In this lesson we are going to talk about applications of Taylor Polynomials. The idea here is that we are going to write down some Taylor Polynomials, and plug in some values of x and we will see how close we get to the original function values. We are going to see a couple of tests that we are going to be using to check how accurate we are. The first one is already mentioned previously - Alternating Series Error Bound. The second one is new and is called Taylor's Remainder Theorem. The whole idea will make more sense after we go through some examples.
Alternating Series Error Bound:
is a series that satisfies Alternating Series Test, and we use the
partial sum sn as an estimate of the total
sum S (the true answer), then our error E could be
positive or negative, but its bounded by |an+1|:
|E| ≤ |an+1|
Taylors Remainder Theorem: Suppose
you use the Taylor polynomial Tk (x)
around a to estimate f (x) at a value of x near
a. Find the maximum value of f (k+1) on the
interval between a and x (or an upper bound for it).
Call that M. Then the
Error ≤ .
Hints and tips:
The point of these formulas is
that (among other applications) we will use Taylor polynomials to
estimate the values of f (x) for some functions that
would otherwise be difficult to compute. Then we will use these
error formulas to determine how accurate our estimates are.
General principles on accuracy of Taylor Series:
If they dont converge, then theyre no good at all.
But they usually converge, since 1/n! gets small quickly.
They are very accurate
when x is close to the center a, since (x − a)n
gets small quickly. When x is far from a, (x −
a)n gets big and they arent very accurate.
This is why we use different
center values of a depending on what values of x we want to
use the series for.
When you use Taylor Series to
estimate values of f (x), you need to center the
series around a value of a for which f (a), f ′
(a), f ′′ (a), and so on, will be easy to
In Taylors Remainder Theorem,
the difficult part is often coming up with the constant M .
Here are several points to remember:
Since M only needs to be
an upper bound, not an exact value, it is OK to estimate any
quantities in your expression for | f(k+1)
|. However, it is important to be certain that your estimate is
higher than the true value, not lower.
|sin x| ≤ 1, and
similarly for cos x.
|a − b| ≤ |a| +
To estimate high, make the
numerator as big as possible and the denominator as small as
Sometimes we are given the error
tolerance ahead of time and we must use these error formulas to
reverse-engineer the number of terms we need to be accurate to
within the given tolerance. (For example, an engineer building a
bridge might know that she needs calculations accurate to one
ten-thousandth of a meter. Or a calculator might have a display of
ten decimal places, so the manufacturer must design the internal
algorithms to produce answers with ten digits of accuracy.)
In both formulas (but especially
for Taylors Remainder Theorem), if you are given the error
tolerance, it may be hard to solve the formula explicitly for n
or k. You may need to use trial and error to find the right
n or k. (Remember that they are whole numbers.)
A common mistake is to think that
the Taylor polynomial Tk (x) has k
terms. k refers to the degree, not the number of terms. So,
for example, the Taylor polynomial T4
(x) for f (x) = sin x centered around a
= 0 is T4
(x) = x − x³&frasl 6 , because the term
of x4 is zero.
Taylor Polynomial Applications
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This is educator.com and we are here to talk about applications of Taylor Polynomials.0000
The idea here is that we are going to write down some Taylor Polynomials, and plug in some values of x and we will see how close we get to the original function values.0007
That will probably make some more sense after we study some examples, but I want to give you a couple of tests that we are going to be using to check how accurate we are.0020
The first one is one that we have seen before, if you look back at the lecture on alternating series.0028
We learned the alternating series error bound.0033
What that said is that if you have a series that satisfies the conditions for the alternating series test, then if we just take a partial sum, as an estimate of the total sum, then our error can be positive or negative, but it is bounded by aN+1.0037
In other words, the first term that we do not take, the first term that we cut off.0059
So, if you are a little rusty on that, you might want to go back and look at the lecture on alternating series.0068
That is where we did some practice with error bounds for alternating series.0074
The second series we are going to use is a new one.0078