I kind of got smoked this week on KA. I only ended up watching 4 videos from the section Lagrange Error Bound and didn’t make it through a single exercise. I got destroyed trying to work through he 1 exercise from that section… Even though I made little progress through KA itself, it wasn’t for a lack of effort! In total I definitely studied for more than 5 hours this week by watching videos on Taylor Series and Taylor Polynomials from other YouTube videos. I’m still struggling with them, but I now at least have a better idea of why they’re useful, their notation, how they work (but not necessarily WHY they work 😔), the difference between a Taylor Series vs a Taylor Polynomial and how the remainder/error of a T.P. works. So all in all I would say I’m happy with the progress I made in terms of what I learned about T.S. and T.P. in general, but am sad that 1) I know that I still have a lot left to wrap my head around and 2) that I didn’t make any real progress through KA.
I started the week by watching these two videos I found on YouTube:
I can’t remember exactly but I think I Googled something along the lines of, “what’s the purpose of learning Taylor Series and Taylor Polynomials?”. Being that part of the title in the first video literally includes “This is why you’re learning Taylor Series”, it seemed like a good place to start. From there I found the second video which was on the side of the first video in the ‘recommended videos’ section. The second video is from the channel 3Blue1Brown which I really like. Between the two videos, what I took away is that the reason Taylor Series/Polynomials are useful is because sometimes it can be more-or-less impossible to solve certain functions at specific points but that Taylor Series/Polynomials can give such a close approximation to those values that the difference essentially becomes irrelevant. The first video stated that T.P.’s are often used in engineering and for things like force applied to bridges from cars. Although I still am pretty lost when it comes to T.S. and T.P., these videos helped me ‘buy-in’, so to speak, to the importance of learning them.
From KA, on Wednesday I wrote out the notation used in T.P. for an nth-degree T.P. and the notation for the error/remainder that goes along with it:
By getting a better understanding of the notation and by watching so many videos, one thing that became more clear to me this week is that a Taylor Polynomial, Pn,a(x), approximates a function, f(x), at a specific point which is most often denoted with either a or c (where c stands for ‘center’), and that the difference (a.k.a. the error) between the polynomial and the actual function at a point close to a/c is the remainder/error, Rn,a(x). I’m now able to visualize this concept in my head which is making it all seem much easier to grasp. 😮💨
I kept trying to grind away through the 4 KA videos from this section from Wednesday to Friday but wasn’t making much progress. I also kept looking for other videos to watch on YouTube to help me give a clearer picture of how T.P.’s work. I eventually came across this video:
This video was SUPER helpful and gave me a much better understanding of the fundamentals of what’s going on with Taylor Polynomials. I spent all of Saturday watching this video and copying out the first example given from this video. It made a huge difference in helping me understand what’s going on with all of this. Here are my notes from that example:
As you can see from the screen shots of Desmos, the function ln(x) at 1.1 and the T.P. at 1.1 are equal to 0.09531018 and 0.095308333 respectively. The difference between those two values is 0.09531018 – 0.095308333 = 0.000001847 which is indeed less-than 0.000002. On thing that the teacher in the video said that helped me appreciate why T.P.’s can be useful was that without a calculator, it would be very difficult to solve ln(1.1) but by using a T.P. you can solve it to within a tiny margin of error by with a pen and paper.
After I watched that video and wrote out those notes, I tried to go back and finish the one KA exercise I was struggling with on Sunday morning but still was finding it really tough, and so I didn’t bother trying to get through it before starting to write this post. Now that I have a better understanding of the notation for T.P.’s and what’s going on with them in general, I think I’m in a good position now to get through that exercise early next week. I’ll probably have to take my time on the first few questions on it but I think I’m pretty close to really getting a good idea of what’s going on.
This coming week I’m hoping I can quickly get through the Lagrange Error Bound section so that I can then at least finish the following section, Power Series Intro, before the end of the week. The latter section only has 2 videos and 1 exercise so, assuming it’s not too difficult, I think I have a fairly good shot at getting through it. If I can get through the two exercises from each section that’ll put me at 64% of the way through Series (currently at 1,120/2,000 M.P.). I think it’ll be nearly impossible for me to finish off the unit before the end of the year since even if I manage to get through both sections I’ll still have 22 videos and 4 exercises left to finish… But either way, I’m confident that I’ll get through it all soon enough and think I might even actually understand it all once I do. 😂