It was another disappointing week working through KA. 😒 I made it through the final article from the section Tangent Planes and Local Linearization and then made it through the first four videos in the following section, Quadratic Approximations, but that’s it. I don’t have much of a grasp on what was taught in the latter section either which makes it feel worse. I only worked on KA for ~30 mins each of the first few days of this week which is a big part why I’m so disappointed. I tried to make up for it on Saturday by working for more than 2 hours but, even still, I’m not too thrilled with my effort this week. Regardless, I still learned a few new things, all of which had to do with quadratic approximations of multivariable functions. I was able to understand what Grant was talking about when it came to quadratic approximations, but I had a hard time memorizing how to solve these types of questions and didn’t understand at all why they worked. Hopefully next week (a.k.a. the start of my FIFTH. YEAR. working on KA! … 🤯) I’ll have a more productive week and make better progress.
I got through the second article on Local Linearization on Tuesday. It was more of the same from what I worked on and wrote about last week, so I’m not going to go into more detail here. That said, here’s an example question I worked through from that article:
I really like the vector notation used in the final screen shot above. I still don’t have the notation fully memorized and so I find it confusing to interpret, but I’m starting to get a vague idea of what it’s denoting. Comparing the vector notation to my two pages of notes, the vector notation is much more concise so I can definitely see the value in learning how to read and understand it properly. I’m pretty motivated now to get better at using vector notation to make things simpler for myself going forward.
I got started on the following section, Quadratic Approximations, on Wednesday. As far as I can tell, in the same way that a tangent plane looks like a sheet of paper on top of a 3D function and represents the function’s slope at a given point, a quadratic approximation does the same thing except it hugs the 3D function a bit tighter at the given point as it uses parabolas (not sure if that’s right) across the partial derivatives of x, y, and z. (You can see what I’m talking about if you scroll down to the second last screen shot at the bottom of this post. The quadratic approximation is the white 3D shape which is an approximation of the actual function which is the checkered-blue 3D shape.)
Like I said at the beginning, I don’t really understand how the equation for quadratic approximations work, but here are a few screen shots from second and third videos from the section where Grant explained it and some notes I took trying to work through and understand the equation:
The good thing is that the equation is relatively easy to memorize. The bad news is that its derivation is kind of intense and, although I was able to follow along with Grant and work through it with him, I have zero intuitive sense for how/why it works. Nonetheless, here’s an example question Grant worked through in the fourth video and my notes below it:
Even though there’s a lot to these quadratic approximation questions, I feel relatively comfortable working through all the partial derivatives and understanding the general concept of what’s going on. This makes me think that although I definitely need more practice working through these questions, it (hopefully) won’t take me too long or be too difficult for me to start to get a handle on how to work through them and why they work the way they do.
Every time I begin a new year in KA I start a new Word doc. This past year it says I wrote 64,329 words in 193 pages and, if I add that to the previous three years, that puts me at a total of 263,149 words and 733 pages of notes. 😳 I’m currently 45% of the way through Multivariable Calculus so it might be closer to the five-year mark before I actually finish calculus. It’d be great to get through it before the end of this calendar year which I think is a reasonable goal to give myself, although probably a bit ambitious. If I’m going to meet that timeline, I need to start making some bigger strides each week. This coming week it’d be great to get through the current section I’m in plus the following one. I’ll need to watch nine videos, complete four exercises, and read two articles to do that. Again, ambitious but I think possible! It’d definitely be nice to start my fifth year off on a positive note. 🤓