I decided this week that instead of doing a minimum of 1 hour of KA per day that I’d aim for a minimum of 1.5 hours per day. I managed to do it and, as a result, had one of the most productive weeks I’ve had in a long time! I FINALLY finished the Applications of Integrals unit test (praise Jesus 🤲🏻) and am now 10/30 questions through the Calculus 1 course challenge AND got all 10 questions correct so far! That said, I did go back into my notes and used Google to review a handful of the questions of the course challenge which I’m not thrilled about. Usually I wouldn’t do this but, since so much of Calculus 2 overlaps with Calc.1 and I’m going to have to review a bunch of calculus as I work through Calc.2, I’m ok with ‘cheating’ and reviewing my notes before submitting my answers on this course challenge.

On Tuesday, I began the week on question 10/19 of the unit test. I had already gotten 2 questions wrong at that point so I knew I was going to need to redo the test. I managed to get through all 10 questions on Tuesday and didn’t get any wrong. I was happy that once I finished the test I realized that my overall M.P. score didn’t drop below 80% meaning I didn’t need to redo any exercises which was a relief. It took me until Friday to get through the unit test for the third time but finally passed it with a perfect score! Here are 3 example questions from the test:

Question 1

I was annoyed by the question because I’m pretty sure up to now I haven’t been given any question like this in Applications of Integrals and I was a bit caught off guard. Finding the area underneath f(x) seemed pretty straightforward to me but, as you can see from my notes, I couldn’t figure out how to find the upper bound of the integral. I got to the point where I knew that –1/2 = sin(x) but I wasn’t really sure what to do with that. I reviewed 30-60-90 triangles and remembered that if –1/2 = sin(x) then x = 30° but I didn’t realize that meant that upper bound of f(x) was 210°, a.k.a. 7π/6. I ended up using Desmos to double check the upper bound and, after that, had no problem solving the question.

Question 2

This was a question that shouldn’t have been too difficult but I hadn’t seen many questions phrased like this so I found it a bit tricky. I was happy that I was able to figure it out on my own without using a calculator except for the very last step where I had to figure out what 1/e^0.4 and 1/e^0.2 were both equal to.

Question 3

(Note: I started a new notebook which is why the “Week 146” title is at the top my notes from this question.)

This question, 16/19, was the last question I worked through on Thursday and it took me ~1 hour to answer. (😳) In retrospect it seems like a fairly straightforward question where I needed to use the washer method in terms of y to come up with the volume of R being rotated around the y-axis. At this stage, I was getting close the end of the test and was feeling a lot of pressure to get it correct so I began second guessing myself. In the end, I wasn’t very confident submitting my answer but I ended up getting the question correct even though I was very unsure of myself at that point.

As I mentioned up top, I finished the unit test on Friday and began the Calc.1 course challenge that same day. I was very nervous starting the challenge (and, for that matter, am still nervous about it heading into next week) since I literally started Calc.1… 64 WEEKS AGO (…😳…) in Week 82, so I was sure that I would forget how to answer a bunch of the questions. So far, I haven’t been given any questions where I felt completely at a loss for what I was being asked, although there have been a few questions that I needed to review. In any case, here are 6 questions that I’ve worked on so far:

Question 1

This was the very first question I worked on which threw me off because it seemed too simple and I couldn’t understand how it related to calculus. I knew that a secant line was just a line going through the two points and knew that the slope, m, would be calculated with the basic rise/run formula, a.k.a. (y₂ – y₁)/ (x₂ – x₁), but I assumed there was something more that I was missing and wasn’t confident submitting my answer. Nonetheless, I trusted that I had done it properly and ended up being correct.

Question 2

For the most part I was able to make sense of the notation in this question when I first looked at it. It’s been close to a year since I worked through limits though and I wasn’t 100% sure that I remembered how they work so I looked up a few articles on Google to double check. The reason why the first answer isn’t correct is because there’s a jump discontinuity at h(–4) and the first answer implies that the limit from BOTH sides of h(–4) is –2, however the limit from the left, a.k.a. the negative side of h(–4), is –6. The second and third answers are correct as they both refer to the limit approaching h(x) from a specific side.

Question 3

I always struggle with questions like this and find the comprehension of them difficult. This question took me 15-20 mins to think through so I was pretty happy with myself when I ended up getting it correct. The first clue I picked up on was that the question said “the tank decreases” which made me think that solution needed a negative sign in front. The second clue that stood out to me was that it said, “at a rate proportional to z” which made me think that there would be a variable being multiplied by z to act as a scaler. This made me think that k*z would be appropriate. Finally, the part that said, “and inversely proportional” made me think that the solution should contain a 1/(20 + 2t). I was about 60% confident in myself when I answered this question and was relieved that I got it right.

Question 4

I had to look this question up. I couldn’t remember how the separation of variables method worked, although I was actually on the right track in my head before I looked it up and might have been able to come to the correct solution on my own. This question didn’t actually ask me to solve the equation but, after looking up of the separation of variables method works, I went ahead and solved it just to practice:

(Note: In the bottom left corner of my notes you can see where I expanded ln(–(e^x + C) -> ln(–e^x + C) but you can see that the (+) in front of C didn’t change. The reason why C doesn’t turn negative is because we don’t actually know what C is so we simply leave it as + C which implies it could be + (C) or + (–C).)

Question 5

This is another question I had to look up because I couldn’t remember how the squeeze theorem worked. If I didn’t look it up and had to guess one of the three choices, I think I likely would have gone with the third answer and gotten it correct but I wouldn’t have known exactly why the third choice was true. I Googled the squeeze theorem and found the following example question which, as you can see below, I worked through on my own, as well:

Reviewing this theorem, the squeeze theorem seems much simpler to grasp now then it did when I was working through it a few months ago. As far as I can tell, it simply states that if there’s a function (ex. b(x)) that appears to have a limit at a certain x-coordinate but you’re unsure, if you know that a(x) ≤ b(x) ≤ c(x) and can determine that both a(x) and c(x) have the same limit at that x-coordinate, then you can state that b(x)’s limit at that x-coordinate MUST be equal to the limit of both a(x) and c(x).

Question 6)

This was also a question that I had to look up. I couldn’t remember what the Mean Value Theorem was but, after looking it up, I remembered that it states that if a function is continuous over a certain interval and is differentiable along that interval, there exists at least 1 point where the slope of a tangent line at a certain point is equal to the average slope of the function along that interval. (To be honest, I forget why this is important but I can visualize it in my head, so I have that going for me which is nice.) Here’s an image that explains the M.V.T. and is a good representation of what I visualize in my head:

I am PUMPED to finally be working through the Calc.1 course challenge. I’ve decided that if I manage to get >90% on the challenge I’m going to move forward into Calc.2. Typically, I’d aim to get 100% on the challenge but, since so much of Calculus 2 overlaps with Calc.1, I think it’s likely that the Calc.2 course challenge will cover many of the same questions from this challenge so that’ll give me a chance to boost my Calc.1 score up to 100% when I inevitable work through the Calc.2 course challenge. Looking ahead to Calc.2, I’m already 61% of the way through it! I’m quickly approaching the 3-year mark of working on KA and am doubtful that I’ll get through Calc.2 before I reach that point, BUT I do think it’s still possible. Three years isn’t a short amount of time to do anything but, even still, I’m happy and proud of myself that I’ve managed to start with basic arithmetic and learn calculus in just 3 years. 👍🏻 🤓