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# Intuitive Fundamental Theorem of Calculus

### 2015 December 19. Saturday.

## The Derivative

In calculus, we first learn the derivative operator. It takes a smooth function and an `x`

-value and it returns the slope of the function at that `x`

-value.

Whats the slope of `f(x) = x`

at `x = 5`

? Well `1`

of course. It’s `1`

everywhere!

How about `g(x) = ln x`

at `x = 17`

? If you draw it really accurately you see that it’s `1/17`

. (This has to do with `ln x`

and `1/x`

kind of being flips of eachother)

## The Integral

The integral takes a not-necessarily-smooth function and an `x`

-value, and returns the area between the function and the x-axis between 0 and that point.

How much area is under `h(x) = Sqrt(3-x^2)`

from `0`

to `3`

? That’s a quarter of a circle of radius `3`

, so the area is `pi*r^2 / 4 = pi*9/4`

which is about `7`

.

More simply `k(x)=5x`

can be evaluated at `x=7`

by drawing a triangle of height `35`

and width `7`

to get an area of `122.5`

.

## The Theorem

The fundamental theorem of calculus states that these two operators are inverse functions; the derivative of the integral of `f(x)`

is always that same `f(x)`

. What?

To understand this, consider at what rate the area is changing. If the function is tall at some `x`

, then the area grows quickly there. If the function is short, then every inch forwards has nearly no gains to the surface.

Another way of stating this: The rate of change of the area is the function.

## Links

Kalid Azad’s more thorough article on the same topic

Public Domain Dedication.