The fundamental laws that govern the universe
speak to us only in the language of mathematics.
The key to understanding this language is calculus.
Calculus allows us to see the true beauty of nature.
Calculus makes the predictions of modern physics possible.
The variable Y represents distance in space.
The variable X represents moments in time.
The purple line represents the relationship between space and time.
This line shows what the distance in space is for every moment in time.
Let us focus on what happens during just one unit of time.
The distance traveled in one unit of time is what we call speed.
The speed is represented by the green rectangles.
Suppose we add the areas of all the rectangles together.
The areas all these rectangles add up to the distance travelled.
Now let us consider a different scenario.
Again, consider what happens during one unit of time.
Again, the speed is the distance traveled in one unit of time.
As before, let’s add all the rectangles together.
As before, the areas add up to the distance traveled.
Up till now, we have only been dealing with cases
where objects move without changing speed.
Now let’s consider a case where we change speed and sometimes move backwards.
Again, the speed can be represented by a group of rectangles.
This graph shows that the speed is no longer constant.
Every moment in time now consists of a different speed.
A negative speed is represented by yellow rectangles.
A negative speed means that the object is moving backwards.
Let us once again add the areas of all the rectangles together.
The yellow rectangles have a negative area, and cancel out the green rectangles.
The area that is left once again represents the distance traveled.
Let us change the graph for the speed of the object.
Before, the graph of distance versus time consisted of sharp corners.
Now the graph of time versus distance is much smoother.
Now the graph is a smooth continuous curve.
Let us once again add all the rectangles together.
No matter what kind of graph we have,
the rectangles will always add up to the distance traveled.
This is the fundamental theorem of calculus.
Consider a very small increment of time which we will call “dx.”
Consider the distance traveled during this time, which we will call “dy.”
The speed is the distance traveled divided by the time taken,
which is “dy” divided by “dx.”
We can therefore re-label the speed as “dy” divided by “dx”,
and we refer to it as the derivative of y with respect to x.
If we extend this white line, then “dy” divided by “dx” will be the line’s slope.
For every moment of time, “dy” divided by “dx” is the slope of the extended white line.
As the slope increases, “dy” divided by “dx” increases.
When the white line is horizontal, the slope is zero.
When the object moves backwards, the slope is negative.
Let us again consider the situation where we had large step increases in speed.
The purple line will now no longer be smooth, and will again have sharp corners.
Each time we pass a sharp corner, the slope changes abruptly,
and this is why “dy” divided by “dx” changes abruptly.
Now let us return to the case where the speed changes smoothly.
Let us give a new name to “dy” divided by “dx”.
Let us call it f(x).
For every value of x, we have a corresponding value of f(x).
For the yellow rectangles, f(x) is negative.
The distance traveled during a very small moment of time is the
speed multiplied by the time, which is “dy” divided by “dx” multiplied by “dx.”
Since “dy” divided by “dx” is called f(x), another way to write down
the distance traveled during this moment of time is f(x) multiplied by “dx”.
This equation, f(x) multiplied by “dx”, also happens to be the
area of the rectangle representing what happens during this moment of time.
The area of the rectangle is length multiplied by width,
which is f(x) multiplied by “dx.”
Therefore, the area of this rectangle is exactly equal to
the distance traveled during this moment in time.
The area of the entire graph is the sum of all the individual rectangles.
Therefore, the area of the entire graph is
the sum of all the different value of f(x) multiplied by “dx.”
We use this symbol to indicate the
sum of all the different values of f(x) multiplied by “dx.”
We call this the integral of f(x).
The integral of f(x) represents the area of the entire graph,
with the yellow rectangles having negative area.
In this case, we are interested in the area of the graph
between the time where X is zero, and the time where X is 10.
We represent it like this, and we call this the integral of f(x) from zero to ten.
The integral of the graph of the speed will always be equal to the distance traveled.
This is the Fundamental Theorem of calculus.
This is the foundation allowing us to write equations
describing how all objects in the universe behave.
This is the language in which the fundamental laws of the universe are written in.