Need Help Understanding what FTC means

[young charm]

Hi
Ik this might be really basic but can someone please explain what these equations mean conceptually? Like how is f(t) becoming f(x) and all

[robust birch]

The Fundamental Theorem of Calculus is stating in natural language (as opposed to with mathematical notation): Derivatives and integrals are inverse operations of each other. Analogous to if I have any number, I don't have to know what it is, if I add three too it and subtract the result by three I'll get back to what I started with, or if I have any number, I don't have to know what it is, if I multiply it by three and divide the result by three I'll get back to what I started with, if I have any function, I don't have to know what it is to know if I integrate it and differentiate it again, I'll have the function I started with.

[shell bay]

okay, lets take any function f(t)
now what if you wanted to measure area from a to any number you wish?(variable)
That will be the function g(x). g(1) = area from a to 1, g(2) = area from a to 2, and so on.
Clear?

Now FTC steps in: If you take the derivative of this area function, you will get the value of the integrand at x
So that means... how much the area function is changing at x, is the value of integrand at x!

how cool is that!

Does it help?

We took different variables to avoid confusion
if we used same variable x for both area and integrand, it will be something non intuitive
f(t) just represnts any function
x is used for area function
clear?

Just one side note:
make sure f(t) is continuous. FTC would die if thats not the case

[barren grail]

The integral is the inverse operation of the differential, and the differential is the inverse operation of the integral.

If both operations are applied to a function, the result is the original function itself, i.e., they cancel.

[tough mural]

t is just use as a place holder or dummy variable

[young charm]

Yup yup

[young charm]

Oh i needed this
It makes so much sense
Thank you soo muchh

Thank you sm guys it helped a lot

How do i close this?

[odd vector]

.close or .solved

[young charm]

.close

[timber sand]

Solved

Post marked as solved by @young charm.

Use .unsolved if this was a mistake.