Proof of fundamental theorem of calculus

The fundamental theorem of calculus is very important in calculus (you might even say it's fundamental!). It connects derivatives and integrals in two, equivalent, ways:

\begin{aligned} I . & \frac{d}{d x} \int_{a}^{x} f (t) d t = f (x) \\ I I . & \int_{a}^{b} f (x) d x = F (b) - F (a) \end{aligned}

The first part says that if you define a function as the definite integral of another function

f

, then the new function is an antiderivative of

f

The second part says that in order to find the definite integral of

f

between

a

and

b

, find an antiderivative of

f

, call it

F

, and calculate

F (b) - F (a)

The AP Calculus course doesn't require knowing the proof of this fact, but we believe that as long as a proof is accessible, there's always something to learn from it. In general, it's always good to require some kind of proof or justification for the theorems you learn.

First, we prove the first part of the theorem.

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Proof of fundamental theorem of calculusΔείτε μεταγραφή του βίντεο

Next, we offer some intuition into the correctness of the second part.

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Intuition for second part of fundamental theorem of calculusΔείτε μεταγραφή του βίντεο

Finally, we prove the second part of the theorem based on the first part.

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The fundamental theorem of calculus and definite integralsΔείτε μεταγραφή του βίντεο

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