This week I finally – FINALLY! – made it through the Applications of Integrals unit test. I passed it with a perfect score on my second attempt on Thursday. I’m relieved to be moving forward but, at the same time, as frustrating as it was, it’s probably a good thing that it took me so many attempts to get through it. I actually have a pretty strong grasp on how to determine the volume of objects using integrals now, plusI got a bunch of practice with calculus, in general, but also with algebra and arithmetic. That said, I’m pumped to finally be getting started on the 2nd last unit of Calc.2, Parametric Equations, Polar Coordinates, and Vector-Valued Functions, which is the first full unit I’ll be working through since I’d already completed the majority of the other units I’ve been working through since I started in Calc.2.
Nothing came up this week on either attempt at the unit test that was new or very notable, but here are 3 questions I worked through just for funsies:
Question 1
The calculus on this question was pretty straightforward but I wanted to add it here because I was happy that I remembered and applied properly the log rules at the end of this question. I didn’t add the first part of the log rules in my notes so here’s how they work from the second last step:
- –10 – 4ln(–2) + 30 + 4ln(–6)
- = 30 – 10 + 4ln(–6) – 4ln(–2)
- = 20 + 4(ln(–6) – ln(–2))
- = 20 + 4(ln(–6/–2))
- = 20 + 4(ln(3))
- = 20 + ln(34)
- = 20 + ln(81)
Question 2
This question took me longer to get through than it probably should have. When I started this question, I was mistakingly trying to calculate the width of each triangle with y instead of x. In the past few weeks I’ve gotten better at quickly figuring out which variable to use, but, as what happened here, I still get it mixed up sometimes. Once I realized I needed to use x to calculate the width of the triangle, for some reason I didn’t trust myself that the height, z, of each triangle was x√3 so I had to work through 30-60-90 triangles to prove it to myself. You can see where I did that at the bottom of the second page of my notes. From there I worked through the calculus as per usual but for some reason wasn’t confident with my answer when I got to the end. I held my breath and submitted my answer and thankfully got it correct.
Question 3
For whatever reason I don’t seem to have a problem remembering the formula for the length of an arc which is b∫a (1 + (dy/dx)2)1/2 dx. The video I watched where sal explained how Pythag’s Theorem relates to an arc’s length seemed very clear and stuck in my head. Even still, as I did at the top of my notes for this question, I usually work through the formula step by step to help ingrain it into my memory. This question was notable because I didn’t know that the notation for (ln(x))2 could be written as ln2(x). This seems somewhat intuitive since (sin(x))2 can similarly be written as sin2(x).
On Thursday after I finished the unit test, I got started on the following unit, Parametric Equations, Polar Coordinates, and Vector-Valued Functions. There are 20 videos and 14 exercises in this unit and I managed to get through the first 3 videos and ~1.5 exercises by Saturday. Based on the few videos I’ve watched so far, this unit seems to be about using one variable (ex. t for time) within two related but separate equations. The very first video I watched set up a question where a car travelled off a cliff at a certain velocity and used v(t) as the x-component for velocity and g(t) as the y-component for gravity. Using t in both functions and combining them would output the exact (x, y) coordinates of the car as it shot off the cliff.
As of now I don’t really understand why the math in these questions work but I’m starting to get a feel for how, i.e. the method used, to solve these questions. Since I don’t really understand why they work, I’m not going to try and explain the math, but below are 3 questions I worked through from the exercises. (Hopefully I’ll be able to give an explanation as to why they work next week):
Question 4
Question 5
Question 6
There are now 103 days left in 2022. I think it might be a long shot, but my goal is to get through Calc.2 before the end of the year. This seems potentially doable but, based on my experience, I could easily come up against some concept that takes me a month to wrap my head around how to solve the math behind it, let alone why the math works. Regardless, I think it’s a good goal to aim for and will it’ll hopefully push me to grind through Parametric Equations, Polar Coordinates, and Vector-Valued Functions (80/1,500 M.P.) asap. I’m happy and oddly excited that the very first video I watched in this unit gave me a physics problem and I’m hoping that the other videos and exercises will be like this too. These types of questions seem much more interesting and relatable to me than questions where I’m just ask me to work through trig or algebra equations. If this unit starts getting into physics, I’ll actually feel like I’m learning math that can be applied to some purposeful/useful scenarios which will be much more rewarding!