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!
