FINALLY!!! 😤 I finally made it through the Series unit test this week. It took me two tries to get through it this week but only ended up getting two questions wrong on my first attempt. Going through the questions, I felt like I had a pretty good understanding of what I was being asked and, for the most part, could visualize the math and the functions/dot plots of the series I worked through. I also had a pretty solid grasp on all the notation and what it was denoting. (Actually, now that I think of it, I don’t think there was any notation I saw on my second attempt that I didn’t fully comprehend. ☺️) There were definitively some questions that I still had a hard time understanding but I was able to remember the patterns/formulas of how to solve them even if I didn’t understand why they worked. I’m hoping that over the next few months some of what I still don’t understand will become clearer. After finishing the unit test, I only made it through one question on the Calc.2 Course Challenge. I got it correct but kind of cheated. It’s the last question at the end of this post. Regardless, I’m PUMPED to finally be moving on to the course challenge and hoping it won’t take me to long to get through.
I made it through 16/20 questions on the unit test on Tuesday which I thought was a good start, apart from getting 2 out of the first 4 questions wrong. 😠 Having such a bad start, for a minute I felt like my progress learning about series was going backwards. Then I got 13 questions correct in a row and felt much better. It took me from Wednesday to Friday get through my second attempt at the unit test so, if I add both attempts together, I ended up getting 33 questions correct in a row! Although I don’t understand everything about series, getting that many questions correct back-to-back makes me think that I must know SOME things about series. In any case, here are three questions I worked though between the two attempts:
Question 1
I didn’t actually fully solve this question. This is the type of question I don’t like, where you have to have part of the solution memorized ahead of time. One key to finding the solution to this question is that you have to have the Maclaurin polynomial for ex memorized. To double check what that polynomial was, you can see that’s what I did in the top half of my note. In the bottom half you can see that I then wrote out the first 4 terms (including 0) of the series. I got to the point where the series would alternate between positive and negative but then ran out of space and didn’t want to start a new page. I knew that if x was actually –x then the series would stay negative meaning that the solution was e–x.
Question 2
This is a question that I don’t really understand why it works. The question asks to find what is the first term in the index where the series is within +/– 0.0001 of the actual value of the series. The way I think about this question is that each successive term is alternating above and below the x-axis and as you go further down the x-axis each term is getting closer and closer to y = 0. This means that when you get to the 10th term all the terms after are going to be less-than 1/10,000 since they’re all getting closer and closer to y = 0. The hard part to understand is that from there the sum of each two successive terms will partly cancel each other out since one will be positive and one will be negative meaning that their sum will be substantially closer to 0 than the individual term before…
…..
OK, I’m having a really hard to putting this into words but I can somewhat visualize what’s happening in my head. That said, I don’t understand why the error for this question is 9 and not 10. It seems to me that since the 10th term in this series would equal –1/10,000 = +/– 0.0001 then the answer should be 10. 🤔
Question 3
Again, this is one of those questions where you have to have part of the solution memorized. To solve this question I had to first know that the Maclaurin polynomial for sin(x) = (1 – x3/3! + x5/5! – … ). I don’t really understand why it’s ok to the simply divide both sides of the equation by x (i.e. I can’t visualize what’s going on) but the algebra of how to do it is pretty straightforward, as you can see from my notes.
When I finished the unit test on Friday, I started the course challenge right away but only made it through the first question. It was a trig identity question (a.k.a. my arch nemesis of questions). This was it:
Question 4
I had no intuition when I first saw this question as to which answer was correct. My gut told me that it wasn’t “None of the above” since the denominator in the integrand looked a lot like (c2 – a2)1/2 so I assumed there’d be a trig identity which would work to simplify the question. Through trial and error, I then inputted some of the answers into the integrand in place of x and tried to solve/simplify the expression. I never got to a point where I thought I fully knew which of the solutions was correct, but when I used x = (13)1/2(sec(θ)) I was able to get the integral into ∫ (13(sec(θ))/(13)1/2(tan(θ)) dθ which was the simplest solution I could come up with so I assumed it was correct. I threw the integral into Symbolab and the answer it gave me used sec(θ) so I was more confident with my answer but therefore kind of cheated. Turns out I was correct but with a bit of an asterisk.
First off, I didn’t realize that dx doesn’t just turn into dθ but it actually goes to sec(θ)tan(θ)d(θ). In the bottom half of the second page of my notes you can see where I worked through the calculus to solve for dx. You simply use the chain rule and have to know that the derivative of sec(x) = sec(x)tan(x) so it was pretty straightforward after I saw KA’s answer but I definitely didn’t realize I needed to do that before answering the question.
My problem with this question is that even once I get to the ‘correct’ answer which was ∫ (13(sec(θ))/(13)1/2(tan(θ)) sec(θ)tan(θ)dθ which can be simplified further to 13∫ sec3(θ)dθ, I still don’t see how this solution makes the question much easier to solve. I put the solution into Symbolab and this is what it gave me:
So even though you can ‘simplify’ the integrand it doesn’t seem like it makes finding the solution that much easier…
It’s a bit crazy to think that after ~3.5 years I’m finally working through the Calc.2 Course challenge. I’m assuming it will take me at least a few weeks to get through but hopefully I can be moving on to the next course, Multivariable Calculus (100/4,800 M.P.), by April. That course is half the size of Calc.2 so hopefully it’ll take me half as long to get through but being that each successive subject/course has been getting harder and harder, that may not be the case. Whenever I do manage to get through that course, the final calculus course is Differential Equations which is only made up of videos and doesn’t have any exercises, so I’m assuming that course will be pretty quick to get through. I’d like to get through everything in less-than 6 months, i.e. the 4 year mark. As far away as that sounds, I’m sure there’s a lot for me to learn between now and then so I’m assuming it will be a pretty tight deadline. Either way, the light at the end of the calculus tunnel is starting to get brighter! Fingers crossed this course challenge doesn’t take too long! 🤞🏼