Questions about the fundamental thoerem of calculus

[EchoRush]

TL;DR Summary

Some questions/thoughts on the fundamental theorem of calculus?

As you can see form my previous posts, I am in my first university level calculus class ever. It is going very well, and through the class I am asking good questions and trying to actually make connection with the stuff we arr doing - not just doing the math just for the sake of passing - I am actually interested in math.

So, let me set it up for you. We began talking about limits (and everything that goes with them) and then we started with derivatives (and everything that goes with them) and then chain rule and then applications of this/related rates and then we studied integrals. All of this was leading up to her telling us about the fundamental theorem of calculus...that was today.

I was thinking to myself before class today "okay, here it is, I am going to have this big ah-ha moment when i put it all together". Then we were told what the theorem is. Needless to say, I was disappointed. I found out that the theorem basically says that taking a derivative is the inverse function of taking an integral and vice versa. If that is not correct, then please correct me.

The reason I am disappointed is because during the class time when we started talking about integrals and how to do them, it becomes fairly obvious that it is the opposite of taking a derivative. What I mean to say is way back when the founders of calculus came up with this theorem it must have been a HUGE deal to discover this because today when you see it after doing derivatives and integrals it is sort of like "duh, yeah". My question is, did this stump the fathers of calculus at first? Were they confused on what is the opposite of a derivative is? Today when you see it after taking a calc class, it is no more surprising than figuring out multiplication is the inverse or dividing or likewise with addition/subtraction. My question is was it a big mystery to the founders of calculus?

 

[PeroK]

If you are disappointed by the FTC you must be hard to please.

 

[jedishrfu]

There's a sequence of videos on the 3Blue1Brown channel called the "Essence of Calculus" and you should watch it as they will give you more insight to what you are learning now.

 

[Infrared]

Can you explain why you think the fundamental theorem of calculus is obvious?

You compared the situation to division being inverse to multiplication. But that is true by definition, whereas integration is usually defined (in a first calculus course) as a limit of Riemann sums, which doesn't make mention to differentiation at all.

 

[jedishrfu]

Sometimes "obvious" things must be proven be we know they are obvious.

Ramanujan ran into similar issues where he thought something was obviously true but hadn't considered some other aspect that would counter it. The most well known is the $\pi(n)$ counting function that he thought was correct but Hardy saw that he hadn't considered the effects of complex zeros in the formula.

https://mathoverflow.net/questions/288410/what-did-ramanujan-get-wrong
Nice WIRED article on Ramanujan:

https://www.wired.com/2016/04/who-was-ramanujan/

 

[FactChecker]

1) "Fundamental" is not the same as "most difficult to understand".
2) Other than linearity (which is a common property), can you think of a more basic property for calculus? And this property ties the two main subjects of calculus together. Otherwise, those two subjects would probably not be considered to belong in the same subject called "calculus".
3) You may want to read about the long development of calculus here to understand how hard it was.
4) You might want to read some of Newton's original writings to understand how obscure the thinking was. (Leibniz may have been easier to understand, but I have never seen his writings.)

 

[fresh_42]

(Leibniz may have been easier to understand, but I have never seen his writings.)

I'm not sure that you want to ...
https://www.maa.org/press/periodica...leibnizs-papers-on-calculus-integral-calculusand here are all volumes of the journal
https://gdz.sub.uni-goettingen.de/id/PPN372545874

 

[WWGD]

I'm not sure that you want to ...
https://www.maa.org/press/periodica...leibnizs-papers-on-calculus-integral-calculusand here are all volumes of the journal
https://gdz.sub.uni-goettingen.de/id/PPN372545874

Still likely better than reading about fluxions.

 

[Delta2]

You might find it interesting to read my insight on the fundamental theorem of calculus
https://www.physicsforums.com/insights/a-numerical-insight-for-the-fundamental-theorem-of-calculus/
For me , back when I was high school student and was originally exposed to derivatives and integrals for the first time, it wasn't so obvious that integration and differentiation are reverse operations, only when I read the proof of the FTC I came to realize some things.

 

[fresh_42]

I like Stokes' version
$$
\int_C d\omega = \int_{\partial C} \omega
$$
and all of a sudden it is all but obvious anymore, and placed in the center of the theory again, where it belongs, due to its central meaning for calculus.

 

[HallsofIvy]

If you say that the "Fundamental Theorem of Calculus" is "obvious", exactly what were you taught as "integration"? It is obvious that "anti-differentiation" is the inverse of "differentiation" but the point of the FTC is that the integral, as defined by the Riemann sums, IS given by the "anti-derivative".