Hey it’s Professor Dave, I wanna tell you about calculus.
Calculus is a subject that is cloaked in darkness.
Many students quit math entirely when they get up to this point, mainly out of fear.
But in the grand scheme of things, calculus is actually considered intermediate math.
There are dozens of topics that are more abstract and more difficult to comprehend.
We will get to some of those in due time as well, but as far as calculus is concerned,
as long as you apply yourself, there is no reason why you can’t learn this subject.
Just like trigonometry and algebra before this, it’s just about learning some new
operations, along with the associated notation, which won’t be so scary once we understand it.
So first things first, what exactly is calculus, and why did it come about?
Remember that at the beginning of this series, we learned how humans developed arithmetic
out of necessity, in order to trade goods with one another.
So what challenge could possibly have necessitated the development of calculus?
To answer this, let’s visit a few moments in the history of mathematics.
As we saw in our study of geometry, the ancient Greeks were responsible for developing our
understanding of shapes and their characteristics.
One thing they figured out was how to find the area of a polygon by splitting it up into
triangles.
A polygon contains two fewer triangles than its number of sides, and they knew how to
get the area of a triangle, so this was no problem at all.
The trouble came when trying to find the area of a curved shape, like a circle.
Here, there are no vertices from which to draw lines, so this can’t be split up into triangles.
For this reason, they developed something called the method of exhaustion.
This is where they would inscribe a triangle into the circle, and then gradually increase
the number of sides of this polygon.
Four sides, then five, then six, and continuing from there.
As we can see, the shape is starting to look more and more like a circle, so the area of
the polygon must be getting closer and closer to the area of the circle, though it will
never quite get there.
In fact, the polygon would need to have an infinite number of sides in order to have
an area that is identical to that of the circle, so we can say that in the limit of infinity,
the area of the polygon equals the area of the circle, where n represents the number
of sides in the polygon.
Though the Greeks did not fully grasp the concept of limits, they were able to conceive
of this rudimentary technique, and prove the formula for the area of a circle this way.
It is precisely this kind of logic that we will use to do calculus, when we try to find
the area under a curve, meaning the area of the section in between a function and the x-axis.
Just like with the circle, we can approximate this area, only this time by drawing rectangles,
and to get a more accurate approximation, we simply make them narrower,
and narrower still, to more closely follow the curvature.
In the limit of infinitely thin rectangles, we could find the precise area under this curve.
So clearly, there is something about curvature that specifically requires calculus.
We can easily calculate the area of a rectangle, because the distance between opposite sides is fixed.
But when a value such as the height of a figure is changing from each point to the next, the
techniques that we learned in algebra and geometry just don’t work any longer, and
we need something more sophisticated.
During the 17th century, this more sophisticated branch of mathematics finally came about.
The first to make considerable progress with this was French mathematician Pierre de Fermat,
but everything truly came together with Isaac Newton, who along with Gottfried Leibniz,
is credited with developing modern calculus.
The interesting thing is that Newton developed calculus simply out of necessity, in order
to have the tools he needed to solve problems in physics regarding celestial motion.
Leibniz did similar work just a few years later, independently of Newton, and it is
actually his notation that we still use today.
So what was it that Newton understood that changed everything?
Well one realization came from looking at falling bodies.
He understood that if you drop an object, its speed will increase every single instant
until it hits the ground.
But during that time, the object must have some definite speed at any given instant.
He knew of no mathematics that could adequately calculate these instantaneous velocities,
as something had to be developed that could describe the difference between the value
of a function, in this case position, and the rate of change for that function, in this
case velocity.
So as we can see, this concept of rate of change was the other of the two important
paths that would bring about this mathematical revolution, as its exploration comprised the
development of differential calculus.
This is in contrast with the first approach we discussed, with the area under a curve
and the rectangles.
That is called integral calculus.
But whether differential or integral, both concepts involve the idea that we can do something
infinitely many times and get a finite answer that is useful, and both concepts involve
dealing with things that are infinitely small or infinitely close together.
Although it took until the 17th century to get this down in a rigorous and formal way,
it’s still pretty much the same idea that was being pondered all the way back in ancient Greece.
Even back then, a form of Zeno’s paradox said that a person can never walk towards
a wall and touch it, because they would first have to get halfway there.
And then halfway from that point.
And then halfway again.
And so on.
Because this remaining distance can be split in half infinitely many times, they will never
get to the wall.
Of course we know that a person can indeed go and touch a wall, so this paradox was just
a foreshadowing of our modern understanding that we can do something infinitely many times
and get a finite result, like some of the infinite series we looked at that actually
have finite sums, or calculating the area under a curve, in the case of the infinite rectangles.
This notion of seeing what something does in the limit of infinity is the central idea
of calculus.
It led us to the development of new operations, differentiation and integration, which will
be the major focal points of our study.
But don’t be intimidated, from exponentiation to logarithms, we’ve picked up totally new
operations many times by now, it’s just a matter of defining the operations and learning
the symbols we use to execute them.
So buckle up, take a deep breath, and let’s learn some calculus!