I managed to get through six videos and two exercises this week which sound pretty good but in reality, I put in a poor effort this week. 😒 I probably only worked on KA for around three hours. I watched a few of them twice, but each of the videos were only about five minutes long, and the exercises were super simple. My excuse is that I’ve had a lot of other things on the go lately—I’ve mentioned a handful of times that I’m working on another project which I’m SO close to getting across the finish line which I’ll talk about soon—and so I haven’t been dedicating as much time and effort into KA. As I’ve said before, I have this superstitious feeling that I NEED to finish the math section of KA to prove to myself that I can accomplish what I set my mind to (i.e to have this other project be successful 🤞🏼). It’s weird that it seems like the closer I get, the more resistance I have to finishing it off. 🤔 In any case, I need to stop being such a baby and stop making excuses. I’m still somewhat happy with what I was able to get through this week, so I guess it could have been worse. 🤷🏻‍♂️

Video 1 – Worked Example: Slope Field From Equation

This video worked through an example of determining which slope field matched the given differential equation. I was able to figure it out in two seconds just by looking at the point (2, 2) knowing that, given the equation, the slope would be 0. Since only the bottom left slope field had a horizontal line at (2, 2), I knew instantly that it was the correct solution.

Video 2 – Worked Example: Forming a Slope Field

This video was very straightforward. Sal simply took the differential equation, inputted the three sets of coordinates to calculate the slope at those points, and then drew a little dash at each point with the appropriate slope. As simple as this video was, I still appreciated working through it to get more practice understanding slope fields and getting a better intuitive grasp on what’s going on with them.

Exercise 1 – Slope Fields & Equations

This exercise worked on questions exactly like the one above. The questions were very straightforward but, again, it was useful to help me wrap my head around what’s going on:

Question 1 

Question 2 

Video 3 – Approximating Solution Curves in Slope Fields

In this video, the D.E. and the slope field were shown from the start. Sal looked at the points (1, 1) and (1, 6) and calculated the slope of each point. He mentioned that since there’s no x-component in the D.E., the slope at any given point stays consistent as you more horizontally in the x-direction. He talked through the “flow” of the slope field and drew out the green, orange, and purple lines to help explain it. He also talked about how the D.E. asymptotes at y = 4 and y = 0.

Video 4 – Worked Example: Range of Solution Curve From Slope Field

I was able to figure out the solution to this question without watching Sal work through it, but I still find it confusing. I understood that the question was asking me to start at (0, 6) and determine what the slope field ranges from across the y-axis that includes that point. I also remembered that the proper notation for the solution would include the curved and squared brackets, with the squared bracket meaning the solution included the point y = 6. The thing that still makes me still feel a bit confused is where it says the “solution curve”. I understand what that means theoretically (I think) but I feel like I don’t have enough experience thinking of a solution as an equation for it to completely make sense yet.

Exercise 2 – Reasoning Using Slope Fields

The questions from this exercise were also just like the question from the previously mentioned video. They were both pretty simple and didn’t require me to do anything other than think through the solution in my head.

Question 3

Question 4 

Video 5 – Euler’s Method

As a random side note, I always read “Euler” as ‘you-ler’ but really it’s pronounced ‘oil-er’. Just thought I’d mention that. Anyways, I think written notes do a good job of getting across the gist of what Sal discussed in this video and of describing my understanding of oilers method, so I’ll leave it there.

Video 6 – Worked Example: Euler’s Method

I found this video and question difficult to understand. I tried to work through it on my own before watching Sal and got it wrong. As I watched him solve it, I was able to follow along with the logic and the math, but wouldn’t have intuitively figured it out on my own. I think I understand theoretically what’s going on, but I can’t visualize what’s going on with the equation and taking one step along the x-axis and then determining the slope at each point. In any case, after I watched Sal do it and after I wrote out my notes above, I got a better grasp on how to solve this type of question even though I didn’t make much progress understanding why the math works.

And that was all I got through this week. Like I said, getting through six videos and two exercises seems pretty good, but given how easy everything was, I know I should have gotten through more. 😞 Nonetheless, I’m happy that I’m making progress and am starting to see the light at the end of the tunnel. My goal is to get through Differential Equations and finish off the Multivariable Calculus Course Challenge before the six-year mark in September. I feel like that shouldn’t be too difficult, but even still, I need to start bumping the number of hours I’m spending each studying each week back up to at LEAST five. 🤬