I had a pretty rough week this week. 😔 I only got through 3 videos on KA and didn’t get through a single exercise. On the plus side, I did get a much stronger grasp on how to interpret r when using polar coordinates and a stronger intuitive understanding that r can be thought of as the hypotenuse of a right triangle when using polar coords. I ran into one of the toughest questions I’ve worked on in KA in a long time at the end of this week which stopped me from passing the 1 exercise I worked on, but also helped me learn quite a bit about polar coords. Looking back through my notes, overall I’m happy with how much work I got through this week and what I learned but I’m a bit bummed that I didn’t make much headway in getting through this unit, Parametric Equations, Polar Coordinates, and Vector-Valued Functions.
The reason why I had a rough week wasn’t because of my lack of progress on KA. I’m not sure if I’ve mentioned this in a previous post, but at the end of August I injured my right, ring finger when a tennis ball hit the top of it and the tendon came off the last knuckle. It didn’t hurt at all but it started to swell and, after Googling it and talking to my doctor, I realized I needed to wear a splint for 6-8 weeks for it to heal. This week I took the splint off because it looked like it was better but it definitely hasn’t healed and I’m incredibly depressed about it. I literally don’t know what to do. It’s very hard to concentrate on KA, or anything for that matter, when all I can think about is that it’s been 2 months and I haven’t made any progress fixing this stupid fucking finger. I’m at a complete loss for what to do and am feeling very hollow and apathetic about everything.
Nonetheless……
My week started out by going through the first video in the section Arc Length: Polar Curves. In this video I learned what the formula is to find the arc length of a curve using polar coords which is a∫b ((f’θ)2 + (fθ)2)1/2dθ. I thought the proof for this formula was going to be pretty easy to work through since I found the proof for the formula used to find the arc length of a curve with Cartesian coords pretty straightforward, but I was wrong. Here’s the proof:
I’m not going to explain the algebra/calculus from my notes above so hopefully you can decipher my chicken scratch and make sense of it. The formula itself is pretty simple to use. Here’s an example of how to use this formula that came from the 2nd video of this section:
For the most part, the questions from the exercise in this section weren’t too difficult. I have a better understanding of how they work now, but when I first started this exercise I didn’t have much of an intuitive understanding of what was going on, but the formula was pretty straightforward to use so it wasn’t hard to get the correct solution. Here are two examples questions from this exercise:
Question 1
Question 2
After getting through the first few questions in this exercise I thought it was going to be smooth sailing to to the finish line. That was until I ran into this question at the end:
Question 3
I still haven’t fully worked through this question so I’m not going to try and solve it here. It took me a long time just to understand how/why the circle created from the function r = 4cosθ looks the way it does. At first I thought since cos(0) = 1 and cos(π/2) = 0 that r = cosθ was just some type of unit circle based on the Cosine function. I realized that wasn’t the case though since cos(π/3) = 0.5 but the circle created from the function r = 4cosθ crossed π/3 at a point that was 1 unit out from the origin along the x-axis but it should have been 2 units away. I couldn’t figure out what was happening so, it ended up being pointless, but I made the following note to try to figure out what was going on:
Even though this note doesn’t exactly relate to r = cosθ, I still think it’s useful. It turns out that what’s actually going on with the circle created by r = 4cosθ is that r is the hypotenuse of a right triangle and as you input θ values from θ = [0, 2π], the result is a circle. Here’s a page of my notes that explains it:
Once I figured that out, the next thing I had to understand was why r = secθ = 1. I still find this one a bit confusing but here’s the algebra as to why I believe secθ = 1:
I’m not 100% if my note above is correct but that’s my understanding of it at this point. Like I said, I haven’t fully worked through the rest of this question so I’m going to wait until next week to go through the explanation of how the whole thing works.
I think it’s pretty unlikely that I’ll get through this unit, Parametric Equations, Polar Coordinates, and Vector-Valued Functions (1,090/1,500), before the end of this coming week. Even still, I feel happy with what I’ve been able to learn over the last few months and really feel like I have a fairly good grip on what I’ve learned. I’m sure I still have a long ways to go before I fully understand it all but I can tell that I’m making progress and expanding my knowledge of what I’ve learned even if that hasn’t been reflected in my progress through the KA units. This entire journey has been an absolute marathon so if it takes me an extra week to get through this particular unit, it won’t be the end of the world.