Daily Definitions (12/5)

The last subject we covered in class was The Fundamental Theorem of Calculus which brings this course full circle with a proof that proves all the basic things we used in every Calculus class we have been in. I have stated the definition:

Fundamental Theorem of Calculus: 

(i) If f: [a,b] \rightarrow \mathbb{R} is integrable andF'(x) = f(x) \forall x \in [a,b] then \int_a^b f = F(b)-F(a)

(ii) If g: [a,b] \rightarrow \mathbb{R} is integrable, define G(x) = \int_a^x g for x in [a,b]. Then, G is continuous on [a,b]. Furthermore, if g is continuous at a point c in our domain, then G'(c) exists.


Notice that with some work, we can prove that if we have a continuous function, then the integral of that function is: \int_a^b f = h(b)-h(a) for some h where h' = f.

Thus, proving the computation of a proof if the function is continuous has now become 100x.

Note that this proof also allows us to prove many of the properties of integrals.

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