I started this week pretty strong but ended off on a bit of a low note. I made it through the Differential Equations unit test on Tuesday which was a relief but a was a bit stressful going through it. I felt fairly confident with my understanding of the questions but was slightly unsure of myself since I’d made so many careless mistakes on the previous three attempts of the test from last week. After finishing the test on Tuesday, I began Applications of Integrals on Wednesday. By Thursday I finished the 3 videos and 1 exercise I needed to get through in that section and then got started on the unit test. I had a strong start to the test but got the 7^th question wrong on a somewhat careless mistake with algebra and then got the 9^th question wrong on a question that, after spending ~1.5 hours on, I still don’t understand. 😡 Nonetheless, I still made some solid progress this week and was happy with what I remembered from Applications of Integrals. I’m definitely on the right track, it’s just taking a really long time to get there.

As I mentioned, I finished the Diff. Eq. unit test on Tuesday and didn’t find it too difficult although I did take longer than I probably should have answering many of the questions. I still don’t have a super strong grasp on why differential equations work the way they do, but they’re slowly becoming more intuitive for me and I definitely understand them much more now than I did a month ago. Here are 3 questions from the test:

Question 1

I made a note to add this question to this post because I got this type of question wrong on one of my previous attempts. I still don’t have a great understanding of WHY this question works, or what it means, but I have a pretty solid grasp on how to solve it. I was happy that I could easily separate the composite function into individual functions, i.e. breaking it down into a(x), a’(x), b(x), and b’(x). 

Question 2

This was another type of question I got wrong on my last attempt at the test. The process to solve it was still fresh in my mind which is what helped. I’m sure that 6 months from now I’ll have a hard time remembering how to solve this type of question, but I was still happy that I was able to used the log rules properly even having ln(0.5)/13 as the exponent on e which I wouldn’t have been able to understand a year ago.

Question 3

I made a note on this question that I didn’t know how to solve it right off the bat but I managed to work through it in a few minutes without much difficulty. I have a decent understand of Euler’s Method so I was able to apply what I knew about it to this question and work through it step by step. Boom.

I got back to the unit Applications of Integrals on Tuesday and only had one section that I needed to get through titled Arc Length. This section, as you’d expect from the title, covered the formula used to determine the length of a function that is non-linear. This is how it works:

It’s a pretty simple formula and also easy to derive and understand its derivation. Once I learned the formula, it wasn’t too tough to apply it to the questions that came up in the exercise. Below are 3 questions from this section. The first one is from one of the videos in the section and the two after are from the exercise:

Question From Video

Question 4

Question 5

The note I made from this question actually had nothing to do with answering the question itself but went through how you can use the log/exponent/power rule to bring down the 2 on x² inside of ln(x²) to set it as 2 * ln(x) in order to solve its derivative OR you can decompose the composite function into separate functions and use the chain rule to determine its derivative. I made a note about this just because I hadn’t made that connection before between these two rules even though I knew how to use both methods.

I got started on the unit test on Friday but only got to the 10^th question out of 20 and already got 2 questions wrong. I can’t remember exactly but I feel like it’s been a long time since I last worked on this unit and so I was happy/surprised that I was able to remember how to solve most of the questions even though I got two of them wrong. Here’s the second question I worked on which I was glad to get correct:

Question 6

I didn’t initially remember what to do to solve this question but started with the fact that volume = base * width * height. From there I remembered that I’d always set the base as the integral, i.e. in this case b = ₁∫⁴ dy. Knowing that the area of a circle is πr² and that the radius would be the distance between the function e^x and x = –4, I then worked out that I needed to figure out what e^x was in terms of y so that I could use that function in the integral.

The last question I worked on, question 9 on the test, I actually solved but then didn’t trust my answer and submitted something else and got it wrong. I still don’t have any idea what’s going on with this question which is super frustrating. Here it is:

I must be missing something here, but I can’t understand how the VOLUME of an object would have a smaller volume than the surface area of one of its faces. As you can see, I didn’t solve the question using the same algebra that KA’s answer used, but I got to the same solution nonetheless. I’m really annoyed that I don’t understand what’s going on. I’m assuming I’m going to need to redo some of the exercises before I can attempt the unit test again so I’m hoping when I work through some of the questions that I’ll eventually figure out what’s going on here.

I’m really hoping I can get through Applications of Integrals (1,980/2,000 M.P.) this coming week. The 3-year mark of me working on KA is September 2^nd which is only 12 days away! I’d definitely like to have started on the first of the final two units from Calc.2 by then. The first, which might have the longest title of any unit I’ve ever worked on, is called Parametric Equations, Polar Coordinates, and Vector-Valued Functions (0/1,500 M.P.). The following unit, Series(0/2,000 M.P.), is the final unit of the course which is crazy to think that I’ll be done calculus once I get through it.

SO. CLOSE!!!