If I was to grade myself for my effort this week I’d give myself somewhere between a B and a B+. I made it through five videos, one exercise and 3.5 articles, and probably studied for ~7–8 hours. Part of me wishes I would have made more progress but I worked through some tricky-ish concepts (some of which sunk in, some didn’t) so, all in all, I’m happy with what I was able to learn. Everything I worked through this week pertained to the SPDT, the Second Partial Derivative Test. This test tells you if a critical/stabile point on a 3D function (i.e. a point where the gradient = 0) is a local max/min or a saddle point. Working through SPDT questions gave me a lot of good practice doing partial derivatives and a much stronger intuitive understanding of what a second derivative represents. So the quantity of what I learned this week wasn’t a lot, but I’m happy with the quality of what I learned and the clarity that came from it. 😬
I began this week watching the third, fourth and fifth videos from the section Optimizing Multivariable Functions. These three videos introduced and explained what the second partial derivative is used for and how/why it works. Here are some screen shots from those videos and a page from my notes that talks about the formula:
In the second and third screen shot above you can see that the 3D function has been sliced across the x-axis and y-axis, respectively. As you can see, both partial derivatives show an upwards facing parabola. This indicates that the second derivatives for x and y (which I think of as the function’s acceleration) is positive, i.e. the concavity of the function is positive across both axes. However, it turns out that this is actually a saddle point because even though the second partial derivatives in the x- and y-directions are both positive, the second partial derivative in the diagonal axes are negative.
I’m not going to try to explain the SPDT equation, fxx(x0, y0)fyy(x0, y0) – (fxy(x0, y0))2, because I don’t completely understand why it works. In terms of how it works, if the product of the second partial derivative of x and y at (x0, y0) is positive and greater-than (fxy(x0, y0))2, then there’s a local maximum or minimum. If it’s negative or less-than (fxy(x0, y0))2, then it’s a saddle point.
The final two videos in this section went through an example question of how to find critical points in a two input, one output function and use the SPDT to determine if the critical points are maximum or minimum points, or saddle points. Here are a few screen shots from those videos and the notes I took as I worked through the question:
The process to solve these questions is lengthy but not too hard. The first thing you need to do is find the partial derivatives of the function, set each partial derivative to equal 0, and then solve both equations to find what values of x and y are equal to 0. You then input those critical points into the SPDT and find out if the output is positive or negative at each critical point. If the output is negative, it’s a saddle point. If it’s positive, then you have to go back and look at either fxx or fyy to see if they’re either positive or negative. (It doesn’t matter which second derivative you look at because if it’s a max/min point it means the concavity for both partial derivatives will be the same.) If the second derivatives are positive, it’s a local minimum point because the function is – the way that I like to think about it – accelerating, i.e. it’s increasing across that axes, i.e. it has upwards concavity. It’s a minimum point if the reverse is true and the second partial derivative is negative.
The one exercise I did this week had me work through questions just like the one above. Below are three example questions I did. I didn’t understand the part of upwards/downwards concavity based on the sign of the second partial derivative until after working through a few of the questions which is why I got the second question wrong.
Question 1
Question 2
Question 3
On Thursday I got started on the following section, Optimizing Multivariable Functions (Articles), which had five articles in it and no videos or exercises. The 3.5 articles I made it through helped me better understand critical points and gave me a bit more practice with the SPDT, which was useful. I screen-shotted (is that a verb?) two things that I thought were noteworthy. The first made the SPDT easier to understand by showing how it relates to the Hessian matrix:
I think I’d actually made that connection between the SPDT and the Hessian matrix before reading this, but seeing it made the whole thing click for me. The other thing I made a note of was much less intuitive and had to do with the math behind the quadratic approximation formula. As you can see below, the linear algebra used in the quadratic approximation formula is a bit tricky and combines the Hessian matrix with a given set of coordinates, (x0, y0). Here it is:
I can somewhat follow along with what’s going on but find the math a bit overwhelming. I’m happy that I can keep up with what’s happening to a degree, but can only do so by going through it one step at a time and definitely can’t see the big picture of what’s happening in my mind. In the same way that I can now pretty easily visualize the unit circle and how it relates to a sine wave and how it’s associated with different angles and their x- and y-values, I’m confident that one day I’ll also be able visualize the math from above in the same type of way. It might not be for awhile though… 😐
I have eight videos and 4.5 articles left to get through before I can start the unit test for Applications of Multivariable Calculus (400/500 M.P.). I doubt I’ll be able to get through it all this week but think there’s a slim chance. My fingers are crossed that I’ll be able to finish it off which would mean I could get started on the unit test first thing next week and potentially get through the entire unit by ~Sept. 20th. After that, there are two more units left to go in Multivariable Calculus that have 2,200 M.P. combined. My goal is to get through MC before the start of the New Year which will be tough but potentially within reach. As I’ve said a million times before, I’m not too concerned with how long it takes me to get through MC or the math section of KA, in general, but, nonetheless, the race is on! 🏎💨