Questions about the fundamental thoerem of calculus

In summary: The fundamental theorem of calculus is a theorem that states that taking a derivative is the inverse of taking an integral and vice versa.
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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?
 
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If you are disappointed by the FTC you must be hard to please.
 
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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.

 
  • #4
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.
 
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  • #5
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/
 
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  • #6
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.)
 
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  • #10
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.
 
  • #11
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".
 
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FAQ: Questions about the fundamental thoerem of calculus

What is the fundamental theorem of calculus?

The fundamental theorem of calculus is a fundamental concept in calculus that establishes a relationship between differentiation and integration. It states that the integral of a function can be evaluated by finding the antiderivative of that function and evaluating it at the upper and lower limits of integration.

How is the fundamental theorem of calculus used in real-world applications?

The fundamental theorem of calculus is used in various fields such as physics, engineering, and economics to solve real-world problems involving rates of change and accumulation. For example, it can be used to calculate the area under a curve, which has applications in finding the distance traveled by an object or the amount of work done by a force.

What are the two parts of the fundamental theorem of calculus?

The fundamental theorem of calculus has two parts: the first part, also known as the fundamental theorem of calculus (FTC) part I, states that the integral of a function is the antiderivative of that function. The second part, also known as the fundamental theorem of calculus (FTC) part II, states that the derivative of an integral is the original function.

Can the fundamental theorem of calculus be extended to higher dimensions?

Yes, the fundamental theorem of calculus can be extended to higher dimensions through the use of multivariable calculus. In this case, the integral becomes a double or triple integral, and the derivative becomes a partial derivative with respect to multiple variables.

Are there any limitations to the fundamental theorem of calculus?

While the fundamental theorem of calculus is a powerful tool in calculus, it does have limitations. One limitation is that it only applies to continuous functions, which means that it cannot be used to evaluate integrals for functions that have discontinuities. Additionally, it may not be applicable in some cases where the function is not well-behaved, such as when the function is not differentiable.

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