Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach. In this first part--part one of five--you will extend your understanding of Taylor series, review limits, learn the *why* behind l'Hopital's rule, and, most importantly, learn a new language for describing growth and decay of functions: the BIG O.
Expansion Points
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Del curso dictado por University of Pennsylvania
Calculus: Single Variable Part 1 - Functions
calificaciones
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De la lección
Taylor Series
This module gets at the heart of the entire course: the Taylor series, which provides an approximation to a function as a series, or "long polynomial". You will learn what a Taylor series is and how to compute it. Don't worry! The notation may be unfamiliar, but it's all just working with polynomials....
Conoce a los instructores
Robert GhristProfessor
Mathematics and Electrical & Systems Engineering
Welcome to calculus. I'm Professor Greist.
We're about to begin Lecture 6 on Expression Points.
We've seen that the Taylor series provides a good approximation to a
function near zero. What happens if we wish to focus our
attention at some other point. In this lesson, we'll consider changing
the expansion point. This will lead us to a broader definition
and interpretation of Taylor series. As we have seen, Taylor expansion gives
an excellent way to approximate a function for inputs near zero.
However, in many applications, zero is not the most interesting input that you
can think of. Many examples of this, besides finance
and economics, where we care about nonzero values.
What would be nice is a way to Taylor-expand a function that works well
for inputs that are not necessarily close to zero.
There certainly is. This is an important definition.
The Taylor series of f at x equals a, is the sum, k goes from 0 to infinity, the
kth derivative of f, evaluated at a, divided by k factorial times quantity x
minus a to the k. This is not a polynomial in x, but rather
a polynomial in the quantity x minus a, where the coefficient in front of each
monomial term is the kth derivative of f evaluated at a, and then divided by k
factorial. A little bit of terminology is in order.
The constant term is called the zeroth order term.
Next, comes the first order term, the second order term, the third order term,
and so on. These correlate to the degree of the
quantity x minus a. If we were to change things, perform the
change of variables defining h to be the quantity x minus a.
Then the above series would become a polynomial series in h.
However, it would not be giving you f of h, but rather f of x, which is a plus h.
This series in h is telling you that if you want to know values of f that are
close to a, then you can substitute in a small value for h.
And if you're looking for an approximation, you can ignore some of the
higher order terms. Let's look at an example.
Compute the Taylor series of log of x. Now, we know that we cannot do that about
x equals 0. So let us do it about a more natural
value. Let's say about x equals 1.
To compute this Taylor expansion, we're going to need to start taking some
derivatives. If we looked at the function log of x,
the zeroth order term is obtained by evaluating at x equals 1, and that gives
us, simply 0. The derivative of log of x, you may
recall is one over x. Evaluating that of x equals 1 gives us a
coefficient 1. Now, taking the derivative of one over x,
gives us minus x to the negative 2. We evaluate that and continue
differentiating. As we go, evaluating each term at x
equals 1. After computing sufficiently many
derivatives, you start to see a pattern. It requires a little bit of thinking, and
a particularly formal way of thinking called induction.
But, with a bit of effort, one can conclude that the kth derivative of log
of x is negative 1 to the k plus 1 times k minus 1 factorial times x to the
negative k. Evaluating that at x equals 1 gives us
simply the coefficient, negative one to the k plus 1 times k minus 1 factorial.
Now, to get the full Taylor Series, we need to divide this coefficient by k
factorial. In so doing, we see a not too unfamiliar
series. For log of x as x minus 1, minus x minus
1 squared over 2, plus x minus 1 cubed over 3, et cetera.
The coefficient in front of the degree k term is negative 1 to the k plus 1, over
k. And our summation goes from 1 to
infinity. Indeed, if we let h be x minus 1, then we
obtain the very familiar series, log of 1 plus h equals sum k goes from 1 to
infinity, negative 1 to the k plus 1, h to the k over k.
We've seen that before with an x instead of an h, but it works the same.
Do keep in mind that Taylor Series are not guaranteed to converge everywhere.
Indeed, if we look at the terms for log of x and take a finite Taylor polynomial,
then the higher and higher degree terms only provide a reasonable approximation
to log of x within the domain of convergence.
We know from our last lesson that that domain is going to be for values of x
between 0 and 2. Outside of that, these terms are
providing worse and worse approximations of log of x.
What do we do if we want to approximate log of x outside of this domain?
Well, you need to do a Taylor expansion about a different point, someplace close
to where you want to approximate. It's a bit easy to get confused with all
of the different notation associated with Taylor series.
So, let's review. One way to think about the Taylor
expansion about x equals a is to write f of x as a series in the quantity x minus
a. Another way to do it is to write it as a
series in the quantity h, where h is equal to x minus a.
In which case, you're coming up with an approximation for f of a plus h.
One way to think about this is that the more derivatives of f you know at a point
a, the better an approximation you get at x or at a plus h.
This is a perspective that you're going to want to keep with you for the
remainder of this course. The principle is that successive
polynomial truncations of the Taylor series approximate increasingly well.
We've seen that before. In this lesson, the point is that where
you do the Taylor expansion matters. If you are expanding way over here and
trying to get information about way over there, then you're going to need a lot of
derivatives to do that. On the other hand, if you approximate
about the correct expansion point, you might not need so many derivatives in
order to get the job done. Let's look at an explicit example.
What would it take to estimate the square root of 10?
Well, we would have to look at the Taylor series of the function square root of x.
If we expand that about a point, x equals a, then I'll leave it to you that the
first few derivatives work out to a Taylor series of root a plus one over 2
root a times quantity x minus a minus 1 over 8, times the square root of a cubed,
times quantity x minus a squared plus some higher-order terms in x minus a.
Well, to compute the square root of 10, what are we going to do?
Let's say I expand about 1 because 1 is a simple value.
I know the square root of 1, that is simply 1.
That makes the coefficients of the Taylor expansion easy to compute.
On the other hand, when I'm actually trying to estimate the square root of 10,
based on this, then I get 1 plus one half times 9 minus one eighth times 9 squared
or 81, plus higher order terms. How good of an approximation is that?
Well, that gives me a value of negative 4.125.
Now, I know this is not the square root we're looking for.
That is a bad approximation. So, what if we were to compute an
expansion about x equals 9? Then, this is something for which I also
know the square root. The square root of 9 is 3.
The coefficients are easy to work with. And if I plug in a value of x equals 10
into this Taylor series, I get an approximation of 3 plus one sixth minus 1
over 216. This gives an approximation of 3.1620
some other stuff. The true answer agrees with this up to
the first four digits. The expansion point, in this case,
certainly matters. Now, one thing to be cautious of, is that
if you're computing a Taylor series of a composition, you must expand about the
correct values. If you have a function, f composed with
g, and you want to expand it about some input x.
Then, you must expand g about x. But you must expand f, not about x, but
about g of x. And it is that term in particular that
causes problems. In an explicit example, we'll be able to
see how this works. Compute the Taylor series of e to the
cosine of x about x equals zero. Well, even the cosine x is a composition.
What do you what do you do first? First, you take the cosine of x, then you
exponentiate it. If we are to compute the Taylor series
about x equals 0, then we must expand cosine about 0, but not e to the X.
Cosine of X about 0 is very simple. This one, we know.
However, for the second term, the exponential, we must expand that about an
input of 1 because that is what gets fed into it.
Cosine of 0 is 1. Well, the Taylor series of e to the u
about u equals 1 is easy to compute. There's nothing difficult there.
However, what we must then do is substitute into it this series for cosine
of x. That is u is 1 minus x squared over 2
plus x to the 4th over 4 factorial plus higher ordered terms.
That is [COUGH] a little bit too much algebra to fit on this slide, and so I'll
leave you to do that. Though I might point out a little bit of
help here, that this can be rewritten as e times e to the cosine of x minus 1.
You'll find that to be a little bit easier to compute the Taylor series of.
This lesson marks the end of our introduction to Taylor series.
But by no means does it mark the end of our use of Taylor series.
In our next lesson, we will consider one of the primal objects of calculus, the
notion of a limit. And we'll consider it using the
understanding gained from Taylor series.
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