FINALLY, after 82 weeks, I’ve made it to calculus. It took me three days to get through the Precalculus course challenge this week, which is the longest amount of time I’ve spent working on a single test/challenge, but I finished it with 29/30 questions correct! For the most part I felt pretty good about my understanding of everything that was covered. I generally knew what I had to do to answer each question, however I did double check quite a few of my answers using Desmos before entering them. I also had to look up a few formulas and definitions but that only happened a handful of times. The worst part about the challenge was that the only question I got wrong was the final question! Even still, I’m really happy with getting ~97% and pumped to finally (FINALLY!) be starting calculus.
Before I did the course challenge, I had 3 videos left to watch in the unit Probability and Combinatorics. Each video dealt with a specific question which combined probability and combinatorics and required me to think slightly outside the box to solve. I found these questions pretty difficult and didn’t manage to solve any of them perfectly before watching Sal do it but I had the right idea for each of them. The silver lining was I had no issue understanding what Sal did as he went through them. After going through those 3 videos, I then moved straight onto the course challenge on Wednesday morning.
The first question I had to double check before answering was a question dealing with complex numbers. I was asked to simplify the expression (√2 + √2i)^6. Embarrassingly enough, the first thing I had to think about was what the value of i was. I eventually remembered that i^2 = -1 and then worked through the question in the following way:
- (√2 + √2i)^2 = (√2 + √2i) * (√2 + √2i)
- √2^2 + √2^2i + √2^2i + √2^2i^2
- = 2 + 2i + 2i + 2(-1)
- = 2 + 4i -2
- = 4i
- (√2 + √2i)^6 = (√2 + √2i)^2 * (√2 + √2i)^2 * (√2 + √2i)^2
- = (4i) * (4i) * (4i)
- = (16i^2) * (4i)
- = (16 * -1) * (4i)
- = (-16) * (4i)
- = -64i
The next tricky question I came up against asked me to multiply matrices. I was pretty sure the way to do this was by multiplying the row from the first matrix by the column of the second matrix for each corresponding cell of the product. This turned out to be correct but I still had to double check my notes before submitting my answer.
The most difficult question I went through on Wednesday showed me a diagram of a square that was transformed on a coordinate plane with a 180° rotation. I was asked to figure out the values of the vector that was used for the transformation. I remembered that the vector would multiply each vertex of the square and would need to come first in the equation. I Googled “180° vector rotation” and was able to figure out how to do it. I’m not going to go into the details of it but here’s my work from this question:
I finished Wednesday with 12/12 questions correct which I was happy about but surprised that I only managed to get through 1/3 of the the challenge.
The very first question I went through on Thursday, question 13, showed me a hyperbola and asked me to come up with the exact formula for it. It took me a second but I was able to remember the formula to determine the coordinates of the foci and also the general formula for the hyperbola itself. That being said, I marked down in my notes that I likely wouldn’t have been able to solve this question if I hadn’t just recently worked on hyperbolas and had the formulas memorized. I also used Desmos to confirm my answer before submitting it.
The next difficult question I went through on Thursday was question 17. I was asked to find the inverse matrix of a matrix that was given. It took me a minute to remember that I needed to multiply the given matrix by a certain formula to come up with it’s inverse. I managed to remember ~90% of the formula but when I double checked my notes I realized I had missed one part of it. I multiplied the given matrix by the inverse formula to come up with what I thought was the correct answer. I also remembered that if multiplying a matrix by it’s inverse results in the product being the identity matrix. I multiplied my answer with the given matrix which resulted in the identity matrix so I was sure I solved it properly.
Question 18 asked me to find the conjugate of (3i + 8) and I couldn’t remember what that meant. I looked it up in my notes and remembered the conjugate of a complex number is the same number except that the (+/-) sign in front of the imaginary part is the opposite sign, i.e. the answer was (-3i + 8). From what I remember, you can add the complex conjugate to the original complex number so that the imaginary part of the sum equals 0 and so you’re left with only a real number (which is double what the real part of the original complex number was).
The only other question I went through on Thursday that I had difficulty with asked me to do polynomial long division. I had the right idea of how to answer the question but had to double check my notes to remember the correct process. Part of the reason why I second guessed how it worked was because the question was unlike all the other polynomial long division questions I’ve ever worked though in that the quotient was a 1st degree integer with no variable attached to it and the remainder was a 3rd degree polynomial. Once again I won’t go into the details of the math but here’s the question from my notes:
I finished Thursday with 23/23 questions correct which I was really happy about but, again, surprised it was taking me so long.
Friday turned out to be the easiest day not just because I had fewer questions to work through but because I found the questions themselves easier than the ones from Wednesday and Thursday. I still had to double check my notes a few times but for the most part had the questions completely figured out before doing so. The only question I got wrong was the very last question, question 30:
For some reason I thought he question was asking me what 8 + (-5) + (-5) equaled and so I inputted -2. As you can see from the screen shot, clearly my interpretation was wrong as there was no addition in the question (*face-palm*). Once I looked at how to properly solve the question it seemed pretty obvious. I rushed through the question and may have been able to get the correct answer if I had spent a bit more time thinking about it but, nonetheless, I got it wrong but was still happy with my final score on the challenge.
I began the course Calculus 1 on Saturday starting with the first unit Limits and Continuity. The first and only thing I got through was the definition of a Limit:
As you can see from the photo, a limit in calculus is a point on a function which you get closer and closer to without ever actually reaching that specific point. In the photo I used the word “straddle”, as in the limit straddles either side of the point, but another way to describe this idea is that the limit gets infinitely closer to the point on the function without ever reaching that point. As you can also see from the bottom right corner of the photo, a limit cannot exist at a point on a piecemeal function where the function splits. It is possible, however, to have a limit on a function at a point where the function is undefined or where the value of the point on the function does not equal the value of the limit:
That’s all I managed to get through this past week from the unit Limits and Continuity (80/3500 M.P.) but I was still happy with what I learned. This unit is huge so my goal is to get through it in ~2 months. The course Calculus 1 itself has 8 units in it and is 17, 100 M.P. long so I won’t be surprised if it takes me to the end of the year to get through this course. It’s weird to think that I’ll be spending the next ~8 months studying a calculus and working through a single course without really knowing where this is all leading but I’m still enjoying it and feel like it will all eventually lead to something useful. My goal has always been to learn calculus so it’s sick that I’m finally taking my first steps into the subject. I just hope it doesn’t take me another 82 weeks to get to the other side of it.