This is calculus, the single variable. And I am Robert Ghrist, professor of
mathematics and electrical and systems engineering, of the University of
Pennsylvania. We're about to begin lecture one on functions. Here we'll review
several functions that you've already encountered in mathematics before -
trigonometrics, polynomials, exponentials. And we'll end the lesson with a mysterious
formula. Calculus is really all about functions. You're probably used to
thinking of functions in terms of their graphs, where you plot X versus Fx). of X.
A slightly different perceptive things of X as an input and F of X as an output. And
it's not a bad idea at all to visualize a function as a machine that takes in X, and
returns F of X. From that perspective, certain terminology is concerning
functions. A clear, the collection of all possible inputs is called the domain. The
collection of all possible outputs is called a range. Now this is single
variable calculus, which means that our domains and ranges are specially simple.
The reals, or some subinterval thereof. Certain operations on functions are
critical to understand. Perhaps the most important is that of composition. The
composition of two functions F and G, is defined to be the function that takes as
its input X, and returns as its output G of X fed into F, that is read F of g of X.
One thinks of this as first you do G, and then you plug the output of G into the
input of F. Lets look at a simple example, the square root of one minus X squared,
can be thought of as the composition of two functions F and G. If G is the
function one minus X squared, then what would F be? F would be the function that
takes an input and returns it's square root. Another operation on functions that
is very important is the inverse. The inverse is written F with a superscript
negative one. That does not mean the reciprocal of F. This denotes the inverse.
The inverse is the function that undoes F. By which I mean, if you plug X into F what
output do you get? You get F of X. If you plug F of X in to f inverse what you will
get is X, F inverse undoes F. Now, notice that you can run this machine both ways if
you plug X into F inverse. What you get when you plug that into F is X back. Again
let's look at a simple example if we consider the function X cubed then its
inverse is going to be X to the one third or rather the cube root of X. Why is that?
If we take the cube root of X cubed we get X. If we take the cube of the cube root,
we get X back again. Now, notice the symmetry that you see in the graphs. The
graphs of F and F inverse are always going to be symmetric about the line Yx. equals
X. That is, the line where the input and the output are the same. Certain classes
of functions are critical to calculus. Perhaps the simplest is that of the
polynomials. A polynomial is a function of the form of a constant plus a constant
times X plus a constant times X squared, all the way up to a constant times X to
the for some finite N. That top power is called the degree of the polynomial. We
can also write, a polynomial, using a summation, notation. The sum, K goes from
zero to N of constant. C Sub K, times Monomial term, X to the K. That summation
symbol, is going to be used very frequently in this course. You might as
well get used to it now. Another class of functions very simple to work with are the
rational functions. Rational functions are functions of the form P of X over Q of X
where each is a polynomial. These are very simple to work with, for example 3X minus
one over X squared plus X minus six. The only thing you have to be careful of with
a rational function is the denominator. When the denominator takes a value of
zero, then the function may not be well defined. Other powers, besides those of
positive integers are useful. Let's review, what is X to the zero? And we all
know what that is, that's equal to one. What is X to the negative one half? We'll
recall a fractional power denotes roots. for example, X to the one half means the
square root of X. The negative sine in the exponent means that we take th e
reciprocal. So the answer is one over root X. What is X to the 22 sevenths? Well,
that means we take X to the twenty-second power, and look at the seventh root of
that. What is X to the pie? When you take an irrational power, it seems as though we
ought to be able to make sense of that by some sort of limit. Well, let's, let's
keep that in the back of our heads for the moment and turn to some other matters.
Let's see, the trigonometric functions are of primary importance in calculus and the
rest of mathematics. You should be familiar with the, the basic trigonometric
functions, sine, and cosine. One of the things you're going to have to keep in
mind, that remember, is that cosine squared plus sine squared equals one.
There are several ways to think about that geometrically. For example, if we look at
a right triangle with hypotenuse one, the, the sine of the angle theta gives you the
opposite side length. The cosine gives you the adjacent side length to that Theta. We
could embed that picture intouh, a, a diagram for the unit circle. Where we see
that cosine of feta and sine of feta returns the X and Y coordinates and the
point on the unit circle with angle feta to the X axis. That explains the nature of
the formula cosine squared plus sine squared equals one. It's simply the
Pythagorean Theorem for the X and Y coordinate that triangle. There are other
trigonometric functions as well. Besides sine and cosine, you should be familiar
with the tangent function, the ratio of the sine with the cosine, as well as the
cotangent function, its reciprocal, cosine over sine. The secant function is the
reciprocal of the cosine, and the cosecant function, is the reciprocal of the sine.
All four of these have vertical asymptotes at regions where the denominator term goes
to zero. The inverse trigonometric functions are likewise very useful. One
must be a little bit careful with these however. One often writes sine negative
one in the exponent. To denote the inverse, but this can cause confusion.
Students might think that th is, is one over the sine, that is the cosecent.
That's a bad idea. I recommend instead, using the terminology arc sine. For the
inverse of the sine function. The arc sine function, takes on values between negative
Pi over two and pi over two, and has a restricted domain going from negative one
to one. The arc cosine function, likewise, has a restricted domain from negative one
to one, but is chosen to take values, between zero and Pi. The arc tangent
function, unlike these two, has an unbounded domain. It is well defined for
all inputs. But it has a restricted range between negative Pi over two and Pi over
two. Our last class of important functions concerns the exponential. You should of
seen the exponential function E to the X in your previous precalculus and calculus
background. You know what it's graph looks like. You know also it's inverse. The log
function or more properly than natural logarithm function. These are logarithm
base E. These two are inverse to one another. That means their graphs are
symmetric about the diagonal line Yx. equals X. You know that for example, E to
the zero equals one, because anything to the zero equals one. And therefore, you
know also that log of one equals zero. Now, you've seen these functions before.
But if I ask you, what exactly do I mean by E when I say log base E, or E to the X.
What is meant by that? You could answer that E is the number for which the natural
log of E equals one. But then, what do we mean by the natural logarithm? Well of
course E is simply a number a particular irrational number location on the real
number line. B why is that number so special? Where did it come from? Why is it
so useful so prevalent in mathematics? Before we answer that question. We need to
go over some properties associated to the exponential function, E to the X. Let's
recall some of the algebraic properties first. For example, E to the X times E to
the Y equals E to the X plus Y. In like manner, E to the X raised to the Yth power
equals E to the X times Y. Both of these properties follow from the, the simple
properties for exponents. E to the X however, has some very unique properties
concerning derivatives and integrals. In your previous exposure to calculus, you've
seen a little bit concerning derivatives and integrals. And you may have seen the
following two important examples. The derivative of E to the X equals E to the
X. The integral of the E to the X DX likewise is E to the X. Oh plus a
constant, don't forget that. We'll be talking more about these ideas and these
properties later, but for the moment keep them in mind, we'll see them again next
lecture. Now there's one formula that ties together all of the functions that we've
looked at in this first lesson trigonometrics, exponentials, and that is
Euler's formula. Euler's formula states that E, e to the IX equals Cosine of X
plus I times of sine of X. You may or may not have seen this before? Let's think for
a moment about what it means. It's a wonderful formula if we can make sense of
it. First of all, a little bit about the, the terminology used. The I in the
exponent is the imaginary number, square root of negative one. It's the number that
has the property that I squared is equal to negative one. This is not a real
number. That doesn't mean that it doesn't exist, it just means it is not on the real
number line. Euler's formula concerns the exponentiation of an imaginary variable.
What exactly does that mean? How is this related to trigonometric functions? The
graphs of these two don't seem to be alike at all. That is something, that we will
answer, in our next lesson. Our introduction to functions is complete. But
the mystery Euler's formula remains. In our next lesson, we'll resolve that
mystery, by coming to a deeper understanding, of what the exponential
function is and does.