SUNY Geneseo Department of Mathematics
Friday, March 31
Math 223
Spring 2023
Prof. Doug Baldwin
(No.)
GROW STEM is offering a workshop on “Bystander Intervention in Scientific Settings” on recognizing and intervening to minimize microaggressions in STEM work/education settings.
Tuesday, April 4, 5:00 PM, ISC 115.
If you’re interested in applications of calculus and vectors to computer graphics, consider Math 384, “Computational Graphics,” for next fall.
I teach it, TR 9:30 - 10:45.
It looks at the math behind graphics, particularly an approach called “ray tracing.”
Prerequisites are Calculus 3, Linear Algebra 1, and an introductory programming course.
Problem set 9 is ready.
Mostly on directional derivatives and gradients, with a bit more.
Work on it next week, grade it the week after.
See the handout for more details.
Based on “Critical Points” and “Second Derivative Test” in section 3.7.
The definition of a critical point: a point at which either some derivative doesn’t exist, or all derivatives equal 0.
The definition of a saddle point: a point at which all derivatives are 0, but the function at hand doesn’t have a local minimum or maximum.
The definitions of local minimum/maximum: a point at which the value of the function is smaller/larger than the values at all points in some disk around that point.
The second derivative test and its interpretation (see the book and the example below for details).
Fermat’s Theorem: basically, local minima and maxima can only occur at critical points (just like Fermat’s Theorem for single-variable functions).
Why does the discriminant work to tell minima from maxima from saddle points? And why is 0 an “inconclusive” value for it?
I don’t know for sure, and will see if I can find out more.
One observation though deals with the ∂2f/∂x2 ∂2f/∂y2 term: that term is positive exactly when f has the same concavity in the x direction as in the y direction, and having the same concavity in both directions would seem to be a prerequisite for having a minimum or maximum. Having different concavities, on the other hand, is characteristic of the classic saddle point. So this thinking about the sign of ∂2f/∂x2 ∂2f/∂y2 aligns well with how to interpret the sign of the overall discriminant. Not sure what the role of the ∂2f/∂x∂y term is though.
Find and classify the critical points of f(x, y) = x3 - 3x - y2 + 4y + 6.
Start by finding the partial derivatives and figuring out where they are 0 (and, in general, undefined, but that doesn’t happen with this function):
So there are critical points at (-1, 2) and (1, 2).
Now use the second derivative test to figure out what happens at each critical point. For this, we need to start by finding the second derivatives and plugging them into the discriminant formula:
Finally we can evaluate the discriminant at each critical point. Interpreting the result is a two-step process: first, look at the sign of the discriminant to tell extrema from saddle points. Second, if the discriminant is positive, look at the sign of ∂2f/∂x2 to distinguish minima from maxima:
In terms of deciding whether an extreme is local or global, there are a couple of tricks you can use. If an extreme value is at the only critical point, then it has to be a global minimum (maximum), because to have a smaller (larger) value elsewhere you’d have to have another critical point where the function can change its direction of growth. But a saddle point will do to “change direction of growth,” as someone pointed out in class, it doesn’t have to be another local minimum or maximum. You can also look at the function’s end behavior, e.g., if it goes to positive or negative infinity as one or more variables grow, then no finite value can be other than a local minimum or maximum.
Absolute extreme values and closed regions, i.e., including boundaries in extreme value calculations.
Please read “Absolute Maxima and Minima” in section 3.7 of the textbook.