I once again had another successful week on KA! I’m SO happy to say that I not only made it through the final unit test from Calculus AB this week, but also (somehow) managed to pass the Course Challenge on my first attempt, AND got 100% on it. 😳 Too be fair, I did double check the majority of my answers on Desmos and Symbolab before submitting them, but I there was only one question that I didn’t have the correct solution to before checking. I was honestly a bit shocked that I managed to pass the Course Challenge, let along get 100% on it. Also, this was the first time ~6.5 years of working on KA that I got a perfect score on a Course Challenge. 🥹 I know I shouldn’t have used Desmos and Symbolab a much as I did, but given that my goal six years ago was simply to “teach myself calculus”, I think it’s fair to say that even if I’d only got 29/30 questions correct without using Desmos or SL, I still more-or-less have achieved that goal. The icing on the cake is that I got started on the Calculus BC Course Challenge (the final thing I have left to do) and made it through five questions, getting all five correct! For some reason the five questions were all super easy which was weird to me, but I’m not complaining. So, all in all, it was a solid week for me on KA, and I honestly can’t believe how close I am to finishing this all off! 😭

It took me a few attempts to get through the final unit test (from Unit 8, Applications of Integration) but managed to finish it off on Friday morning. Most of the questions weren’t too hard (although there were a handful of tricky ones, for sure), but there were 19 questions on the test and I’d often make a careless mistake at some point and would have to restart. But all in all, it wasn’t too bad.

Here are six questions from the test:

AP College Calculus AB – Unit 8 ­– Unit Test – Applications of Integration

Question 1

This was the second question that came up on the test which I found fairly difficult. I figured that the expression in the integrand would be (b * h)/2 = (b * b/2)/2 based on the shape being made up of isosceles triangles. (I was actually pretty happy/proud of myself that I was able to come up with the proper expression.) But even once I got the expression figured out, then I found doing the actual integral pretty hard. I got to –(e^–2 – 1)/8 but assumed that was wrong. I put it into Symbolab and it said the integral was e² – 1/8e², so it seemed like my solution was actually pretty close to what SL said. It took me a few minutes of multiplying my solution out but I eventually realized my answer was actually correct which makes sense because I’m an absolute gangster. 🔫

Question 2

There were lots of questions on this unit test like this where they just asked me to solve a straight-up integral. I managed to solve this pretty easily on my first attempt (the notes above were my “rough” notes) which I was pretty happy about. I did double check that ∫sec²(x) did in fact equal tan(x), which I was pretty sure was correct but just wanted to be extra sure. So, again, I sort of cheated but did have it correct in the first place…

Question 3

I got this question wrong not thinking through what KA’s answer shows, that _a∫^b g’(x) dx = g(b) – g(a), meaning that you can add g(a) to both sides to solve for g(b). When I looked at KA’s answer, I quickly understood how the algebra works, but I couldn’t picture what was going on, let alone why the algebra works. I spent about 15 minutes reviewing the Fundamental Theorem of Calc. which I sometimes do by putting f(x) = x³ and g(x) = 3x² (i.e. f'(x)) into Desmos, finding the average slope of f(x) and then thinking through how the average slope of f(x) equals 9 which is the average height of g(x). This means that the area under g(x) from 0 ≤ x ≤ 3 is 9 * 3, i.e. height * width. (I’m guessing that wouldn’t have made much sense to anyone reading this, but it’s the way I understand the FToC.) After I did that review, I definitely got a better grasp on the FToC but still had a hard time visualizing what was going on with _a∫^b g’(x) dx + g(a) = g(b). 😔

Question 4

I wasn’t sure what to do initially on this question but figured I needed to take the integral of ₄∫⁸ 1/x dx to get the average slope. This lead to ln(8) – ln (4), but then I wasn’t sure what to do. I looked at it for a minute and then all of a sudden I spontaneously blurted out, “you have to divide it by 4.” This is what I mean when I say I don’t really know what I’m doing but I can somehow sort of just figure it out…

Question 5

This was a crazy question. The images of my hand written notes above were all terrible rough notes I made trying to figure out what was going on. It took me about 15 minutes just to come up with the expression for the integrand but then didn’t know how to integrate what I had — which was (–by/a +b)(–by/a +b)(3^1/2/2)(1/2) — so I threw it into SL and it gave me the solution. Even though I used SL, I was still happy I got it correct as I was certain that I didn’t set up the integral properly. 

Question 6

This was the final question on the test and I was having SUCH a hard time with it because I was sure that I’d set up the integral properly but when I checked my solution on Desmos, we had different answers. I was so annoyed/frustrated since a) it was the last question, and b) it seemed so simple. After about 10 minutes of being annoyed, I realized I was integrating the difference between f(x) and g(x) in the third quadrant (from –5 ≤ x ≤ 0), not in the fourth quadrant (from 0 ≤ x ≤ 2). Once I switched bounds of the integral, I finally got the correct solution. 😮‍💨

I finished the unit test on Friday morning and then got started right away on the Course Challenge. I got through the first 15 questions on Friday and the remaining 15 questions on Saturday. Here are four questions from the test:

AP Calculus AB – Course Challenge

Question 7

This was the 12^th question on the test. It took me a second to remember/think through what I needed to do to solve it, but once I remembered the technique, it wasn’t that difficult. I’ll let my notes do the talking for me on how you solve these types of questions, but I’ll mention that I got a handful of questions like this wrong on my first attempt at the Course Challenge, so it was definitely satisfying getting this one correct.

Question 8

This was the 15^th question on the test and it took me around 10 – 15 minutes to work through. I was pretty pumped when I got the solution since it was also one of the types of questions I got wrong on my first attempt at the CC.

Question 9

This was the question I would have gotten wrong had I not thrown the equation into Desmos to look at the function. When I did, it was pretty clear that the vertical line the question was asking about was at x = –2. Even though I used Desmos, the silver lining is that I knew I needed to use a system-of-equations to solve it, and that 2xy – x³ would need to equal 0 since there’d be no change-in-‘x’ on a vertical line, and I that the numerator, 3x²y – y², wouldn’t be equal to 0 since ‘y’ WOULD change, but I didn’t know what to do after that. 😔

Question 10

This was the final question on the test. It was pretty straightforward in that clearly I just needed to find the slope between x = –6 and x = –3 and assume that would be the best estimate for f’(–4). But even though it was pretty simple, I was still pooping my pants when I clicked submit on my answer knowing that it would determine whether or not I got 100% on a CC for the first time ever. 😰 But luckily I got it correct and was PUMPED. 💪🏼

Like I said, I did get through a handful of questions on the Calculus BC Course Challenge on Sunday morning, but they were all so easy that none of them really seemed like they were worth adding to this post. Nonetheless, I did take screenshots on two of the questions:

AP Calculus BC – Course Challenge

Question 11

This was the first question on the test. I was confused looking at it because it seemed way too simple. I assumed that the Calc. BC CC would be way harder than the Calc. AB CC so I was thrown off looking at this question. It seemed obvious to me that solution was lim_x–>π[cos(x)] = –1 since cos(π) = –1, but because it seemed so simple (and also because I don’t have a ton of confidence with limits), I assumed there must have been something I was missing. I checked SL and it turned out that I was correct in thinking that the solution was –1. 🤷🏻‍♂️

Question 12

I didn’t do the algebra properly here but did figure out the solution on my own. (But of course double checked Desmos…) I assumed that when looking at x –> –∞, the –x in the numerator and 3 in the denominator of the expression would not be relevant since the other terms were higher degrees. I also guessed that, assuming the –x in the numerator would become irrelevant, the (4x⁴)^1/2 would equal 2x² which would then lead to the expression essentially being 2x²/2x² and equalling 1 as x went to negative infinity. Although my algebra wasn’t done properly, it turned out I had it correct.

And that was it for this past week! Like I said, I’m SO pumped to finally be doing the last Course Challenge and final test of all of the Math: High School and College section. I’m guessing the questions on this CC are going to get harder and that I’ll probably get more than three of them wrong and need to redo a few unit tests and then reattempt the CC, BUT, nonetheless, I’m guessing that I’m a few weeks away from being done. (And unlike the last time I thought I was done, this time it will actually be true.) I mentioned in my post from Week 346 that when I finish, I’m going to get started on the Computer Science section of KA which I’m very excited about. I mentioned that I started watching a lecture series from Harvard called CS50 about computer science. I also a bought a book called CODE: The Hidden Language of Computer Hardware and Software, so between that book and CS50, I’ll think I’ll be well primed to get started on the CS section of KA. And from what I’ve seen in the CS50 videos so far, the math in CS seems pretty simple, not to mention that I feel WAY more confident in my ability to learnt can I used to, so I’ve got all of that going for me which is nice.

But before I get ahead of myself, I still need to finish off the Math section of KA. So, as always, fingers crossed I can have a productive week and get it done sooner rather than later! 🤞🏼