I feel like I’m SO close to having the Fundamental Theorem of Calculus figured out but I’m still not quite there. 😔 I didn’t actually spend any time this week working through the exercises I need to redo in Integrals before attempting the unit test again because the next exercise I have to redo has to do with the Fundamental Theorem of Calculus so I spent this entire week looking at different websites and watching non-KA YouTube videos to try and figure it out. Having spent the past 6 days trying to understand the relationship between integrals and derivatives, I feel much more comfortable with the notation used for both types of functions which has been a big help overall. I’ve decided not to go back to the Integrals exercises until I have a solid grasp on the FToC so hopefully it clicks for me soon!
Here’s a page from my notes that helped me wrap my head around the derivative and antiderivative notation:
Before this week, I always used to think of this notation as b(x) being the primary function with b’(x) being the derivative. I still think that conceptualizing it this way it useful but this week I started to grasp that ∫ f(x) = F(X) where f(x) = b’(x) and F(x) = b(x). The difference when using the latter type of notation is that it more accurately refers to F(X) being the ANTIDERIVATIVE of ∫ f(x). It seems like it is going to be important to be able to think of functions going in both dedications (i.e. going from what I would call the ‘primary’ function, b(x), to its derivative, b'(x), but ALSO to be able to go from lower case f(x) when it’s inside an integral to its antiderivative, F(x)).
On Tuesday I Googled, “eli5 Fundamental Theorem of Calculus” and clicked the first link which led to a reddit post. (For all of you non-redditors, eli5 means explain like I’m 5.) The first comment in the reddit post linked me to this Wikipedia page:
https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#Geometric_intuition
I read through the majority of this article on Tuesday and Wednesday and found it a bit confusing but helpful. Even after going through it, I’m still not completely sure how the FToC works and am still confused by a lot of what’s covered in the article. Regardless, here’s a page from my notes that goes over what’s said early in the Wikipedia page:
I don’t really understand the point of the “Excess Area” and how it relates to the FToC. This diagram helps me to visualize and understand that the area under a function at a specific point is essentially the height of the function at that point multiplied by a tiny change in x (in this diagram above, the tiny change in x – a.k.a. d/x – is written as b). Not including the excess area, the formula written in the center of the page means:
- A(x + b) – A(x) = ~f(x)*b
- “The area underneath a function, above the horizontal axis and between x and x + b (where b is a tiny, TINY number) is equal to the height of that function at a certain x * b.
Again, this specific concept hasn’t helped me much with wrapping my head around the FToC but I think it probably relates to it in some fundamental way and I just haven’t made the connection. Further down the Wikipedia article I found the following GIF:
This GIF has gotten me a lot closer to understanding the FToC (I think). My basic understanding of the GIF is that the area in the bottom graph (labelled as S) is equal to the height of the function in the top graph. As the area S increases, the value of the y-coordinate in the top graph increases accordingly. I don’t understand why this is the case, but I’m pretty certain that this is essentially the FToC and explains how antiderivative relates to the area underneath its corresponding function. (Not sure if that even makes sense…)
After going through the Wikipedia page on Wednesday, I then watched a video from a playlist of videos I’d saved on YouTube called Essence of Calculus from a channel named 3Blue1Brown. I watched the 8th video in the series titled Integration and the Fundamental Theorem of Calculus | Chapter 8, Essence of Calculus which helped make integrals seem a bit more clear to me. After watching the video and still not fully understanding the FToC, I decided to start from the beginning and watch every video in the playlist in hopes that by the time I got back to the 8th video it would start to make a bit more sense.
In the 2nd or 3rd video of the playlist I realized something that made derivatives make a lot more sense to me, that d in the notation dx refers to delta-x (a.k.a. Δx or “change in x”) and implies that x approaches 0. In this sense, dx is something like a limit where x gets closer and closer to 0 but never actually gets there. I’m not sure if the following notation actually makes sense, but now I think of it as:
- dx = limx ->0 Δx
After having that mini epiphany, I then went through another one of the videos where the host used the area of a square/rectangle to explain the power rule of derivatives which also helped me further understand what dx actually is:
The function that the host was talking about was f(x) = sin(x) * x2. In my notes, you can see a square at the top of the first page where the top side is labelled as sin(x) and the right side is labelled as x2 and therefore the hypothetical area of the square equals sin(x) * x2. The host then explained that a derivative is a tiny, TINY change in x which he represented by drawing a sliver of extra area added to the bottom and side of the square/rectangle. The video explains that the extra area added IS the derivative and that to calculate that extra bit of area you need to find the area of each tiny rectangle and then add them together:
- d/dx[f(x)] = d[sin(x)] * x2 + d[x2] * sin(x)
This turns out to be the power rule. The host also mentions that there’s a tiny extra bit of area that shows up in the bottom right corner of the diagram in my notes but as x approaches 0 that little bit of area becomes negligible, especially when compared to the longer rectangles.
I’m not going to go through the math, but here are a few pages from my notes that I made when I watched one of the videos where the host talked about implicit differentiation and related rates:
Although this question didn’t help me much with understanding the FToC, it helped me understand that there are times when an object moving along what might be considered the x-axis but that it can at times be useful to reimagine the distance the object is moving on the x-axis as a distance versus time function where the object is away from the origin is then measured on the y-axis in the distance/time graph.
At the end of the week, I ended up back on the Integrals video I watched initially on Wednesday. Here are 3 pages from my notes which I think have helped me to better understand the FToc:
The third page above is more-or-less what the GIF from the Wikipedia article was explaining, that an antiderivative is equal to the area underneath a function. I’m still having a hard time making the connection between the area underneath a function and the slope of its antiderivative. As I said, I feel like I’m very close to having it figured out but, as of now, it’s still a mystery to me. 😔
It’s my mission this coming week to understand the FToC. I feel like once I understand it all of integrals and derivatives will become MUCH easier to understand and calculus, in general, for that matter. I’m hoping I can finally figure it out early in the week so that I can redo and pass the remaining 4 exercises in Integrals (2,980/3,200 M.P.) before the end of the week. Even if that ends up happening, it’s highly unlikely that I’ll finish this unit before the end of February which is a bit crazy considering I’ve spent almost 5 months of working on it… Nonetheless, I’m just as motivated as ever to get through it and move forward! 💪🏼