WeBWorK Problems

Classifying critical points in multivariate Calculus via contour map: A bad idea?

Re: Classifying critical points in multivariate Calculus via contour map: A bad idea?

by Christopher Heckman -
Number of replies: 1

> If we are talking about 3/16*x^8 -17/96 * x^6 - 277/480 x^4 - 1/2 x^2 - 1/10,

I'm not talking specifically about it, because the numbers don't match up exactly. (They did for the original function, but I lost that formula.) I'm talking about a function that meets the four conditions that I posted. I go into this a little more in depth in the next reply.

> And what I meant about adding more contours is that you could add more contours like for z=-1/2 to show even more clearly that the function is climbing towards a peak.

Yes, that's what I thought you meant, too. However, that still won't work, because then the contour map won't show whether the function decides to start decreasing once it gets inside the z = -1/2 contour. If the innermost contour is z=a and the next is z=b, with a>b, there's nothing to prevent the function from decreasing to (a+b)/2 once it's inside of the inner contour.

You need an infinite number of contours to show that this is not happening; no finite number will do.

> If the instructor has assigned this problem, it is fair for everyone to assume there are not important hidden features that are omitted because of the particular granularity of the contour map.

In this case, it's not pathological. But do we want to teach a technique that doesn't give the right answer 100% of the time? In math, the answer is no; otherwise, we'd allow the derivative of 2^x to be x * 2^(x-1), because "most of the time they'll see polynomials, and 2^x is a pathological example."

And what if the instructor doesn't notice that there is a potential problem?

> After my earlier post, I realized I left out what I think is the most important counter point to your proposal to remove such problems. No one has the right to do that.

Not even if it's blatantly wrong? What if I wrote a multiplication problem where the "correct" answer to (-2)*(-3) was -6 and uploaded it to the library? 

Maybe the problem can't be removed, but it should come with a warning.

In reply to Christopher Heckman

Re: Classifying critical points in multivariate Calculus via contour map: A bad idea?

by Robert Mařík -
I also support to keep the problem in the library. Ch. Heckman is right, anything may happen inside the innermost contour. However, to solve real world problems we never have analytical representation of functions. And even in this case we want to interpret numerical data. Imagine any output of finite element simulation. For these problems we typically assume that if the value of a quantity increases when approaching a point, there is no sudden drop inside the innermost contour.

If I remember correctly, there is a section in PGML file for comments and these comments are visible when browsing OPL library. So it is possible to keep a message there. Something like "The problem assumes that the function values can be predicted from the contours. Do not use this problem if you are not satisfied with this assumption."

Of course, the graphs for Library/Michigan/Chap15Sec1/Q21.pg could be better (include colors, do not include numbers on axes). And Library/Michigan/Chap15Sec2/Q03.pg could mention the tolerance used to evaluate the answer. But this is another story.