WeBWorK Problems

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

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

by Christopher Heckman -
Number of replies: 7

This concerns several problems in the Calculus 3 (multivariable Calculus) course, under minimization and maximization.

There are problems such as

Library/Michigan/Chap15Sec1/Q21.pg

Library/Michigan/Chap15Sec2/Q03.pg

(and maybe more) where a student is given a contour map (the level curves) of a function of two variables. Several critical points are plotted, and the student is asked to classify those points (local minimum, local maximum, saddle, etc.).

It is impossible to classify critical points in this manner. For instance, suppose that

G(x) = -17/96 * x^6 - 277/480 x^4 - 1/2 x^2 - 1/10   and   f(x,y)=G(x^2+y^2).

Then f has the following properties:

* If you draw the contour map for f(x,y)=-1, you get a circle (radius 1).

* If you draw the contour map for f(x,y)=0, you get a circle (radius sqrt(2)).

* Contour maps for f(x,y)=R (where R is an integer) are bigger circles if R > 0, and nonexistent if R < -1.

This would indicate that you have a local minimum at (0,0); however,

* (0,0) is actually a local maximum.

It's clear what the intention is, and that the author(s) wanted to find an alternative way to give a "classify the critical points" problem (other than a formula for the original function), but this is not a proper way to do it, in my opinion. I feel that we should not be teaching techniques that don't work, and that these problems should be removed from the library.

What does everyone else think about this?

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In reply to Christopher Heckman

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

by Alex Jordan -

Since the problems cited are contributed from Michigan, my first thought is that it would be up to those contributors to keep or remove problems from their corner of the OPL.

But then if there is a problem where a contour plot is misleading, I would suggest adding more contours until it's not misleading anymore. Or change to a different function. If a contour plot suggests some pattern in the contours, that pattern should really be there. I wouldn't want to view the few contours that are present as minimal information where any pathological happenings are allowed in the regions in between.

I don't entirely follow your example. With f(x,y)=0 or some positive number, there is no contour. The function -17/96 * x^6 - 277/480 x^4 - 1/2 x^2 - 1/10 is negative.

In reply to Alex Jordan

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

by Christopher Heckman -

Adding more contours doesn't help; you get the same problem at a smaller scale.

I appear to have mistyped the example; when I copied and pasted, I missed the x^8 term. Adding 3/16*x^8 doesn't give the right radii, but the general idea is correct:

* The contour where f(x, y) = 0 is a circle.

* The contour where f(x, y) = -1 is another circle (inside of that one).

* There are no other contours inside of the first circle.

* Inside of the circle where f(x, y) = -1, the function increases instead of decreasing (but never reaches the value of 0).

Finally, you said "I wouldn't want to view the few contours that are present as minimal information where any pathological happenings are allowed in the regions in between." (1) This isn't that pathological of an example, when you consider the four important properties, and (2) how do you know you're not working with a pathological example in the problem?

In reply to Christopher Heckman

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

by Alex Jordan -

> The contour where f(x, y) = -1 is another circle (inside of that one).

If we are talking about 3/16*x^8 -17/96 * x^6 - 277/480 x^4 - 1/2 x^2 - 1/10, the contour for  f(x,y)=-1 is two circles, one with radius roughly 1.2 and one with radius roughly 1. At the inner circle the function has already started growing back towards a maximum as you move closer to (0,0).

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. Are you restricting to integer levels? If more granularity were needed to reveal hidden features, you can use more granularity. (But for the function cited, I would just choose a different function.)

> how do you know you're not working with a pathological example in the problem?

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. Depending on your point of view, the exercise is not about accounting for all possible things that might be in there, it's just about helping students grow their visualization skills.

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. The problem has been submitted to the OPL with a creative commons license, which can never be revoked. While the OPL curators could still remove a problem, that would be a very bad thing to do. Just consider an instructor who likes the problem and is using it, and then their WW server admin updates the OPL, and suddenly that instructor's problem set would be broken.


In reply to Alex Jordan

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

by Christopher Heckman -

> 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.
In reply to Christopher Heckman

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

by Paul Seeburger -

I think these two problems are actually quite pedagogically rich!  And I would definitely not want to see them (or other graphically based problems) removed from the OPL.

There are definitely different teaching approaches preferred by different professors.  WeBWorK should allow us each to find (and develop) problems that align well with the way we teach.

One of the real strengths I have found in WeBWorK comes from asking conceptual questions that cannot be done easily by an online solver.  Visual problems like these help student to need to think more carefully about the concepts they are learning and about what the graphs can tell them, rather than only being able to use an algebraic formula to determine the answers procedurally.  I find that this geometric intuition is not too difficult to develop, and it gives students a deeper, more meaningful insight into what's really going on for many of these concepts.

These contour plot problems line up well with the types of questions I ask my students to solve in my multivariable calculus class (in addition to solving them analytically from algebraic functions, of course).  In my way of thinking, if we could not meaningfully guess the location and nature of the critical points and absolute extrema on these contour plots, it would be like claiming that we could not read this kind of useful information from a topographical map, but would need a formula in order to determine the exact answers.  Although we may be able to argue other possibilities, these are quite limited, and it is fairly clear what behavior is implied by these contour plots.

Inside closed contours on a smooth surface, there is either a relative max or min (or possibly both, although this is not common) .  Where a contour crosses itself on a smooth surface, you get a critical point as well, and from the contours you can tell whether it behaves like an ordinary saddle or has some other behavior.

But I know that not everyone sees things this way.  So my point, is really that I value graphical problems in WeBWorK much more highly than skill-based problems in WeBWorK.  I would not want to remove any of these types of problems unless they are incorrect (in which case I would vote to fix them).  I consider them harder to come by and thus a richer resource than the procedural problems that are easier to come by or create.

In reply to Paul Seeburger

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

by Christopher Heckman -
Re: "I think these two problems are actually quite pedagogically rich! And I would definitely not want to see them (or other graphically based problems) removed from the OPL.": In principle, I agree with you. However, in this particular case, I disagree.

Contour maps might tell you a lot about points on the contour, but my point (no pun intended) is that it can't tell you enough, if the point is not on the contour.

I clarify things (I think) in my response to Alex Jordan's response above.