I made some solid progress this week working through KA. I passed the unit test on Wednesday (🥳) but did have to look up two or three questions before answering them. There was only one question where I would have gotten it wrong if I hadn’t looked it up, however, and it was from a section that I never really understood, so I wasn’t too disappointed with myself. Plus, as I’ve mentioned before, I always better understand what I’ve worked through once I move forward into the following units/courses/subjects so I think moving forward even though I didn’t 100% know every question on the test is the best strategy for me to understand what I’ve learned up to this point anyways as opposed to redoing the test and reviewing the few questions I got wrong. (Longest sentence ever.) I then started the following unit, Applications of Multivariable Derivatives, and just about made it through the first section, Tangent Planes and Local Linearization, but fell just short. Overall, I’m really happy with my effort this week and the progress I made. WOO! 

Here are six questions I worked through on the test:

Question 1

Question 2

Question 3

Although I didn’t write out any math to answer this question, I wanted to add it here and mention that this question was useful practice for me to think through how f(,x, y) = z and visualize how the 3D representation of the different functions would look. Actually, I only ended up thinking through the first equation since that happened to be the correct answer, but it was useful practice doing the math in my head and visualizing what would happen when inputting different values of x and y.

Question 4

This is one of the questions I had to look up before answering which is what the screen shot of the video is from. As you can see, the math itself isn’t too difficult. What’s tricky is just visualizing/understanding what’s going on (i.e. understanding the dreaded why of why this question works). To be honest, it still seems a little confusing to me but I believe it’s just saying that when you have a vector inside a multivariable function that outputs a scalar value, the derivative of that vector equates to a derivative in the output. (I don’t know if that makes sense and even if it does, I could definitely be wrong.)

Question 5

(I just realized I didn’t capture the top half of the question in my screen shot. As you can see from my notes, the vector in question was p(t) = (3sin(2t), 3cos(2t), t²).)

I had to look this question up before I submitted my answer but actually had done the math correct before looking it up. I vaguely remembered that I needed to throw all the partial derivatives of the components of the vector into a square root, square them each, and add them together but I couldn’t remember why. I realized after looking it up that it’s because velocity is speed in a specific direction so by finding the absolute value of velocity you are simply left with speed. This is done by squaring and square-rooting (if that’s a word) each component and finding their sum.

Question 6

This was question 20 on the test and I cheated by looking it up. Even after I looked it up, I still had (and still have) no clue what was going on. These questions are very confusing to me and I don’t understand them at all. I’m hoping that as I move forward into the next units and courses I’ll have a better idea of the linear algebra operations and how they are used to change functions into/from scalar, 2D and 3D fields. (I don’t know if any of that made sense…)

As I said, I started the following unit, Applications of Multivariable Derivatives, on Thursday. There were three videos, one exercise and two articles I needed to get through in the first section. I got through everything but the last article and, for the most part, was able to wrap my head around what was being taught. Here are some notes I took on what the tangent plane is plus a video that represents what a tangent plane on a 3D function looks like:

Here’s a screen shot from the video where Grant explained how to find the formula for the tangent plane and my note from that video below:

On Friday I started and finished the exercise from this section. I was pretty confident in my understanding of how to solve the questions (i.e. I was able to use the formula to find the missing values) but I had no clue why the formula worked for me. Nonetheless, here are three questions from that exercise:

Question 7 

Question 8

Question 9 

Lastly, on Thursday as I was watching the first few videos in this section, Grant was talking about the 3D function from the screen shots below and said something like, “I should probably give you the formula for this function.” I paused the video at that exact moment and tried to think through what the function would be in my head. After looking at the 3D image for a few minutes I thought it might be something like f(x,y) = –(x²/2 + y²) + 3. It turned out that the function was actually f(x, y) = –(x²/3 + y²) + 3 so I was pretty much exactly correct apart from the ½ actually being 1/3. I thought it was sick that I was able to understand what I was looking at well enough to essentially come up with the exact formula for the shape in my head, something I never would have been able to do a month or two ago. Here are two screen shots from that video:

I’m really excited to now be working through Applications of Multivariable Derivatives (80/500 M.P.) and am hoping, given how small it is, it won’t take me too long to get through. It’s also crazy to think that I’ll be starting year five a week from now. (YEAR. FIVE…?!? 😳) Looking back, I’m really happy with how far I’ve come and what I’ve been able to learn up to this point. It’s hilarious for me to think that initially I was hoping to get through calculus in a year. Even though it’s taken me four times as long (and counting) I understand and have an appreciation for why learning math doesn’t happen overnight. It’s also interesting that the longer I work on KA, the more interested and committed I become in wanting to keep learning. Working through KA has turned into the weirdest but healthiest and most worthwhile addiction I’ve ever had.