For the first time in what feels like forever, I actually had a pretty successful and productive week on KA! 🥳 I started the week off well by getting through the Limits and Continuity unit test on Tuesday on my first attempt. After that, I began working on the Integration and Accumulation of Change unit test and was pleasantly surprised with what I remembered and how easy I found the questions. That being said, I didn’t actually pass the unit test, BUT I still was happy with my effort and felt like I did pretty well. The end of this math journey is so close that I can TASTE it. 🤤 The one bad piece of news is that I realized I have a bit more work to finish everything off than I though which I’ll explain at the end of this post. But even still, I think I’m only a few months away from reading the finish line, especially if I can have a few more productive weeks like this past one!
Here are three questions from the Limits and Continuity unit test:
Calculus BC – Unit 1 – Limits and Continuity – Unit Test
Question 1
This was the question that stumped me at the end of last week which I left to do at the beginning of this week. It took me ~10–15 minutes to get through, and the photo of my notes makes it look like I just sailed through it, but there were three pages of rough notes that I didn’t take photos of. 😬 For some reason I couldn’t figure out the algebra to get (x + 2) in the numerator and denominator after multiplying the expression in question by (3 + (6x + 21)1/2) / (3 + (6x + 21)1/2). It took me a while, but I eventually figured it out and didn’t even need to consult Symbolab.
Question 2
I was a bit surprised when I got this question correct but even more surprised after realizing that I solved it essentially the same way KA did. Before seeing their solution, I assumed that KA’s answer would have used some fancy algebra to solve it. When I did it in my head, I just thought through what the value of the expression would be as x got closer and closer to 2 from the (–) and (+) sides (ex. 1.9, 1.99, 1.999 and 2.1, 2.01, 2.001, etc.). Doing that, I was able to determine that the function would shoot up to ∞ in both directions. Pretty simple, which is why I thought KA’s solution would have been more rigorous, but apparently not.
Question 3
The reason why I was able to solve this question on my first attempt with the same type of algebra KA used is because I had seen a question just like this last week and had the algebra fresh in my mind. I knew that I needed to multiply the numerator and the denominator by one-over the highest ordered value of x (i.e. 1/x1/4) and then substitute ∞ in for all the x’s. You can see that’s what I did to show that the limit was 0.
Like I said, I finished the unit test on Tuesday making it through 14/30 questions in the morning and then finishing off the remaining 16 questions in the afternoon in about 40 minutes. Considering how long the was, it was a huge relief!
After I got through the Limits and Continuity unit test, I was shocked and bummed out when I realized the Integration and Accumulation of Change unit test was 34 questions long. 😳 For some reason I looked back in Calculus AB and realized the name of the exercise that I needed to level up was in the both calculus courses:
You can see that the exercise shows up in both AB and BC courses, in the same unit, Unit 6 – Integration and Accumulation of Change. However, the Calculus AB unit test is 30 questions long, whereas the Calculus BC unit test was 34. So, I decided to do the Integration and Accumulation of Change unit test thinking that when I pass that unit test, it might count for both courses. I don’t know if that’s true, but I’m hoping it is!
As I said, I didn’t actually pass this unit test this week. I kept getting 8–10 questions correct but then would eventually get one wrong. Considering how long it’s been since I’ve done this type of math, I was actually pretty happy with that. Here are six questions from the test:
Calculus AB – Unit 6 – Integration and Accumulation of Change – Unit Test
Question 5
This was the first question that came up on the unit test. I was completely shocked and PUMPED that I was able to figure it out without cheating. It took me about 30–40 minutes to work through and I sort of did it backwards. I don’t think anyone would be able to understand my notes and my method of madness to solve it — they were supposed to just be rough notes that I wasn’t planning to add here but I figured I might as well. After looking at my derivatives cheat sheet (which I guess you could consider cheating) I realized that the integrand looked like the shape of the derivative of arcsin(x), ∫1/(1 – x2)1/2). With that in mind, I used algebra to get the expression in the denominator of the integrand to look something like (1 – x2)1/2. Honestly, I can’t remember off the top of my head how I did it and don’t feel like digging into it now but regardless, I did it and that’s all that matters. 💪🏼
Question 6
After working through that tricky trig identity problem above, this question was a welcomed relief. I wasn’t sure if I was supposed to figure out the linear functions that created the shape and throw them into and integral. I could have done that but I also knew that I could just solve it by breaking the area into shapes (a square and two triangles) and then find their individual areas, so that’s what I did. As you can see from KA’s answer, they essentially did the same thing but split up the area into a trapezoid and a triangle. Same difference.
Question 7
I had to spend about three minutes reviewing the reverse power rule to solve this question, but it came back quickly. It was pretty satisfying and enjoyable working through this question, being able to solve it so easily. Even though it’s not that difficult of an integral, I was still pumped that I remembered how to solve it without any difficulty.
Question 8
I got this question wrong which was annoying in general, but more so because I spent a lot of time making notes on it. I thought I needed find the mid-point of the function between each x-value and then find sum up their values. I misread the question and didn’t see that it said “right Riemann sum”, so technique was wrong. This was the first Riemann sum question I came across on the test though, so I may have gotten it wrong even if I had seen that it was asking for a right Riemann sum.
Question 9
I got this question correct and was happy that I remembered how sigma notation worked, but I didn’t really know what I was doing when I answered this question. As you can see from my notes, I just started writing out the expanded equation inputting 1–4 for each value in the index, but I quickly realized the equation looked just like the third option of the solutions. Initially I thought I was going to need to do some algebra at some point or something, but that turned not to be the case. 🤷🏻♂️
Question 10
You can see from my notes that I clearly didn’t solve this question the same way KA did. After looking at KA’s answer, I watched the video it recommended to understand why their method worked, which I did, but it didn’t help me understand. 😕 I spent almost an hour reviewing the Fundamental Theorem of Calculus and was able to remember what it is and more-or-less how it works, but it didn’t help me completely understand why KA’s answer works. It was good review but still disappointing that I couldn’t figure it out. 👎🏼
Question 11
I was 14 questions into the test when I got this question wrong. 😞 I initially thought I should try to use polynomial long division to simplify the integrand (which turned out to be correct), but I couldn’t remember how to do it. I tried a few other methods to simplify it but couldn’t make any progress. I eventually threw the integral into Symbolab and they gave me the wrong answer! 😡 (Which was probably for the best considering I was cheating…) Once I saw KA’s answer, it took me about 20 minutes to review and remember how polynomial long division works. I’m not going to try to explain how it works here, but when I finally wrapped my head around it, it was a relief.
And that was it for this past week! I’m optimistic that I’ll be able to get through this unit test soon, hopefully by the end of this upcoming week. 🤞🏼 I noticed that KA now says I’m 100% complete Calculus BC, which isn’t true because there are a few exercises that need to take from 80 Mastery Points to 100 Mastery Points. I also realized that’s true for a few exercises in Statistics and Probability, as well. SO, my new (and final) plan is to first make sure every single exercise in Math: High School and College is at 100 M.P., and then finish by getting a score of at least 90% on the Course Challenge for Multivariable Calculus. (It’s currently sitting at 70% which I completely forgot about.) Once I get at least 90% on that Course Challenge, I will OFFICIALLY be done. 😭
(Six and a half years later…)