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Colleagues. In a few weeks I will be interviewed for a position at a community college. I got selected for an interview and the teaching demonstration is as follows:

Assume you are teaching a Multivariable Calculus class (Calculus III):

Suppose $(a, b)$ is a critical point of $f(x,y)$. Introduce the Second Derivative Test with an emphasis on why it is valid for finding local maxima, local minima, and saddle points. (A rigorous proof is not necessary.)

Here's what I have planned:

1) Introduce the second derivative test talking about the input points where the gradient is zero.

2) Show a graphical representation of the points where the tangent plane on the graph of f is flat.

3) Talk about the second partial derivative test and what defines a max/min, saddle etc.

4) Reasoning behind the validity for finding local max, min and saddle points - here's where I need help. A rigorous proof is not necessary so how deep should I go if I only have 15 minutes to present?

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    $\begingroup$ Wow, this is really specific. Why they don't just have you TEACH the class is not clear to me, that would be the typical way to handle such an interview - maybe the timing is wrong. $\endgroup$ – kcrisman May 9 '17 at 20:55
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    $\begingroup$ Personally, I would keep with your graphical representation if you've already started this way in 2). Show all the possibilities for the partials/second partials and what they "look like". And show pictures for when the test doesn't work and what goes wrong. Ideally, interactive graphics. $\endgroup$ – kcrisman May 9 '17 at 20:57
  • $\begingroup$ @kcrisman I'm only limited to a whiteboard and marker. I would have to draw them. I wish I had technology though. Great input though i'll consider this route too. Steve's response below is a valid argument. It IS complicated to explain. $\endgroup$ – Gerardo May 10 '17 at 15:25
  • $\begingroup$ Why they don't just have you TEACH the class is not clear to me, that would be the typical way to handle such an interview No, the brief teaching demo is actually totally standard at the community college where I teach, and AFAIK at all other community colleges in Southern California. We typically interview 8 candidates back to back on a Saturday. There is no way we could schedule that many candidates to teach actual classes at times when the whole committee could be there to observe. $\endgroup$ – Ben Crowell May 11 '17 at 19:09
  • $\begingroup$ Same at Illinois community colleges. $\endgroup$ – Chris Cunningham May 11 '17 at 19:30
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I'm going to focus heavily on the constraints of the original question (15 minutes, community college interview committee) rather than on the mathematical basis for the answer. I think the answer below might actually fit into 15 minutes, if practiced.

The short version of my answer would be

Recall why the second derivative test is valid in one variable. Show a two-variable function and point out there are multiple "second derivatives." Precisely show how to execute the test. Explain why it works in a simple case by an analogy to the single-variable case. Complete a simple example.


My detailed plan

  • (3 minutes) Quickly execute the second derivative test on three or four single-variable functions. Conclude that at a critical point, $y'' > 0$ demands a minimum, $y'' < 0$ demands a maximum, and $y'' = 0$ means anything goes, including maximums, minimums, and weird stuff. Do not say "inflection point;" you don't have time to muddy the water and engage calculus lecturers' pet peeves.

  • (1 minute) Note that in a two-variable situation, there are three (four? More than one, anyway. Don't get side-tracked, this detail is not important in this setting. While teaching a 200-level class, can you avoid being side-tracked by things that are above the level of the course or tangential to the topic at hand? This is part of the point of the interview) second derivatives, so things will be slightly more complicated, but still similar to the "easy case" (Do not say the word "easy;" that's useless and doesn't reassure students).

  • (2 minutes) Introduce a specific, very simple two-variable function and its critical point that will be your focus for the explanation -- one where $f_{xy} = 0$. Maybe like the point $(1, 0)$ on the function $f(x, y) = 4x^2 - 8x + 3y^2$. Point out that if someone asks for "the second derivative" of this function, that is not enough information, because there are lots of second derivatives. Find them.

  • (4 minutes) Carefully write the "formula / process" of the second derivative test starting with something like "Given a critical point of the function $z = f(x, y)$, calculate $H = f_{xx}f_{yy} - f_{xy}^2$ at that point." and continuing until the end. Make sure all the gory details are precisely correct here and in an easy-to-reference box even though the rest of your explanation is not going to be very (mathematically) precise.

  • (4 minutes) Explain that the reason this works is complicated, but it makes sense in simple situations, namely when $f_{xy} = 0$. When $f_{xy} = 0$, the "formula / process" (Do not say the word "Hessian." Do you have a Ph.D.? If so, part of the interview is checking whether you actually want this job or if this is something crappy you are settling for.) just checks whether $f_{xx}$ and $f_{yy}$ agree about there being a maximum or a minimum. When they agree, you get the single-variable answer, when they don't agree, you get a saddle point, and when either one is zero, anything goes. These pictures are easy to sketch, or you can do them with your fingers by making U shapes with the thumb and forefinger of both hands and combining them together.

Quick diagrams and captionsl

  • (1 minute) Go back to your originasimple (Don't say "simple!") two-variable function and calculate H, showing that its critical point is a local minimum.

  • (After time expires) Break character; talk to the committee. Reassure the mathematicians that you know $f_{xy}$ is important and that this logic isn't completely right, but that in the time constraints given, this would be your approach. Reassure everyone in the room that your next step would be to complete a more thorough example problem in detail. Reassure the administration in the room that in an actual classroom, you would use technology to show more accurate diagrams, but that the sketches are also valuable since it shows the students that the idea is in their grasp.

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    $\begingroup$ If you want a pretty example which fails to fall with in the scope of the test I like $f(x,y) = cos(x^2+y^2)$ since $cos(u) = 1-u^2/2+ \cdots $ it's easy to see that $f(x,y) = 1+\frac{1}{2}(x^2+y^2)^2+ \cdots$ hence the Hessian Steve talked about is vanishing at the critical point $(0,0)$. The way I understand the second derivative test is as an application of multivariate power series paired with the real spectral theorem. In other words, Steve is on the money. See page 249 of supermath.info/CalculusIIIspring2017.pdf $\endgroup$ – James S. Cook May 11 '17 at 18:42
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    $\begingroup$ I guess I want to emphasize that Steve is on the money but that neither of Steve's approaches likely fit in the 15 minute time window, since they are too on the money? $\endgroup$ – Chris Cunningham May 11 '17 at 19:30
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    $\begingroup$ Right, you want to ignore the deeper stuff, unless, one of the faculty asks questions which allow such discussion. Probably what you sketch in your answer is good, given the proposed audience sometimes less is more. I'm pretty sure I lost a potential job once from lecturing to the faculty present rather than the hypothetical students. Make sure you know the audience they have in mind for your talk. $\endgroup$ – James S. Cook May 11 '17 at 20:41
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    $\begingroup$ Right -- part of the point of the interview is that for a teaching position, you are looking for a candidate that knows the correct details of mathematics but can still convey some important core part of it to an audience that is at a much lower level of mathematical maturity. $\endgroup$ – Chris Cunningham May 12 '17 at 16:22
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    $\begingroup$ +1 this seems very nice for an actual 15 minute teaching demo. Might want to also add in $z=xy$ to indicate that the $xy$ term is important sometimes. I disagree, however, that students at a low level of mathematical maturity could not go through my second explanation. In fact, this seems like a perfect place to start building some mathematical maturity! There must also be a question of goals: we can teach a student to execute the second derivative test, but what is the point of doing this? I would only teach it if it could convey a nice story of some sort, and actually make sense. $\endgroup$ – Steven Gubkin May 15 '17 at 15:16
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Remember that you are interviewing for a position at a community college, not Harvard. That is not to denigrate community college students at all--they are just a different "audience." I assume that the interviewers already know why the second derivative test is valid for testing the points. I would focus on reminding my listeners exactly what a second derivative is (a recipe for the slope of the slope) and give an understandable explanation from that point of view. My guess is that your interviewers already know that you know it--they just want to see if you can teach it to THEIR students. Good luck!

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  • $\begingroup$ I disagree with most of this; the question stated that the applicant was supposed to focus on the validity of the second derivative test. $\endgroup$ – Chris Cunningham Jun 3 '17 at 19:05
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Here is a more down to Earth approach, which incorporates the same ideas as above.

I think this topic really deserves several days to be done right, not just 15 minutes! So you might have to summarize rather than actually try to fit it all in. Or just do the first parts.

A. Start with some motivation: we want to get to a second derivative test for functions of two variables. This is not as easy as in 1 variable! Reteach second derivative test from single variable using a new perspective: the 2nd degree Taylor polynomial says function "looks like" $f(p)+f'(p)(x-p)+ f''(p)(x-p)^2$ near a point $p$. So when $f'(p)=0$ we have a chance at having a max or min. Because we know $(x-p)^2$ is always greater than zero, we can tell whether max or min by sign of $f''(p)$. If $f''(p) = 0$, we get no info and would have to climb higher up on the Taylor series. We want to do this in two variables now. However, that involves 2nd degree Taylor approximation near $(a,b)$: $$ f(x,y) \approx f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)+ \frac{1}{2}f_{xx}(a,b)(x-a)^2+f_{xy}(x-a)(y-b)+ \frac{1}{2}f_{yy}(a,b)(y-b)^2$$

So to tell a similar story with multiple variables, we will need to have a really good understanding of expressions like $ax^2+bxy+cy^2$

B. First get students to graph some basic quadratic forms. Examples, in increasing order of difficulty:

  • $z=x^2$
  • $z=y^2$
  • $z=2x^2+3y^2$
  • $z=-2x^2-y^2$
  • $z = 4x^2-y^2$
  • $z=7y^2-2x^2$
  • $z=3(x+y)^2+4(x-2y)^2$
  • $z = 2(x-y)^2-(x+y)^2$

C. Make sure that they understand when these quadratic forms are nonnegative, nonpositive, indefinite, and when they are only semidefinite. You do not have to use these words, but get the concepts across.

D. Get them to graph transformed versions of these. For example, what does $z = (x-1)^2+2(y-3)^2$ look like?

D Now give them an arbitary quadratic form like $x^2+3xy+2y^2$ and have them put it into the form $(ax+by)^2+(cx+dy)^2$. For fuller explanation of this see tricki.org/article/Complete_the_square

E Return to the Talyor approximation. Give them some functions and have them compute the quadratic approximation. Then have them complete the square to put this into an "understandable" form. Play with lots of examples. Understand how definiteness, semidefiniteness, and indefiniteness play into this. Especially, see some cases where a semidefinite quadratic form has all kinds of different behaviour (based on what happens in the "flat" direction of the form).

F Only after all of this, complete the square on a general quadratic. Show that the "second derivative test" follows.

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In my opinion, this is a really difficult topic because the "appropriate technology" for understanding this stuff is at a higher level than is appropriate for most Calc 3 students.

Anyway here is how I think about it:

Let $f:\mathbb{R}^n \to \mathbb{R}$ be a sufficiently nice function (let's say 3 times differentiable).

Then the second order Taylor expansion of $f$ at a point $p$ is

$$ f(p+\vec{v}) \approx f(p)+\nabla f\big|_p \cdot \vec{v} + \vec{v}^\top H\big|_p \vec{v} $$

where $H\big|_p$ is the Hessian matrix of $f$ at $p$.

So at a critical point

$$ f(p+\vec{v}) \approx f(p)+\vec{v}^\top H\big|_p \vec{v} $$

Since the Hessian is symmetric (by Clairaut's theorem), there is an orthonormal basis which diagonalizes this matrix, by the real spectral theorem.

The second derivative test comes down to looking at the definitness of the Hessian, which you can read off from the eigenvalues. If the eigenvalues are all positive, the term $\vec{v}^\top H\big|_p \vec{v}$ is also positive, so $f(p)$ is a local min. If they are all negative then you get a local max. Some positive some negative leads to a saddle. A zero eigenvalues means you cannot tell, and you probably have to look at higher terms of the Taylor polynomial.

Generally we only treat this the case that $f:\mathbb{R}^2 \to \mathbb{R}$. The reason is because they do not know about eigenvectors/eigenvalues yet. In the two dimensional case, you can use a trick: determinant is the product of eigenvectors. So pos determinant means they both have the same sign, negative means a saddle.

This story is not really told well anywhere that I have seen.

To translate this into some implementable advice, I would say focus on the idea of "best quadratic approximation". Investigate what the graphs of homogeneous quadratic functions in two variables look like. Pay particular attention to understanding things like $\pm(ax+by)^2\pm (cx+dy)^2$. If you can understand these, you should be able to understand anything by completing the square, but completing the square is harder with two variable polynomials if you do not know some linear algebra (unless someone can educate me in the comments about how to do it!)

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    $\begingroup$ Hmm actually tricki.org/article/Complete_the_square makes completing the square in two variables seem pretty simple! Certainly "lower to the ground" than the real spectral theorem. I think completing the square on a general quadratic form would be a pretty good way to make sense of what is really going on in the second derivative test. $\endgroup$ – Steven Gubkin May 10 '17 at 2:22
  • $\begingroup$ Amazing response. I wish I could present it this way. I did find this thanks to your last paragraph khanacademy.org/math/multivariable-calculus/… $\endgroup$ – Gerardo May 10 '17 at 4:47
  • $\begingroup$ I have a derivation of the min/max of an arbitrary quadratic form via the method of Lagrange multipliers in my calculus III notes. This paired with the multivariate power series concept pretty much derives it. I completely agree that the connection to eigenvalues is horribly under-emphasized. Most students who see calculus III will sometime soon see some matrix theory aka "linear algebra" course where a forward looking comment in calculus III might help draw together ideas in some more cogent whole. $\endgroup$ – James S. Cook May 11 '17 at 18:49

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