I said at the end of my post last week that I thought it was going to take me at least 2 weeks to get through Differential Equations. I was SO close to getting through it this week but fell just a bit short. I managed to get through all 17 videos and the single exercise by Thursday but didn’t pass the unit test, even after 3 attempts at it. I was able to get through the questions on the unit test without too much trouble and understood HOW to do them but often didn’t understand why they work. I made 1 or 2 careless mistakes on each attempt as well which resulted in me getting at least a few questions wrong on each attempt. There were also 1 or 2 questions that completely stumped me. Nonetheless, I can tell I’m getting a much stronger feel for how to solve differential equations which I’m happy about.

The first 11 videos I watched this week were all videos I’d seen before and, for the most part, were all very straightforward. On Tuesday I got to the final section of the unit titled Logistic Models which had 7 videos I’d never seen before and an exercise. I found this section a bit confusing and it took me until Thursday to get through it. The key takeaway from this section was a formula known as the Logistic Differential Equation. I feel like I have a surface level understanding of this equation and how it works and know that it’s used to model the growth of populations. Here are two pages of notes I took that talk about this equation:

I drew the graph of the equation on the first page wrong. The function should not go past the y-value y = k. This is a point that’s sometimes referred to as the ‘carrying capacity’ of a population which is the limit that a population can grow to due to any number of factors (ex. limited landmass, access to food/water, etc.). The reason why the function can’t go past y = k is because of the expression within the function (1 – n/k).

After working through the 7 videos from this section, I began the exercise on Wednesday but it took me until Thursday to get through it. Here are two example questions from the exercise:

Question 1

Question 2

Once I understood what the questions were asking, the math used to solve them was fairly straightforward. One important thing I took away from this exercise was a more solid grasp on the fact that a derivative of a function equals the value of the slope of the tangent line to that function at any point along the function. What helped me better understand this concept was, as shown in the second question, that the derivative for an “S” curve function is a parabola. For whatever reason, working through that second question made it very clear how the parabola equals the value of the slope of “S” curve as it goes from 0 to k = 24,000.

I started my first attempt at the unit test on Thursday which has a total of 13 questions in it. I ended yesterday on my third attempt at the test and finished on the 11^th question with two wrong so I’ll have to go through at least once more this coming week. In any case, here are 5 questions from the test that I thought were notable:

Question 3

I sat and stared at this question for a pretty long time before I made sense of it. It seems obvious to me now, but as I was working through it I didn’t realize that the answers were y-coordinates and thought they were (x, y) coordinates. Knowing that’s what they are now, the way the question is phrased, “what is the solution curve y = f(x) for x ≥ 0”, seems pretty clearly to be asking what the range of values are on the y-axis, but I’m so used to seeing (x, y) coordinates and not (y₁, y₂) coordinates that I didn’t understand it. It makes me think there should be separate notation for (y₁, y₂) than there is for (x, y).

Question 4

Question 5

This was a question where I didn’t have a clue what was going on when I first saw it. The way I think through it now is that you have to 1) find the derivative of the potential solution in question (in this example, I had to find the derivative of h(x) = 4 * ln(x) which is h’(x) = 4/x), 2) set that as the derivative to the OG derivative, and 3) put the function in question (h(x) = 4 * ln(x)) back into the OG equation and solve to see if both sides equal each other.

I don’t know if what I said just makes sense. This is a good example of me knowing how to solve this type of question but don’t know why this question works the way it does. I have a feeling that the more I’m able to understand what’s going on with this question and why, the better I’ll be at articulating how to find its solution.

Question 6

Question 7

This is another question where I didn’t and still don’t really understand what’s going on. I’m happy that I can work through the algebra pretty easily on this question but I don’t have an intuitive feel for when or why to substitute each expression in at which point in the process to come up with the correct solution. The silver lining is that I don’t think it’s going to take me much more time to figure it out since I feel like I have a pretty strong grasp on the math behind how to solve this question.

I’m hoping it won’t take me too long to get through the unit test this coming week so I can get past Differential Equations (1,240/1,300 M.P.) and move onto the following unit, Applications of Integrals(1,900/2,000 M.P.). I only have 3 videos and 1 exercise to get through in the latter unit but I have a feeling it’s unit test will be pretty difficult. The unit is all about computing the size and volume of objects which I think I found very interesting and useful thew first time I went through it but also a bit difficult. Even if it takes me awhile to pass the Applications of Integrals unit test, I think it’ll be really useful review and the optimistic side of me thinks that I’ll actually be able to understand it a lot better this time around. 🤞🏼