The fundamental theorem of calculus
What does it mean?
Answers
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function.
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Answer:
Essentially it means that the concepts of integration and differentiation are related.
This was a big deal when it was discovered.
The concept of integration goes back a long way, already existing in the works of Eudoxus and Archimedes in the form of "the method of exhaustion".
The concept of differentiation proved more complicated and subtle, only showing up a couple thousand years later, but it was certainly there in the work of Fermat.
Then later came along Newton and Leibniz. Independently, these two worked on developing a "calculus" of differentiation and a "calculus" of integration. That is, systematic processes / rules / methods for performing differentiation and integration, rather than reasoning from scratch each time. And in doing this, they each identified the "fundamental theorem" which revealed that the two concepts are actually intimately connected.
The theorem essentially tells us that differentiation and integration can, for most practical purposes, be seen as inverse operations.
Nowadays, it might seem difficult to appreciate the "big deal" that the theorem is because students are told from the beginning that these things are inverse operations. But the point is that this way of seeing things is a consequence of the theorem!
This is a special (one dimensional) case of Stokes' Theorem which relates to the idea that integration in n dimensions is connected to integration in n-1 dimensions through a differentation process.