Chapter
IX: DIFFERENTIAL EQUATIONS AND POLYNOMIALS: TAYLOR'S THEOREM
© 2000 M. Flashman
In numerous cases throughout this book, as in the real world, we
find problems where an exact numerical answer is either impossible or impractical
to express as a decimal number. Examples of these situations include calculation
of 
, p,
e, and ln(2). As we have seen there are many ways to characterize and define
these numbers and use these characterizations to find an estimate. For
our discussion in this chapter it is important to notice that each of these
numbers can be characterized as arising from the evaluation of a function
which is the solution to a differential equation. [See Table 1 below.]
|
NUMBER
|
FUNCTION
|
DIFFERENTIAL
EQUATION
|
= f(2)
|
|
|
|
ln(2) = f(2)
|
f(x) = ln(x)
|
|
|
e = f(1)
|
|
|
|
p =
f(1)
|
f(x) = 4arctan(x)
|
|
We have discussed approximations from the very beginning of our work
on the derivative, with Newton's method for estimating the roots of equations,
Euler's method for estimating particular solutions to differential equations,
and numerous methods for estimating definite integrals. In the following
sections we will delve further into the nature and applications of
polynomial approximations based on imformation about the derivatives of
a function.
To provide further motivation for what we shall refer to as Taylor's
approximation theory, let's consider first a simple situation where
an exact value for a function is attainable but perhaps not worth the effort.
EXAMPLE IX.A.1: Suppose 
and we wish to find f(.5).
Of course the exact answer is .25 + .125 + (.5)100, but if
we only need an estimate of the answer it is easy to settle for .375. The
error we make in using this estimate is relatively small [namely (.5)100].
In this example we should note two things.
-
First, the computation is theoretically easy because the function involved
is a polynomial; and
-
second, the estimate was found by evaluating a related polynomial,
.
Taylor's Theory-Objective and Key Ideas: The main concerns
of Taylor's theory for estimating function values are
-
to find estimating polynomials for a given function and
-
to measure the error in using these polynomials to estimate the
desired values.
Two Key Ideas: You may recall from our earlier discussions of estimations
using the differential that when x is close to a, f(x) is approximately
equal to a linear function, f(a) + f '(a) (x-a). Furthermore,
the Mean Value Theorem guarantees that as long as f is a sufficiently
well behaved function there is some c between a and x where f(x)
- f(a) = f '(c)(x-a), so the difference between the function's
value at x and a can be measured using the value of the derivative.
Taylor's theory generalizes these two ideas that use the derivative
to estimate.
IX.A. Estimating ex with Polynomials: An Introduction
to Taylor's Theory.
To focus our discussion more specifially, in this section we will consider
only 
and
begin by trying to estimate f(1)=e illustrating the key ideas of
Taylor's
theory.
At x = 0, we find that f(0) = 1, f '(0) =1, and since
f
'(x)
= f(x) we have that f (n)(0) = 1 for all
natural numbers
n. We now consider our first result, typical of
the Taylor's theory approach to estimation using polynomials and derivative
information.
PROPOSITION IX.A.1: If 
then Pn(x) is a polynomial of degree n
so that 
and Pn(b) is approximately equal to eb. In
fact, if we let Rn = eb - Pn(b)
then 
for
some c between 0 and b.
Estimating e: Before we proceed to justify this result, we'll
apply
this result using n = 5 to estimate the value of e [= e1].
First,
,
so according to the proposition, e is approximately equal to 
(
about 2.716667) and R5=e-P5(1) where 
for some c beween 0 and 1. Thus, e is approximately 2.716667 and the error
in using this estimate is R5. Since c is between 0 and 1, ec
is between 1 and e, so we can deduce that R5, the difference
between e and the estimate, P5(1), is no greater than
e/720. Using estimates of e from our earlier work in Chapter VI we know
that e < 3, so our errror R5 is no larger than 3/720
= 1/240.
A more accurate estimate can be obtained by using a larger value for
n. Try using n=6 and 7 to see the improvement. [This can be done progressively
using a spreadsheet which you can examine
now or later.]
Proof of Proposition: We'll begin our proof by finding Pn'(x).
From this it follows easily that 
.
Now suppose that b is not 0 and let Rn = eb
-
Pn(b).
We need only justify the formula for evaluating Rn.
For convenience we'll write R=Rn and let
.
Then
, while 
from the definition of R.
Furthermore 
But 
, so 
.
Now we apply the Mean Value (or Rolle's) Theorem to the function g,
we can say that there is a number c between 0 and b where g'(c)=0.
Thus 
and 
.
Solving this last equation for R gives 
.
EOP.
Estimating the value of e is not the only use for the previous proposition.
Before turning to the more general Taylor's theory in the next section,
here are two more examples of its application in estimating definite integrals
related to ex.
EXAMPLE IX.A.2: Estimate 
.
Use this to estimate e.
Solution: Using Proposition IX.1 with n=4 for each x
between 0 and 1 we have
where
for some c with 0<c<x<1.
Thus 
[Remember ec<3.] and so 
for all x between 0 and 1. Now we use the monotone property of the
definite integral [Ch. V . ** ] to obtain the estimate: 
,
so
.
Therefore e is approximately 
as in the first estimate we saw in the note after the statement of the
theorem, and the error in using this estimate is less than 
as we also saw previously.
The next example follows a similar analysis but applied to a more difficult
yet still important integral.
EXAMPLE IX.A.3: Estimate 
.
Solution: For each t between 0 and 1, -t2
is between -1 and 0. Let x = -t2. By Proposition
IX.1 with n = 4 for each x between -1 and 0 we have
where
for some
c with 0>c>x.
Since c < 0 , 0 < ec < 1, and since x < 0 we
have that 
.
Substituting -t2 for x ,we see that
Now since 
we have 
.
By evaluating these integrals we obtain
or 
.
Therefore, 
is approximately equal to 
and this is an overestimate by no more than 
.
Comment: In both of these examples we have been able to estimate
a definite integral involving the exponential function by using a Taylor
polynomial of degree 4. It should be apparent that by using a higher degree
for the estimating polynomial, the error term will become smaller and we
will obtain a more precise estimate. The systematic pattern in these polynomials
should allow you to find more precise estimates for the last example without
much difficulty.
Exercises IX.A.:
-
Use the Taylor polyonmial for ex of degree 4 to estimate the
following:
(a) e2 (b) e3 (c) e.5 (d) e-
1 (e) e3.14.
-
Estimate e using the Taylor polynomial of degree n where n is (a) 6 (b)
7 (c) 8 (d) 10.
In each of these estimates discuss the size of the error term Rn.
-
What value of n should be used so that the Taylor polynomial of degree
n will give an estimate of e that is within .000001 of the exact value
of e? Explain your result.
-
Use the Taylor polynomial for ex of degree 5 to estimate
.
Discuss the error in this approximation.
-
Use the trapezoidal rule and Simpson's rule with n = 5 to estimate
.
-
Use the Taylor polynomial for ex of degree 4 to estimate
. Discuss the error in this approximation.
Sketch on the same graph the Taylor polynomials for ex of
degree 1,2,3 and 4 along with ex. Discuss the graphical interpretation
of Rn .
-
What value of n should be used so that the Taylor polynomial of degree
n will give an estimate of
that
is within .000001 of the exact value? Explain your result.