I didn’t have a great week on KA. I only watched six videos and made notes on five, but all in all I’d still give myself a passing grade for the week—but just barely. I think I probably only spent 4 hours studying which I’m not too happy about, but the silver lining is I was more focused than usual when I was studying and the notes I took were pretty legit, so I’ve at least got that going for me which is nice. I’m still struggling to understand a lot of the concepts, especially what I worked through at the end of the week. Nonetheless, I’ve reached the final few videos I need to get through in this unit and have set myself up decently well to finish it off completely next week. Slowly but surely, I’m getting closer to the finish line! 🏁
Video 1 – Exact Equations Example 3
Just like last week, I still haven’t figured out why exact equations work, but I feel pretty good now about how they work. I think my notes above to a decent job of explaining the step by step the process of how to solve these types of questions, so I won’t bother explaining it further. One thing I’ll say though is that I still don’t really understand why figuring out a new expression for M + Ny’ (the expression being Xi—the pitchfork looking thing) is better than simply using M +Ny’. There must obviously be reason, but I don’t know what it is…
Video 2 – Integrating Factors 1
This video and the next were part of the same question that Sal was working on, just split up into two vids. The point of these videos was to show that if you have an equation that’s not exact because My ≠ Nx, you can try to make the expressions equal each other by determining if there’s a function you can multiply into both expressions so that they equal each other. (Hopefully that makes sense.) To figure it out, you start by stating that each expression equals each other, then multiply a placeholder function into both expressions, u(x), then find the partial derivative of My and Nx, and finally do some algebra to determine what u(x) is equal to. In this case it ended up equaling x.
(The second video explains the rest. 👇🏼)
Video 3 – Integrating Factors 2
Once you figure out the function that needs to be multiplied into both expressions so that their respective partial derivatives equal each other, you multiply it into the equation and it becomes exact, at which point you solve it the same way you solve other exact equations. Boom. 🧨
Video 4 – Growth Models: Introduction
This was the first video from a new section titled Logic Models. In this vid, Sal walks through what sounds like a relatively famous equation apparently called the ‘exponential growth and decay function’ that has to do with population growth. The idea being explained is that because there are environmental constraints in general (i.e. there’s a limit to the size of a population a given environment could support), population growth can’t grow indefinitely. The function graphed at the bottom of the screenshot is supposed to represent this concept. It turns out you can model this concept with the exponential growth and decay function, N(t) = Cert.
Video 5 – The Logistic Growth Model
This was the final video I made notes on this week, and it was a continuation from the previous video. As you can see, the Logistic Differential Equation takes the instantiations rate of change function, N’(t) = rN, and adds on the expression (1 – N/k). What this does is sets a ‘ceiling’, so to speak, on how high the population can go. As the population, N, gets close to the upper limit, k, the expression (1 – N/k) ends up going closer and closer to 0, meaning the rate of change will end up going closer and closer to 0 and so the population will plateau at k.
The final video I watched this week, Logistic Equations (Part 1), was very confusing so I decided I’d leave it and start it this upcoming week. 🙃
And that was it for this past week. Not great, but progress nonetheless. I only have five videos left to get through in this unit so, as long as I can somewhat quickly grind through the Logistic Equations (Part 1) video I was just talking about, I think I have a good shot at finishing off this unit this upcoming week. There are also only 33 videos between the next two units which are the final two units in Differential Equations. I’m realizing as I write this how close I am to finishing this off! 😳 If I can do five videos a week, I’ll be done this in course in two months. Then I have to redo the Multivariable Calculus Course Challenge and then I’ll be DONE. Even if it takes me a month to get through the course challenge, I could still be through this all in about 3 months… After 5.5 years, that seems like it’s right around the corner! 🤯