I felt like I wasn’t making much progress through KA this week but, looking back on my notes, I’m realizing that I actually got quite a lot of work done and made some decent headway. (Woo!) On Tuesday I made it through the Probability and Combinatorics unit test which was the final unit I needed to get through in Precalculus. I then restarted Calc.2 beginning with the unit Integration Techniques. I forgot to write down how far along I already was through this unit before I began, but I think it was close to ~60-65% of the way through it. I’m now finished 89% of it and am currently 4/11 questions into the unit test, however I got the first question of the test wrong so I’m going to have to redo the entire thing. In any case, I’m pumped to be back working on Calc.2 and am hopeful I can get through the course in a few months.

The Probability and Combinatorics unit test was only 12 questions long but it took me 7 attempts (SEVEN! 🤬) to get through it. I kept making a single, simple mistake on each attempt leading me to get one question wrong and then needing to redo the entire thing. Although for the most part it was pretty straightforward, there were a few questions that came up where I had to stop and think through them and took me a bit of time to get to the solution. Here are two example questions from the unit test:

Question 1

I found this question hard at first because I didn’t understand what it was asking. (Comprehension on questions like this has always been a weakness of mine. 😔) One thing that I heard Sal say a LONG time ago that I’ve always remembered about probability was that you can always draw a probability tree to make it clear what the exact probability of each outcome is. As you can see from my notes, that’s what I did after spending some time thinking through what exactly the question was asking. 

Question 2

I’m sure that when I was working through the Statistics and Probability course a million years ago I would have had the formula to solve this type of question at the top of my head and it would have taken me 2 seconds to answer. I hadn’t seen a question like this in such a long time, however, that it took me a few minutes to think through. I realized I needed to multiply the number of shots made by the probability of making them and then add up the expected number of shots made from each category. Boom.

As I mentioned, I began Calc.2 on Tuesday and got back to the unit Linear Partial Fractions which was the unit I was stuck on a few weeks ago that led me to going back and working through Precalculus. This time around it was very straightforward and only took me a day to get through the 1 video and 1 exercise from this section. Here’s an example question I worked through:

Question 3

To find the solution to the given integral, the first step is to use partial fractions to simplify the function in the integrand. You can see that’s what I did in the bottom half of the first photo and the top three quarters of the second photo. At the top of the second photo, you can see where I wrote and worked through [@ x = 3] and then [@ x = (–2)] which you do to cancel out the B and A terms respectively in order to solve for the other. Once you solve for A and B, you can then rewrite the integral using those terms (a.k.a. partial fractions?) which makes it relatively straightforward to solve the integral.

I began the final section of Integration Techniques on Thursday which was titled Improper Integrals. This section taught me that there are sometimes integrals that have one or both bounds that are being calculated to infinity and yet the integral can have a finite value. The same concept can also apply to an integral that has finite bounds but goes on in the positive or negative y-direction to infinity. Here are the notes I took:

Two of the key takeaways I learned from this section were the definitions for the two terms convergent and divergent. As I said in my notes above, a convergent improper integral is when it equals a finite value whereas a divergent improper integral is when the integral’s value grows to infinity. (To be honest, I’m not sure if that’s exactly correct or if I’m saying it the right way, but that’s my understanding of it as of now.)

When it comes to improper integrals, in the case of a bound being calculated to infinity, the key is to set the bound to equal a variable such as a and then, after integrating the function, finding the limit of the antiderivative as a approaches infinity. Here’s an example:

I wasn’t planning on adding this question to my post which is why it’s a bit messy but it does a good job of showing how to solve this type of question. As I just mentioned, the first step in this question is to set the upper bound of the integral to a and state that the integral then equals lim­­_a->∞ 0∫^a sin(x) dx. As the equation continues, you get to the point where lim­­_a->∞ (–cos(a)) + 1 and since (–cos(∞)) doesn’t exist (since you’d just be going around the unit circle forever) the improper integral doesn’t have a solution and is divergent. (Or at least I’m pretty sure that’s what’s going on.)

Here’s another example from the exercise I worked through:

Question 4

After wrapping up that section on improper integrals, I started the unit test on Saturday and got through the first 4 of 11 questions but unfortunately got the first question wrong. This was the first question:

It took me about 20 minutes and I had to look at the titles of each section of this unit but it eventually dawned on me that polynomial long division was likely the right technique to use for this question. I more-or-less got to the correct solution using polynomial long division but didn’t really know what I was doing or looking at once I got to the solution and I eventually ended up getting the question wrong. As you can see from my note above, my main mistake was not understanding that the remainder, which was –3, needed to be used as the numerator above the initial numerator, 2x + 4. Had I known/done that, the integral would have become pretty straightforward to solve. I was annoyed that I got the question wrong but I’m hopeful that I’ll find it much easier to 1) recognize quicker that polynomial long division is the technique needed to solve these types of integrals, and 2) remember how to actually do the polynomial long division.

The third question I worked on required me to use integration by parts. I was happy that I quickly realized which technique I needed to use but I couldn’t remember the formula which is a(x)b(x) – ∫ a(x)b’(x) dx = ∫ a’(x)b(x) dx. I looked it up and, after that, the question wasn’t too difficult to solve:

I’m hoping I can get through the unit test for Integration Techniques (980/1,100 M.P.) early this week and then move on to the following unit Differential Equations (1,100/1,300 M.P.). The unit after that, Applications of Integrals (1,900/2,000), is also very close to being done so it’d be great to get through all three units in the next few weeks. I’m a bit concerned that I’ll struggle with the Differential Equations unit test since it was such a long time ago that I worked through that unit in Calc.1, but I’m sure it will be really useful and good review for me to go through. Reaching the end of Calc.2 is within sight which is crazy considering that was my goal THREE YEARS ago. I have no plans to stop using KA once I get through the math section – I want to work through the physics section next! – so, considering how much I’ve learned in the past three years, it’s interesting to think of how much I could learn in three more! 🤓 📚