This week on KA I made it through 6 videos and 3 exercises from two sections of the unit Parametric Equations, Polar Coordinates, and Vector-Valued Functions. I was pretty busy at my work this week which played a part in me falling short of the ≥9 hours of studying KA I’ve been trying to get done each week. I did make a significant amount of progress understanding what’s going on with parametric equations, however! The first section I worked through, Planar Motion, helped me wrap my head around what’s going on with these types of equations and focused on the position, velocity, and acceleration of particles. Also, I got a better feel this week for how to switch between Lagrange notation (ex. a(x), b’(x), etc.) and Leibniz notation (ex. dy/dx, d²x/dt², etc.). I can tell that I still need a lot more practice with everything I improved at this week but I can tell I’m making progress which is reassuring.

The easiest way for me to explain what I learned this week will be to go through some of the questions I worked on. Here are 4 example questions:

Question 1

I was correct last week thinking that the graph of the position of a particle is more like a map indicating the particles position and not a singular function in itself. The way I think about it now is that the graph of the particle’s position is actually two functions put together, the x(t) function and the y(t) function. I didn’t make this clear at the top of my notes above, but now I think of the particles position coordinates as s(t) => (x(t), y(t)), it’s velocity as v(t) => (x’(t), y’(t)), and its acceleration as a(t) => (x’’(t), y’’(t)). Knowing/thinking through these questions this was made the questions seem pretty straightforward. It then just became a matter of plugging in the position equations and finding their derivatives.

Question 2

As you can see, this was a question where I had to find the second derivative of the particle’s position to find its acceleration vector. After having now used the power and sum rules a billion times, finding the second derivatives of x(t) and y(t) was pretty simple. The question asks to find the particles acceleration at t = 1 so, after finding x’’(t) and y’’(t), you then input t = 1 into both equations to output the x and y coordinates for the acceleration vector. (As I write this, I’m a bit unsure if I have that correct and/or if I’m saying it properly.) Finally, you then use Pythag’s theorem to find the magnitude of the particle’s acceleration vector at t = 1.

Question 3

This question came from the 2^nd exercise in the section Planar Motion. Once I got used the pattern of the steps used to solve these questions, they were relatively easy to work through. To solve this question, the first step is to find the derivative of d/dt [x(t)y(t)] which requires the product rule and leads to x’(t)y(t) + x(t)y’(t). You then plug in the values given to you at the beginning of the question in order to solve for x’(t). To finish, you use Pythag’s theorem on x’(t) and y’(t) to find the magnitude of the velocity vector of the particle at (x(1), y(1)) => (4, 4).

Question 4

This question came from the 3^rd exercise in Planar Motion which was about the same thing but going in the inverse direction by using integrals on the acceleration and/or the velocity of a particle to get to its position. To solve, you use the F.T.o.C. on x’(t) and y’(t) from t = [1, 2] and then input the values of x(t) and y(t) between t = [1, 2] into Pythag’s theorem to get the particles displacement. It took me a couple of tries to get through this exercise but I don’t find the concept of what’s happening too difficult to understand. 

I wrapped up this week by watching the two videos from the following section Polar Functions. I was completely lost when I watched the first video and had to Google polar functions to review what was going on. After doing a bit of review, I was able to remember the gist of how polar functions work but I will likely need to do more review the beginning of this week. Often when I come back to a concept that I studied awhile ago it seems relatively easy to understand compared to when I first learned it. I’m hoping and am optimistic that this will be the case with polar functions and they they’ll seem pretty simple once I get through this section.

I now have 4.5 sections left to get through in Parametric Equations, Polar Coordinates, and Vector-Valued Functions(640/1,500 M.P.) which have a total of 6 videos and 7 exercises. I’m hoping I can get through 2.5 sections this week which have a total of 3 videos and 5 exercises. Things at work should be quieter for me this week so I think I have a pretty good shot at getting through all of it. As I said last week, my goal is to finish this unit by the end of October. For once, it’d be nice to actually reach my goal BEFORE the deadline and not after. 🙏🏼