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.

Posted on by Brittney D'Oleo
Categories: Weekly Blogs

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