This was the first week in the last few that I’m disappointed with the amount of work I got through. I did manage to finish the unit test of Limits and Continuity (though not with a perfect score which I’ll get to) and was happy that I began the following unit, Derivatives: Definitions and Basic Rules, though I barely scratched the surface of it. The main reason I didn’t get as much done as I had hoped (drum roll for the most random excuse you’ll ever hear 🥁🥁🥁) is because I have a canker sore the size of Montana on the inside of my bottom lip and it actually may be the death of me. I got it on Tuesday and for the past 120 hours it’s been nonstop pain which made it incredibly difficult to concentrate. Also, the fact that I always eat breakfast when I do KA made the pain 10x worse. It’s the most ridiculous excuse ever but I’ve honestly been on the verge of tears for the past few days because it hurts so much and it hasn’t showed signs of getting any better. It’s been exhausting so, although I’m bummed I didn’t get much done this week, in the big picture I don’t give a sh*t about KA and just want my stupid f*cking mouth to not be in so much pain.

AAGHGGHGHHGGHEFGUVDEHG!!!!! 😡😡😡😭😭😭

Anyways…..

The last thing I had to cover from the unit Limits and Continuity was what’s called the Intermediate Value Theorem (IVT). Here’s a page from my notes that explains it:

In the same way that the Squeeze Theorem seemed pretty obvious, the IVT also seems pretty self-evident. The gist of the theorem is if there’s a function that is continuous between a given x-interval, every point on the function within that interval will ‘map’ to a y-value between f(x₁) and f(x₂). To me this seems pretty straightforward and doesn’t require much more explanation than that.

I started the unit test on Wednesday and was shocked to see that it was 35 questions long. Typically unit tests range from ~9-15 questions long so I was caught off guard and a bit bummed/nervous when I saw how long this one was. Up until this week I had never moved ahead without getting a perfect score on a unit test but I told myself beforehand that if I missed a few questions but generally felt like I had a good handle on everything that came up I would move on. (I’ve been doing this for the course challenges which are usually ≥30 questions.)

I got off to a bad start by getting the fourth question wrong. The question gave me a function and asked me to find the limit at a certain x-value within the function. It was a multiple choice question and gave me three options which were essentially:

• A) Limit Found
• B) Limit sits on an asymptote
• C) Indeterminate form.

This is the flow chart I needed to use:

After using direct substitution, I got an answer that was a/0 (i.e. the limit was on an asymptote) but in my head I thought a/0 was indeterminate form which I quickly entered and got wrong. I was annoyed getting it wrong since I knew how these types of questions worked but got indeterminate form and asymptotes mixed up.

The next question I got wrong was question 10 which had to do with nesting functions, a.k.a. what I think of as function-inception:

When I did this question I thought I was supposed to multiply g(-2) (i.e. 0) by h(-2) (i.e. -4) so I inputted my answer as 0 and got it wrong.  What I should have done was taken g(-2) (again, 0) and inputted that answer into h(x) (i.e. h(0) = undefined since there is a jump discontinuity at x = 0). I was once again annoyed that I got this question wrong since I just mixed up the notation and the process to solve it even though it wasn’t that hard.

The last question I got wrong, question 20, was the only question I genuinely would have struggled to figure out even if I had gone back through my notes to review what was being asked. The question asked me to find the limit of a trig function as θ approached π/2. Here’s the question and the process to find it’s solution:

I actually did go back through my trig identity notes to try and figure out the appropriate identities to substitute in the original function but still got it wrong. I initially did the right thing by swapping tan²(θ) with (sin²(θ)/cos²(θ)) but then made the mistake of thinking cos²(π/2) = 0 which led to indeterminate form. As you can see from the photo, what I should have done was said cos²(θ) = (1 – sin²(θ)) and then used the ‘difference or squares’ form of that identity to solve from there. I’m still having a hard time understanding trig identities and how to know which identity to use at which time when solving these types of questions.

It ended up taking me from Wednesday to Friday to finish the unit test and finished getting 32/35 correct. I felt like question 20 was the only question I should have gotten wrong, however, so I felt ok about moving forward to the next unit.

I only ended up watching two videos in the following unit Derivatives: Definitions and Basic Rules since I was on the verge of a nervous breakdown because of this stupid canker. Here’s the only note I took:

As of now, all I’ve learned is that a derivative is the exact rate of change at a single point on a function. (And to be honest, I’m not entirely sure if that’s accurate.)

I’m looking forward to working through Derivatives: Definitions and Basic Rules (0/2500 M.P.). Based off of how many videos and exercises there are in this unit, I should be able to finish it before the end of May. When I try to determine how long it will take me to get through a unit I don’t always get it right but I find it helpful to motivate me to get through x number of videos/exercises each day to stay on track.

Now, please excuse me while I go rinse my mouth out with salt water and cry for the next few hours.