I had a decent good week on KA, although it could have been better. (But I guess I could always say that…) I made it through two unit tests in Calc. AB; one from the fifth unit, Applying Derivatives to Analyze Functions, and one from the seventh unit, Differential Equation. Neither were easy so I’m happy that I managed to get through those. I started the last unit test I need to redo which was from Unit 8, Applications of Derivatives, but didn’t make much progress on it. I basically ran out of time when I got to it, so that was the main reason why I didn’t make much headway. I was hoping to get through all three unit tests by the end of the week so that I could hit the ground running this coming week with the Course Challenge, but unfortunately that didn’t happen. 👎🏼 Still, I’m happy with the effort I put in as I’m guessing I studied for 10 or more hours, so I’ve at least got that going for me which is nice. And I’m getting SO close to finishing this all off which I’m trying to stay focused on. Even though it’s taking longer than I expected (about 6 years longer…), I’m sure I’ll get there soon! 💪🏼

As I said, the first unit test I got through was from the fifth unit, Applying Derivatives to Analyze Functions. Here are seven questions from the test:

AP College Calculus AB – Unit 5 ­– Unit Test – Applying Derivatives to Analyze Functions

Question 1

The was a pretty straightforward question on concavity, but I thought it was worth adding here because it reminded me that if a question asks for concavity, h’(x) (i.e. the slope) doesn’t factor in at all and is irrelevant in regards to figuring out the solution. I’d made the mistake before of thinking I first needed to use h’(x) to figure out the critical points, but I don’t think that was relevant here (although I could be wrong). Plus, I also think you can just find the critical points using h’’(x).

Question 2

I would have got this question wrong, but I used Desmos to check my answer. 😒 I screwed up h’(x) = –12x(x²– 2x – 2) by factoring it wrong and thinking it was h’(x) = –12x(x² + 2x + 3), so I thought h’(x) = 0 only at x = 0. After looking at h(x) on Desmos, I was able to think through that if the end points were concave down, then the critical points would be relative maximums. I determined at h’’(–2) and h’(4) the end points were (–), i.e. concave down and then knew if I checked h(–1), h(0), and h(3) I’d find out which one was the max which ended up being h(3) = 135.

Question 3

I got this question wrong on the 18^th and final question of the unit test. 😡 I didn’t think that I needed to put f(x) into x² + y² = h². Even after looking at KA’s answer, I still don’t really understand how you can square (h(x))² = a(x) and still find out that x = ½ which equals a minimum point, but the screenshot of Desmos helps me to visualize it a bit.

Question 4

When I got the last question 3 from above wrong, I had to go back and redo an exercise before I could redo the unit test. The exercise was about using calculus for what’s called “optimization” and I got absolutely smoked on it. I managed to get through it but mostly just because after about 10 attempts, I’d seen most of the questions so I knew what the solutions were. 🙃 Once I finished the exercise, I came back to the unit test and was given this question above which was one I saw in the exercise, so I knew not only how to solve it but also knew what the solution was. I think I may have been able to figure this one out on my own even if I hadn’t just seen it on the exercise since it’s not that hard but, in my opinion, it was still pretty difficult compared to most of the other questions from this unit test. 🥵

Question 5

When I saw this question, I didn’t know what to do and initially assumed and wouldn’t be able to answer it, but I thought through that a horizontal line would have dy/dx = 0 and therefore 4(3 – x) would have to be equal to 0, therefore x would equal 3. From there I inputted 3 into the OG equation to get y = 5 just like KA’s answer, but I didn’t double check to make sure that 3(5)² ≠ 0, which KA did. So, in that sense I didn’t do it exactly correct, but I still figured out the correct solution in the end. 🤷🏻‍♂️

Question 6

The picture of my hand written notes above were my rough notes and first attempt at this question. I bring that up just to say that I was pretty pumped/proud of myself that I was able to get the solution to this question on my first attempt. Before I started, I didn’t think I’d get the derivative on the first try and was really happy that I did. It was a nice reminder that even though I’m struggling through Calc. AB more than I thought I would, I have managed to learn and retain some stuff over the past 6.5 years. 😮‍💨

Question 7

Once again, I would have got this question wrong but used Desmos to check my answer. 👎🏼 I screwed up the derivative using the power rule on xe^–2x instead of the product rule. Also, the algebra was a bit tricky for me, but I eventually figured it out on my own.

I’m disappointed that I had to use Desmos as much as I did to get through this unit test, but the silver lining is that, except for the optimization questions that came up, for the most part I knew how to do all the math — there were just a lot of questions that involved many small steps where I could easily make a careless mistake, which is why I would use Desmos to double check everything and don’t feel too bad about using it. (But I do feel a little bit bad… And to be fair, there were definitely some tricky bits of alg., trig, and calc that came up where I benefited from by using Desmos…)

I started the next unit test, Differential Equations, from Unit 7 on Thursday. Here are two questions from that test:

AP College Calculus AB – Unit 7 ­– Unit Test – Differential Equations

Question 8

I got smoked on the question. (My written notes above were from after getting this question wrong.) I’m not going to explain it step by step but will say that, looking at KA’s answer, I can more or less follow along with algebra and calculus for this proof but find it very difficult. I also found substituting ‘x’, ‘g(x)’ and ‘C’ into the final equation very confusing. 😞

Question 9

I managed to get this question correct but only because I’d worked through the same type of question previously, got that question wrong, and then learned the derivation of the formulas. 😑 Looking at KA’s answer, I find the algebra, exponent rules, log rules, etc. pretty difficult. The key words in the question that tell you the equation is dV/dt = kV(t) is “rate” (i.e. dV/dt), “proportional”, (i.e. ‘k’), and “value at time” (i.e. V(t)).

I found that unit test very hard but actually managed to get through it all on Thursday. (It only took me about 2.5 hours and 10 attempts at it…) I started the final unit test from Applications of Integration on Friday but had a bunch of other things going on over Friday, Saturday, and Sunday, so I only spent about an hour in total looking at questions from it. I only took screenshots and notes on one question since I knew I was going to have to do it this coming week and I can talk about it in next week’s post. But nonetheless, here’s that question:

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

Question 10

I’m not going to explain my notes here since I know I’ll have to talk about questions like this next week, but I’ll mention that I find (r₂² – r₁²) confusing. In my mind it should be (r₂ – r₁)², so I’m hoping I’ll get some clarity on why that’s not the case this coming week.

And that was it for this past week. I feel like I’ve got some decent momentum going and am hoping I can get through this final unit test early this coming week. It’s 19 questions long though, so that could make things difficult. Still, I think I’ve got a shot at not only making it through that unit test but potentially also getting through the Course Challenge. 🙏🏼 When I think back to the questions I got wrong on my first attempt at the Course Challenge, I feel much more confident with those types of questions now than I did before, so I think I’ll have a decent shot at getting ≥90% on it on my next attempt. As always, fingers I have a productive week so I can get both the unit test AND the Course Challenge checked off my list! 🤞🏼