I sadly did not get through or even start the Parametric Equations, Polar Coordinates, and Vector-Valued Functions unit test this week, BUT I did have a ‘Eureka!’ 💡moment understanding a concept within polar coords that I was very stuck on which completely made up for not finishing the unit! I’m a bit bummed that I didn’t reach my goal of getting through this unit before the end of October but it’s not the end of the world. Most of the time I don’t reach the deadlines I give to myself anyways and I’m sure that I’m further along than I would have been if I hadn’t set that deadline for myself in the first place. Plus, as I’ve said many times before, I’m much more concerned with getting a solid understanding what I’m learning as opposed to rushing through the material for the sake of getting through it quickly. I’m ~3.16 years into this so adding an extra few days onto that isn’t such a big deal at this point.

I started on the only exercise from the section Arc Length: Polar Curves last week and got through 3 of the 4 questions in it without much difficulty. The fourth question crushed me though and, after working through it for a few hours at the start of the week, I wasn’t making much progress. This is the question: 

The thing that was giving me such a hard time this week was trying to understand why r = cos θ, and for that matter r = sin θ, are drawn in the 1^st and 4^th, and 1^st and 2^nd quadrants, respectively. Here are what they look like:

On both of those screen shots you can see that the bounds are set to 0 ≤ θ ≤ 2π but no matter how high you make the upper bound, the functions look the same. I couldn’t wrap my head around why both functions only stayed in two quadrants. I didn’t figure it out until Thursday when I watched this video:

At 2:45 in the video, the speaker explains that when r = 2 a circle is drawn with a radius of 2. At 4:00, he then explains that if r = –2 the radius is still 2 but it moves in the opposite direction of the angle. This is something I’d known but didn’t have much practice with and didn’t really think about or remember. This is the key as to why r = cos θ and r = sin θ stay in two quadrants. At 7:25, the video goes through how to draw r = 4sin θ from θ = [0, 2π]. The concept of why each function is drawn in only 2 quadrants clicked for me at 8:45 when I realized that sin(7π/6) = –1/2 and therefore r = –1/2 meaning that the radius is shooting backwards from 7π/6 into the 1^st quadrant.

understanding this was the last piece of the puzzle, so to speak, before I was able to work through the rest of the question from above. Here’s that question and then two other questions I worked through from the same exercise:

Question 1

Question 2

Question 3

The last thing I had left to do in the unit (besides the unit test) was get through an exercise from the final section titled Calculator Active-Practice. The questions from this exercise were pretty straightforward. They asked me to figure out the area between the origin and a function using polar coords. It took me about 30 mins to remember how the formula _a∫^b ½(r(θ))² dθ worked but I eventually remembered without having to review it. All I needed to do on this exercise was input this formula into Desmos with the given bounds and r(θ) and then round the solution to 3 decimal places. Here’s a question from the exercise:

I’m excited but a bit nervous to start the unit test for Parametric Equations, Polar Coordinates, and Vector-Valued Functions (1,200/1,500). There are only 15 questions on the test which is good but it’s been about 40 days since I first began this unit and I’m somewhat worried I’ll have forgotten some of the things I first learned in the middle of September. That being said, I’m not too stressed and am optimistic I’ll be able to get through it this week. Whenever I end up getting through it, I’ll then be starting on the FINAL UNIT of Calc.2!!! The unit is titled Series and my guess is that it’ll be pretty hard, but nonetheless I’m excited to finally get to it. I’m so, SO close to finally reaching my goal of learning calculus. My goal now is to get through it by New Years! 🤞🏼🎉🎊