This week was pretty normal in terms of how much work I got through on KA. I began the week ~53% of the way through the unit Analyzing Functions and ended off ~67% of the way through it. Unlike last week, I didn’t learn anything brand new but I did get a much stronger grasp on the differences between a function, its’ derivative, and second derivative and why each are useful in their own way. I also learned how combining the information they each give can can be super useful. In the second half of the week, I started to understand how powerful these concepts are when I started working through optimization problems. As of now, I feel like these questions are way over my head BUT I’ve also had this feeling many times before and am confident that I’ll eventually understand how/why they work as long as I stick to it.

I began this week working through a section called Sketching Curves which only contained 3 videos. Each video gave me a function and asked me find the general shape of each function by figuring out their respective the critical points and inflection points. Working through was good practice taking the first and second derivatives of functions by using the power rule, product rule, etc. It also helped me better understand the differences  between a function, its derivative, and its second derivative and helped me picture in my head their respective graphs and understand the different information they each convey. Here’s a screen shot I took from the following section, Connecting f, f’, and f’’, which does a good job summing up the differences between a second derivative, first derivative, and a function:

In this screen shot, the interesting point to look at in all 3 graphs is point “C” on the x-axis. On the OG function to the right, f, you can see that point C is an inflection point, i.e. the functions switches concavity, meaning the slope at that point goes from decreasing on the left to increasing on the right. This is represented in the first graph, f’’, where at point C the function switches from being negative to left of point C – i.e. where the slope on f is DECREASING – to being positive on the right of point C – i.e. where the slope on f is INCREASING. Although the slope begins to increase at the inflection point on f, the slope is still negative just to the right – and to the left, for that matter – of point C. The fact that the slope is negative directly to the right of point C – even though the concavity is positive – is indicated in the middle function, f’, where the function is still negative on either side of point C. Wrapping my head these 2 concepts has been tricky, but to sum it up in my own words:

• The first derivative, f’, of a function tells you if the slope at any point along a function is positive or negative.
• The second derivative, f’’, tells you whether the if the concavity is positive or negative at any point along a function (a.k.a. if the slope is INCREASING as you move in the positive direction along the x-axis or if the slope is DECREASING as you move in the positive direction along the x-axis).

I finished the sections Sketching Curves and Connecting f, f’, and f’’ on Wednesday and began the section Solving Optimization Problems on Thursday. Between Thursday and Saturday, I only managed to get through 5 videos in this section which indicates how difficult I found these questions. To be completely honest, I still I don’t really understand these questions well enough to explain how they work here, but here’s an example of a question that was given in one of the videos which I worked through:

In this example, the part that feels like magic to me is how/why you’re able to use and connect the two formulas/functions together – i.e. the function for cost and the function for volume. In my mind, they’re calculating two separate things so it seems weird in my mind you can use them together. That said, in a way it does seem logical but, regardless, it still seems very unintuitive. 😞

On a side note, I passed the Algebra 1 course challenge this week scoring 27/30 which means I could be certified to tutor Alg.1, but the video didn’t upload properly so I will have to redo it. 😡 In order to “pass”, you have to get a 90% on the course challenge, meaning 27/30 is the minimum score you can get to pass. I wasn’t thrilled that I got 3 questions wrong so I’m ok with having to redo it but am still somewhat annoyed. I realized that I’ll need to go back and review each unit before working through their respective course challenges as I’ve forgotten some of the terms/definitions/concepts that learned when I worked through them initially. I don’t mind doing this, however, since relearning these things isn’t nearly as it was when I was learning about them initially. My plan is to pass that Alg.1 course challenge early this week and then spend the next ~10-12 days reviewing Alg.2 in order to attempt its’ course challenge on Friday, Oct. 1^st. (I’m pretty busy up until then which is why I’ve decided to give myself ~2 weeks.)

Including today, there are 11 days left in September which doesn’t give me too much time to get through this unit, Analyzing Functions (1200/1800 M.P.), before the end of the month. There are only 5 videos and 3 exercises left for me to get through before I have to do the tough to get through them in time. My plan is to get through all the videos and exercises this week which’ll give me 3 days next week to pass the unit test before the month ends. Wish me luck!! 🤞🏼🙏🏼