It was unfortunately another disappointing week for me working through KA. I got through the 3 exercises I needed to redo before retaking the unit test, but it took me until Friday to get through them and I only ended up getting through 7 questions on the unit test, AND got two of them wrong… 😔 I was able to understand the gist of how to solve most of the questions I came up against this week but many of them had ~10 or more steps involving calc, algebra and arithmetic, and I’d often make a mistake somewhere along the way. As often is the case however, even though I didn’t get nearly as far as I wanted to, I’m still happy with the fact that I made some progress. I think I have a better grasp on how to go about finding the area of certain shapes and then using that to then find the volume of certain objects. I also got a better handle on the F.T.o.C. and how to apply it to certain questions which made them MUCH easier to solve. Finally, I also learned about a new (to me) type of division called ‘synthetic division’ which, at this point, I don’t really understand why it works but am happy nonetheless to add it to my repertoire of tools I can use to solve questions.

The first section I worked through this week, Volume: Triangles and Semicircles Cross Sections, had me use the formulas for the area of a triangle (width * height/2) and for the area of a circle (πr²). They almost always asked me for some variation of the formulas, for instance I’d be told the base of an object was graphed with the distance between the function and the y-axis being the width of an equilateral triangle, meaning that I had to know that the height (i.e. the z-axis) would be equal to √(3)y/2 which comes from 30-60-90 triangles. This section took me until Thursday to get through which was pretty disappointing. Here’s a question from the exercise:

Question 1

I used 23 pages of notes and spent ~2.5 hours trying to fully understand this question. There are many steps to this type of question which is a good example of why I often get these questions wrong since there are so many places I can make a careless mistake. Reading it back over now, I’m able to follow along easily and see why I did each step and why each step works so I’m definitely happy about that. I would have struggled with this algebra a year ago and wouldn’t have had any clue what was going on with the calculus so, even though it’s taking me a long time, I’m definitely making progress on calculus, overall.

The last section I worked through was a section with a weird title called Calculator-Active Practice. The questions I worked through in the exercise from this section made a big difference in helping me wrap my head around HOW the formula for the F.T.o.C. works, which is _a∫^b f’(x)dx = F(b) – F(a). In general, I think I understand WHY the F.T.o.C. works (I always say to myself, “if you know the average slope of a function across a certain interval, then you know the average height of its derivative along that same interval and therefore the area”) but I haven’t/hadn’t had enough practice up to this point using and recognizing the formula to understand how to manipulate it algebraically. Here’s a question from the exercise:

Question 2

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Once I was able to start recognizing the different components of the formula for what they were (i.e. the derivative in the integrand and f(a) and f(b)) using the F.T.o.C. and subtracting or adding f(a) or f(b) from the integrand was straightforward. 

It was a good thing I worked through this exercise because I started the unit test on Saturday and the very first question I worked on was identical to the ones from that exercise. Here it is:

Question 3

The third question I worked on from the unit test was pretty tricky but I ended up getting the correct solution, although I still don’t understand it. Here’s the question:

Question 4

What confuses me is that it seems like the volume of any given object should be greater than the value of any one of its surfaces/sides. If you throw 0.5 into the formula for a semicircle (which in this question is the radius of the semicircle on the y-axis), the value is approximately π(0.5)²/2 = ~0.3926. However, the value of the answer to this question is –π(e^–2 – 1)/16 = ~0.1698. So since 0.3926 is greater than 0.1698, I don’t understand how the value of that surface of the object can be greater than the value of its volume. 🤔

As I mentioned, I was disappointed that I ended the week by getting two questions wrong which were the 6^th and 7^th questions on the test. The 6^th question I got wrong because I got the bounds of an integral messed up in my head and used the x-coordinates of the interval when I should have picked the y-coordinates which was a simple, careless mistake that I shouldn’t have made. The 7^th question, however, was a question I got wrong because I straight up didn’t know what to do. Here’s the question:

Question 5

I spent about 40-50 minutes trying to think through this question by using derivative and antiderivative operations on f(x) and ended up getting nowhere. I managed to figure out what x-intercepts were on my own partly through looking at the graph but also by inputting –1, 1 and 5 into the function which outputted 0. I also threw the function into Desmos which gave me –128/3 which I inputted as my answer which turned out to be wrong as the correct answer was +128/3. When I looked at KA/s solution, I realized I would never have been able to solve this question since I don’t think I ever learned about “synthetic division.” I Googled it and found this video:

This video helped me understand HOW to use synthetic division, but I have no idea why it works. Even after learning about this way of dividing polynomials, it took me awhile to realize that I needed to use the given possible solutions in this question, –5, –1, 1 and 5, and use synthetic division on them. Once you know the x-intercepts you can then put the function f(x) into an integrand and use the bounds x = [1, 5] to find the area in that shaded space. Here’s my work from this question:

I’m a broken record for saying this, but I’m disappointed I haven’t gotten past this unit, Application of Integrals(1,800/2,000 M.P.), especially considering I really wanted to have it done before the end of my third year on KA. Nonetheless, I’m excited to be heading into my fourth year and am hopeful/optimistic that by the end of this year I’ll have a very strong grasp on calculus and will be moving on to bigger, better and more challenging concepts. Also, with this post I’ve now written more than 200K words between all 157 blog posts! (The exact number on Microsoft Word is actually 200,308 words.) Even though I still haven’t learned calculus, I think having written 200k words is an accomplishment in itself and something that I’m pretty proud of. ☺️