Week 184 – Mar. 6th to Mar. 12th

I only made it through 17/30 questions this week on the Calculus 2 course challenge which was pretty disappointing, but on the plus side I only got two wrong! I took quite a bit of time on most of the questions but didn’t come up against any where I felt like I had no understanding of what was being asked of me. I did look up some of my answers on Desmos and Google before submitting my answer BUT, when I did, I always had the correct answer before looking them up so I don’t really feel like I cheated. I’ve been working on Calc.2 since Week 147 (…😳) so I definitely thought I might get a question where I had no clue what was going on. As you’ll see in some of the questions below, in the two questions I got wrong I had a vague idea of what was going on but didn’t know either of them well enough to figure them out. Regardless, I’m still happy with what I was able to remember and that I didn’t get completely smoked this week even though I didn’t get too far through the test. 

Here are six questions I worked through this week:

Question 1

Before working through this question, the first thing I did was graph it on Desmos. The function looks like a very elongated sine curve. I didn’t know what exactly I was supposed to do for this question but, since the given answers looked like F(b) – F(a), I assumed I had to throw the function into an integral and use t = 300 and t = 240 as the respective upper and lower bounds. As you can hopefully decipher from my messy notes, that turned out to be correct although I don’t really understand why this questions works and can’t really visualize what’s going on.

Question 2

This was a question that I looked up before answering but ended up having the correct answer anyways. I figured out that the first 4 terms of the series evaluated to 35121, and 2. I was pretty sure that for the integral test to work all the terms had to continuously decrease, so I assumed that the test did not apply. When I looked it up I was reminded that the conditions that must be met for the integral test to apply are that the series must be 1) positive, 2) continuous and 3) decreasing to infinity, so it turned out I was correct. 

Question 3

As you can see from my notes, I didn’t solve this question the same way that KA did. I’m guessing that my calculus wasn’t technically correct in that I’m not sure if you’re really supposed to say ∫ y’(x) = y(x), but I’m 99% certain that’s conceptually correct since taking the integral of a derivative brings the function back to the OG function. Even if my calculus isn’t correct, I’m actually happy that I have a strong enough grasp on calculus, in general, to be able to work through and visualize questions like this. 😤

Question 4

This question took me a long time to work through. When I initially looked at the integral I had no idea where to start or how to evaluate it, but I then remembered the integration-by-parts technique I used a LONG time ago which seemed like the right technique to use. I was surprised that I was able to not only remember the formula itself but could even derive it from he product rule. The thing that took me so long to figure out was how to combine the factors in order to use IBP. It turned out that I needed to split the integrand up to –2x–2 * ln(x) and that a’(x) = –2x–2 and a(x) = –2x–1/–1 = 2/x. You can then input the functions into IBP and it’s pretty straightforward to solve from there.

Question 5

I got this question wrong but “could” have gotten it correct. I was able to figure out the bounds by inputting θ = 0 and θ = π/3 into 1 – 2cos(θ). I also had a vague recollection that for some reason the integrand for this type of question was usually divided by 2 but couldn’t figure out why that would be the case. I tried to algebraically manipulate the formulas for the area and circumference of a circle using r = (1 – 2cos(θ)) to get something that looked like answer A) from KA but couldn’t figure out how to do it. The best I got to was that using πr2 where r = (1 – 2cos(θ)), the integrand should have been π(1 – 4cos(θ) + 4cos2(θ)) which was obviously wrong. As you can see in the bottom left corner of my notes, the reason why the integrand could take the form of r2/2 is because the fraction of the circle that’s being calculated is θ*r/2πr = θ/2π. Multiplying that quotient by the actual area for a circle, πr2, is what leaves you with r2/2.

Question 6

This was similar to the last question I worked through at the end of Saturday but this wasn’t the exact question I worked on. I accidentally closed the browser before screen shutting the actual question but I ended up getting the question wrong. The screen shots above are from a video that worked through the same type of question and helped me figure out how to solve these questions. This probably makes me sound pretty dumb, but I found it hard to understand that the part of the question that states, “the y-coordinate is increasing at a constant rate of 2 unites per minute” implied that the velocity in the y-direction was 2. 🤦🏻‍♂️ Looking at how KA solved the question and then working through it myself, it all seems pretty obvious now. Using Desmos, I can visualize a dot on (4, 4) where the dot’s moving up at 2 units/minute and to the left 2units/minute. I can then also visualize a hypothetical right-triangle of the dot’s velocity, i.e. a triangle where the velocity of the dot is the hypotenuse of the triangle. I had a really hard time with this question before going back through this video so hopefully if I run into this type of question again I won’t struggle nearly as much.

I think it’s possible that I could get through the rest of the 13 questions on the course challenge without getting more than 1 question wrong which would be sick. My goal for course challenges is to get >90% so if I manage to do that I’ll FINALLY be through Calculus 2 and will able to move on to Multivariable Calculus (100/4,800 M.P.). I have a feeling that course will take a LONG time to get through so it’s not like I’ll be done studying calculus anytime soon, but it’ll still be pretty motivating to be one step closer to finishing it off.