Lecture 3 : Taylor Series

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.