Hello everyone, welcome to the magnificence of mathematics.
I'm your host, Eddie Kingston.
If you've ever worked on a farm and had to put up fencing around a particular area, how
could you go about maximizing the area you could enclose with the fencing materials you'd
have on hand?
On the other hand, a lot of you listeners might remember the formula for the area of
a triangle, 1 half base times height, or that of a circle, pi r squared.
But what if you wanted to find the area of, say, a rectangle with a squiggly top, or some
other arbitrary figure?
These two questions can't possibly be related, right?
Well, it turns out that the answers to these two questions turn out to be not only related,
but more or less the opposite of each other.
They have to do with the two subfields of calculus, namely differential calculus and
integral calculus.
Over the course of the next several episodes, we'll answer these two questions using both
differential and integral calculus and see more about how calculus is used a lot in day-to-day
life.
Back in episode 3, I briefly mentioned the idea of a limit in the context of probability.
Knowing what a limit is, is crucial to understanding differential calculus.
The basic idea of a limit is that you want to know how a function behaves near a point
without actually reaching that point.
For example, imagine you draw up two lines, one vertical and one horizontal, intersecting
at one point.
This intersecting point is called the origin.
Now put little marks evenly spaced from each other and label the origin 0, the first mark
to the right of the origin 1, second mark to the right 2, and so on.
Let the first mark to the left be negative 1, the second to the left negative 2, and
so on.
Repeat the same thing for the vertical line, but with right replaced by up and left replaced
with down.
You are now imagining what's called a Cartesian coordinate system, named after 17th century
French philosopher René Descartes.
Now imagine a straight diagonal line going up and to the right, passing through the origin.
You are now imagining the function y equals x, where the x coordinate matches the y coordinate
at every point.
So if you look at where you are on this line at x equals 2, you'll see that you're also
at the y coordinate y equals 2.
Now imagine you move along this line from this spot and get closer and closer to the
point x equals 1, and you'll see that you're approaching the point y equals 1 as well.
In particular, you are said to be approaching y equals 1 from the right, meaning you start
it at the right hand side of the line and work your way from the right to the left.
Now if you start from the left side and work your way up and to the right of the line to
the point x equals 1, you are said to be approaching the point y equals 1 from the left, since
you're starting from the left hand side and working your way to the right.
Since this is true for any real number a, you can then and only then say that the limit
as x approaches a of this function x is simply that number a.
If we were working with a more complicated function, it might be the case that the limit
as x approaches a number a from the right is a different value than the limit as x approaches
a from the left.
In that case, we say that the limit doesn't exist.
For example, consider the function y equals 1 over x.
Since 1 divided by 1 equals 1, the value of y at x equals 1 is simply 1.
If you consider x equals 1 half, the value of y is 2, since 1 divided by 1 half is 2,
because 2 times 1 half is 1, and multiplication and division are inverses, or opposites of
each other.
Now if you consider x equals 1 third, the value of y is 3 for the same reason.
You can see that as x gets smaller, y gets bigger.
In particular, as we get closer and closer to x equals 0, y just grows and grows without
bound.
We say that the limit as x approaches 0 from the right of 1 over x is infinity, positive
infinity in particular.
Similarly, if you consider x equals negative 1 half, y equals negative 2.
If x equals negative 1 third, y equals negative 3.
So as x gets closer and closer to 0 from the negative, i.e. left side, y gets increasingly
negative.
Hence the limit as x approaches 0 from the left of 1 over x is negative infinity.
So we see that as x gets closer and closer to 0 from these two directions, this function
approaches two different values.
So the limit as x approaches 0 of 1 over x does not exist.
Another way limits have the ability to not exist is if they don't approach anything
in particular.
Consider a function that just gradually oscillates back and forth between negative 1 and 1, touching
every real number along the way.
So at one interval, the function starts at negative 1, and as you move right, it goes
up up up up all the way to 1, and then it goes down down down down down all the way
to negative 1.
And this cycle repeats forever in both directions.
One example of this function is called the sine function, so y equals sine of x.
Now what is the limit as x approaches infinity of this function?
Since it's constantly hitting negative 1, 1, and everything in between, it just goes
on forever, not approaching anything in particular.
It's not getting closer and closer to any certain value in the long run, and it's not
growing and growing to either positive or negative infinity, so we say in this instance
that the limit as x approaches infinity of sine of x does not exist either.
Now that we know what limits are, let's get into what derivatives are, and the heart of
what differential calculus is about.
If you've taken an algebra class in high school, you might remember finding slopes
of lines, rise over run, and all that.
This is just how much the value of y changes divided by how much the value of x changes.
For example, for the line y equals 2x, consider the points 1,2, and 2,4, where in the first
point the x coordinates 1 and the y coordinates 2, and in the second point the x coordinates
2 and the y coordinates 4, through both of which the line goes.
The change in y is just 4 minus 2 equals 2, and the change in x is just 2 minus 1 equals
1.
So the slope of this line is the change in y, namely 2, divided by the change in x, namely
1.
So the slope of the line is 2.
Similarly, for any function y equals m times x, for some number m, the slope is just that
number m.
We'll later see that the slope is also the derivative of functions like these.
Any function you can think of has a corresponding straight line at every point, such that the
line just barely touches the function at that point.
This is what's known as a tangent line, and the slope of this tangent line is also the
slope of the function at that point.
This slope at that point is called the derivative of the function at that point.
What if you wanted a function that told you the derivative at every point?
Tangent lines can be approximated by what are called secant lines, which are lines that
just barely touch the function at two different points.
Let's call the x coordinate of one of the points x, and the y coordinate f of x the
function evaluated at that point x.
So for example, for the function f of x equals x squared, the point x equals 3 corresponds
to the point f of 3, which is 3 squared, which is 9.
Also an important thing to notice is that f of x is the exact same thing as y in terms
of function notation.
Now consider moving along the graph by a tiny, tiny amount.
Now we're going to call this tiny, tiny amount h.
So we call the x coordinate of our second point x plus h, and the y coordinate f of
x plus h.
So the secant line intersects the function at two points, the first of which we call
the x coordinate just x, and the y coordinate f of x, and the second point where we call
the x coordinate x plus h, and the y coordinate f of x plus h.
The slope of this secant line is the change in the y coordinate f of x plus h minus f
of x, all divided by the change in the x coordinate x plus h minus x, which is just h.
As h gets closer and closer to zero, the secant line at a point just becomes the tangent line
at that point.
That is, the limit as h approaches zero of the quantity f of x plus h minus f of x divided
by h is the slope of the tangent line, and this limit is by definition the derivative
of the entire function f of x.
This is noted by either f prime of x, which is what's called Lagrange's notation, named
after French mathematician Joseph-Louis Lagrange, whose works were inspired by the famous English
physicist Isaac Newton, or by dy divided by dx, read just dy dx, which is what's called
Leibniz notation, named after German mathematician Gottfried Wilhelm Leibniz.
Regardless of which notation you want to use, this tells you the derivative of the function
at any point you want.
For example, let's consider the function f of x equals m times x for some number m.
If we plug in x plus h in this function, we get f of x plus h equals m times the quantity
x plus h, which if we distribute that out is m times x plus m times h.
Then we subtract f of x, which is just m times x, to get m times h in the numerator of our
limit.
Divide this by h to simply get m.
Since this function no longer depends on what h is, the limit as h approaches zero of m
is simply that number m, and this agrees with what we had earlier.
For a slightly more complicated example, consider f of x equals x squared.
Then f of x plus h is simply the quantity x plus h squared, which if you remember FOIL,
first outer and last from your algebra 1 class, that gets us x squared plus x times h plus
x times h plus h squared, which is x squared plus 2x times h plus h squared.
Subtract off x squared to get 2 times x times h plus h squared.
Note that we have h as a common factor in this expression, so we can factor this as
h times the quantity 2x plus h, and divide that by just h.
The h's cancel out, and we have the limit as h approaches zero of 2x plus h.
We can just evaluate this limit term by term, and get 2x plus zero, which is just 2x.
So the slope of the tangent line at any point x of the function f of x equals x squared
is f prime of x equals 2x.
Whew, okay, so having gone through all that, let's go back to what we talked about before
with the whole fence thing.
Suppose you have 40 square feet of fencing material, say wood or whatever, that you want
to enclose in a rectangular area.
What would you want the dimensions of the fence to be such that the area you enclose
is maximized?
Well since you have 40 square feet of wood to use as the perimeter of your enclosure,
you want to find the length L and the width W such that 2L plus 2W equals 40, that is
the perimeter of the rectangle is 40, and such that the area L times W is maximized.
If we solve our first equation for L by subtracting both sides by 2W and dividing everything by
2, we get L equals 20 minus W. We can plug this into the expression for area and get
W times the quantity 20 minus W and distribute this out to get 20 times W minus W squared.
Finding the maximum area of the enclosure amounts to finding the maximum value of the
function f of W equals 20 times W minus W squared.
The first step in finding the maximum value is to find the derivative of this function.
We can do this term by term, and we actually know what this derivative is already.
The derivative of 20 times W with respect to W is just 20, because that's the slope
of the function f of W equals 20 times W, and the derivative of W squared is just 2
times W. So the derivative of the whole function is 20 minus 2W.
Now the next step is to set this derivative equal to 0 and solve for W. So we subtract
20 from both sides and divide by negative 2 to get W equals 10 feet. But since we had
L equals 20 minus W, the corresponding length L is just 20 minus 10 equals 10 feet. So therefore,
the rectangular shape that maximizes the enclosed area is just a square with side length 10
feet, since all sides are equal.
This is true regardless of how much fencing you have. Calculus has already been seen to
be super useful in everyday life, and we've barely begun scratching the surface. Next
time we'll talk about differential calculus's counterpart, integral calculus. See you then!
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