It turns out the cold I picked up last week actually WAS Covid. I can still feel it a little bit in my head and I’m still coughing a bit but, overall, I feel WAY better than I did a few days ago. I didn’t have the energy to do KA on Tuesday so I only did four days of KA this week but I still think I likely ended up getting close to 5 hours of work done. It took me until Friday to make it through the final exercise of the unit and then I started the unit test on Saturday. I ended the week getting through 5/19 questions on the unit test but unfortunately got the 5th question wrong. (I made a careless mistake because I didn’t read the question properly… 😑.) Considering how I was feeling at the beginning of the week (and the fact that I’m still not back to 100%) I’m fine with how this week went but I’m definitely ready to start kicking KA’s ass beginning this coming week.
On Wednesday I spent most of the morning reviewing the fundamental theorem of calculus. I used Desmos to review how the derivative of a function tells you the functions slope at any given point and by knowing the AVERAGE slope of the OG function along a certain interval, that tells you the average ‘height’ of the derivative along that interval which you can use to fund the area between the derivative and the x-axis:
One thing I need more practice with in order to get a stronger intuitive understanding of is how/why the area between a∫b f(x) dx = F(b) – F(a). Yesterday (Sunday) I started to rewatch the Essence of Calculus series which I think will help remind and further enlighten me on how/why the FToC formula I just mentioned works. When I first started watching that series of videos months ago I assumed that the further into calculus I made it, the more that the concepts explained in those videos would start to seem straightforward. I believe this is the fourth time I’ve reviewed this series of videos and it does seem to make more sense now than the last time I went through them. Hopefully by the time I’m through Calc. 2 everything that’s taught in those videos will seem completely obvious. 🤞🏼
The questions I worked through in the final exercise of the unit, which was titled Contextual and Analytical Applications of Integration (Calculator-Active), were confusing at the beginning but then pretty easy once I knew what I had to do to answer them. Here are 3 questions I worked through in this exercise:
Question 1
I found the key thing to remember in order to solve these questions is that since you’re given a point on the antiderivative (i.e. you’re given a F(a) value) and since a∫bf(x) dx = F(b) – F(a), then you can simply add the F(a) value back to F(b) – F(A), i.e. giving you (F(b) – F(A)) + F(a), to leave you with the value of F(b). Once I understood this concept, these questions became very simple especially since I just used Desmos to come up with the value of a∫b f(x) dx = F(b) – F(a).
Question 2
This question helped me to understand that if you throw a rate inside an integral (in this case, the rate is how much water is being drained per minute) you can determine the total quantity of whatever’s being measured that’s either added or subtracted from the total over that specific interval (in this case, when you put the rate of water being drained inside the integral with the bounds x = [3, 9], you’re given the amount of water drained in that interval).
Question 3
I still find this question confusing. I think what’s going on in the formula (a∫b f(x) dx)/(b – a) is that it’s telling you the average area of the function f(x) along the interval (b – a), but I’m having a hard time understanding why you specifically need to use an integral to figure this out and why you wouldn’t use another method.
After getting through that exercise, I started the unit test on Saturday. Even though I got the last question that I worked on wrong, so far I’m happy with what I’ve been able to remember and my understanding of what I’ve been asked in the first 5 questions. Here’s question 1 from the test:
I assumed from the start of this question that I was allowed to use Desmos to graph the two functions which turned out to be correct. Even though the graph showed me where the two functions crossed over each other, in the first photo where the number 1 is circled, you can see where I did the math to figure out the x-coordinates of where the functions crossed, i.e. the bounds needed for the integral. I made a mistake in the second step where I accidentally set up the integral as -1∫1 f(x) – g(x) dx but it should have been -1∫1 g(x) – f(x) dx. Once I caught my mistake, I was then able to do the calculus to solve the question.
The fourth question asked me to use the washer method to determine the volume of an object:
It had been a few days since I used the washer method so I was happy that I was able to recall how it worked and was able to get the correct answer on my first attempt without any review. It took me awhile to think through each step of this question but I was glad that I had to take my time and visualize each step because I think it helped me to solidify how/why the calculus works for these types of questions.
The final question I worked on, question 5, was one that I should have gotten correct since I actually fully understood what was going on with the calculus but I didn’t read the question properly which lead me to submitting the wrong answer. Here’s the question:
As you can see from my notes, I worked through the calculus to figure out what the total population was by the end of year 10 including the 2,300 people that made up the population in year 3. I didn’t realize until after that the question was asking me simply to find the amount that the population had grown between year 3 and 10 and NOT factoring in the size of the population at year 3. The question tricked me. 🤬
I’m annoyed that I’ll need to redo the unit test but I’m also ok with it. I think that going through these questions is really going to help me grasp how/why integrals work the way they do so, in the big picture, even though it’s going to take me more time than I’d like it to, doing this unit test at least a second will be good practice.
Once I’m finished this unit test, I’ll be done with Applications of Integrals (1,520/1,900 M.P.) and will FINALLY be able to start the course challenge. I mentioned in a post a while ago that I started Calculus 1 in Week 82 so I’m definitely ready at this point to finish this course and move on to Calculus 2. I’m 58% of the way through Calculus 2 already (!) so there’s no way it’ll take me as long to get through that course as it’s taken me to get through Calc. 1. I still have a ways to go but I’m finally starting to see the light at the end of the tunnel and I am PUMPED. 😤