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Fundamental Theorem of Calculus says

Let $f$ be an integrable function on $[a,b].$ For $x$ in $[a,b],$ let $F(x)=\int_{a}^{x}f(t)dt.$ Then $F$ is continuous on $[a,b].$ If $f$ is continuous at $x_{0}$ in $(a,b),$ then $F$ is differentiable at $x_{0}$ and $F^{'}(x_{0})=f(x_{0}).$

Now my question is why we are saying that this is "Fundamental Theorem of whole Calculus".

It is clear that it connects intergal and differential calculus. But i am not getting the exact concept, why this is so important as its name is fundamental theorem of calculus. Why we says that it is fundamental theorem of calculus? Please suggest me. Thanks in advance.

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  • $\begingroup$ See the fundamental theorem of algebra. Now there also, I can question WHY? Why, is it called the fundamental theorem of algebra? $\endgroup$ – Aditya Agarwal Sep 21 '15 at 10:49
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    $\begingroup$ All of the integration techniques of elementary calculus are based on knowing that integration is anti-differentiation, which is what this theorem says $\endgroup$ – Alex G. Sep 21 '15 at 10:52
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    $\begingroup$ You have two fundamental limiting processes in calculus, the derivative and the integral. And they at the outset have no obvious connection. But the FTC shows the connection, dramatically and clearly. So what kind of result did you have in mind more deserving of "fundamental" in its name? $\endgroup$ – zhw. Sep 21 '15 at 18:48
  • $\begingroup$ You should read this paper by D. Bressoud (2009) . $\endgroup$ – Tony Piccolo Sep 22 '15 at 5:29
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Differentiation and integration are two processes that seem totally unrelated. The first one involves finding the slope of a curve; the second one the area under a curve. The fundamental theorem of calculus relates both of them. The "fundamental" part refers to the expected relation between both processes.

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