ...
Mike Z.'s user avatar
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3 votes

Why do some (pre-) calculus text allow $r<0$ in polar coordinates?

Polar curves are a very old subject. This article, published in 1949 in the American Mathematical Monthly, suggests that the subject goes back to Newton. Newton as an Originator of Polar Coordinates. ...
user52817's user avatar
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3 votes
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Educational resources commonly address slant asymptotes. Why not general polynomial asymptotes?

I think it is just that the related concepts encompassed by little-$o$ and big-$O$ notation are more important than "polynomial asymptotes" and do find many applications. We do teach Landau ...
user52817's user avatar
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3 votes

Intuition for order of operations in compound transformations

I'd say the root cause of your problem is that you're forgetting to include the square root in the sequence of operations transforming $x$ into $\sqrt{2x-6}$: $$ x \overset{\times 2}{\ \to\ } 2x \...
Ilmari Karonen's user avatar
2 votes

Generating function example

How many ways is there to split a given sum of money into coins? How does this number grow? If the available coins denominations are $n_1,n_2,\dots,n_k$, then the answer is given by coefficients of ...
Kostya_I's user avatar
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2 votes

Intuition for order of operations in compound transformations

I think that my previous answer addresses this. If we name the intermediate functions, and make the relationships between these functions explicit, things become a bit easier to understand. In your ...
Steven Gubkin'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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1 vote

Why do some (pre-) calculus text allow $r<0$ in polar coordinates?

There is a clear interpretation of negative angles, and we allow angles to be negative. There is a clear interpretation of negative radii, so it's only natural (or at least "consistent") to ...
Justin Skycak's user avatar
1 vote

Intuition for order of operations in compound transformations

Is there a good intuitive explanation for how to think about the order of steps when constructing these compound transformations? Here's a stab at what it would mean to obey the order of operations. ...
Nick C's user avatar
  • 9,709
1 vote

Intuition for order of operations in compound transformations

You can perform the transformations in any order you want as long as you respect this rule: Always transform completely inside, affecting $x$ directly, or completely outside, affecting $y$ directly. ...
Stef's user avatar
  • 556

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