11 infinite sequences and series. 11.11 applications of taylor polynomials infinite sequences and...
TRANSCRIPT
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11INFINITE SEQUENCES AND SERIESINFINITE SEQUENCES AND SERIES
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11.11Applications of
Taylor Polynomials
INFINITE SEQUENCES AND SERIES
In this section, we will learn about:
Two types of applications of Taylor polynomials.
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APPLICATIONS IN APPROXIMATING FUNCTIONS
First, we look at how they are used to
approximate functions.
Computer scientists like them because polynomials are the simplest of functions.
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APPLICATIONS IN PHYSICS AND ENGINEERING
Then, we investigate how physicists and
engineers use them in such fields as:
Relativity Optics Blackbody radiation Electric dipoles Velocity of water waves Building highways across a desert
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APPROXIMATING FUNCTIONS
Suppose that f(x) is equal to the sum
of its Taylor series at a:
( )
0
( )( ) ( )
!
nn
n
f af x x a
n
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In Section 11.10, we introduced
the notation Tn(x) for the nth partial sum
of this series.
We called it the nth-degree Taylor polynomial of f at a.
NOTATION Tn(x)
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Thus,
( )
0
2
( )
( )( ) ( )
!
'( ) ''( )( ) ( ) ( )
1! 2!
( )... ( )
!
ini
ni
nn
f aT x x a
i
f a f af a x a x a
f ax a
n
APPROXIMATING FUNCTIONS
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Since f is the sum of its Taylor series,
we know that Tn(x) → f(x) as n → ∞.
Thus, Tn can be used as an approximation to f :
f(x) ≈ Tn(x)
APPROXIMATING FUNCTIONS
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Notice that the first-degree Taylor polynomial
T1(x) = f(a) + f’(a)(x – a)
is the same as the linearization of f at a
that we discussed in Section 3.10
APPROXIMATING FUNCTIONS
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Notice also that T1 and its derivative have
the same values at a that f and f’ have.
In general, it can be shown that the derivatives of Tn at a agree with those of f up to and including derivatives of order n.
See Exercise 38.
APPROXIMATING FUNCTIONS
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To illustrate these ideas, let’s take another
look at the graphs of y = ex and its first few
Taylor polynomials.
APPROXIMATING FUNCTIONS
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The graph of T1 is the tangent line to
y = ex at (0, 1).
This tangent line is the best linear approximation to ex near (0, 1).
APPROXIMATING FUNCTIONS
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The graph of T2 is the parabola
y = 1 + x + x2/2
The graph of T3 is
the cubic curve
y = 1 + x + x2/2 + x3/6
This is a closer fit to the curve y = ex than T2.
APPROXIMATING FUNCTIONS
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The next Taylor polynomial would be
an even better approximation, and so on.
APPROXIMATING FUNCTIONS
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The values in the table give a numerical
demonstration of the convergence of the
Taylor polynomials Tn(x) to the function y = ex.
APPROXIMATING FUNCTIONS
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When x = 0.2, the convergence is very rapid.
When x = 3, however, it is somewhat slower.
The farther x is from 0, the more slowly Tn(x) converges to ex.
APPROXIMATING FUNCTIONS
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When using a Taylor polynomial Tn
to approximate a function f, we have to
ask these questions:
How good an approximation is it?
How large should we take n to be to achieve a desired accuracy?
APPROXIMATING FUNCTIONS
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To answer these questions, we
need to look at the absolute value
of the remainder:
|Rn(x)| = |f(x) – Tn(x)|
APPROXIMATING FUNCTIONS
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There are three possible
methods for estimating
the size of the error.
METHODS FOR ESTIMATING ERROR
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If a graphing device is available,
we can use it to graph |Rn(x)| and
thereby estimate the error.
METHOD 1
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If the series happens to be
an alternating series, we can use
the Alternating Series Estimation
Theorem.
METHOD 2
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In all cases, we can use Taylor’s Inequality
(Theorem 9 in Section 11.10), which states
that, if |f (n + 1)(x)| ≤ M, then
1( )
( 1)!n
n
MR x x a
n
METHOD 3
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a. Approximate the function f(x) = by
a Taylor polynomial of degree 2 at a = 8.
b. How accurate is this approximation
when 7 ≤ x ≤ 9?
APPROXIMATING FUNCTIONS Example 1
3 x
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APPROXIMATING FUNCTIONS Example 1 a
1/33
2/31 13 12
5/32 19 144
8/31027
( ) (8) 2
'( ) '(8)
''( ) ''(8)
'''( )
f x x x f
f x x f
f x x f
f x x
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Hence, the second-degree Taylor
polynomial is:
APPROXIMATING FUNCTIONS Example 1 a
22
21 112 288
'(8) ''(8)( ) (8) ( 8) ( 8)
1! 2!
2 ( 8) ( 8)
f fT x f x x
x x
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The desired approximation is:
APPROXIMATING FUNCTIONS Example 1 a
32
21 112 288
( )
2 ( 8) ( 8)
x T x
x x
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The Taylor series is not alternating
when x < 8.
Thus, we can’t use the Alternating Series Estimation Theorem here.
APPROXIMATING FUNCTIONS Example 1 b
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However, we can use Taylor’s Inequality
with n = 2 and a = 8:
where f’’’(x) ≤ M.
APPROXIMATING FUNCTIONS Example 1 b
32| ( ) | | 8 |
3!
MR x x
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Since x ≥ 7, we have x8/3 ≥ 78/3,
and so:
Hence, we can take M = 0.0021
APPROXIMATING FUNCTIONS Example 1 b
8/3 8/3
10 1 10 1'''( ) 0.0021
27 27 7f x
x
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Also, 7 ≤ x ≤ 9.
So, –1 ≤ x –8 ≤ 1 and |x – 8| ≤ 1.
Then, Taylor’s Inequality gives:
Thus, if 7 ≤ x ≤ 9, the approximation in part a is accurate to within 0.0004
APPROXIMATING FUNCTIONS Example 1 b
32
0.0021 0.0021| ( ) | 1 0.0004
3! 6R x
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Let’s use a graphing device to check
the calculation in Example 1.
APPROXIMATING FUNCTIONS
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The figure shows that the graphs of y =
and y = T2(x) are very close to each other
when x is near 8.
APPROXIMATING FUNCTIONS
3 x
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This figure shows the graph of |R2(x)|
computed from the expression
We see that |R2(x)| < 0.0003when 7 ≤ x ≤ 9
APPROXIMATING FUNCTIONS
32 2| ( ) | | ( ) |R x x T x
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Thus, in this case, the error estimate
from graphical methods is slightly better
than the error estimate from Taylor’s
Inequality.
APPROXIMATING FUNCTIONS
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a. What is the maximum error possible
in using the approximation
when –0.3 ≤ x ≤ 0.3?
Use this approximation to find sin 12° correct to six decimal places.
APPROXIMATING FUNCTIONS Example 2
3 5
sin3! 5!
x xx x
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b. For what values of x is this
approximation accurate to within
0.00005?
APPROXIMATING FUNCTIONS Example 2
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Notice that the Maclaurin series
alternates for all nonzero values of x and the
successive terms decrease in size as |x| < 1.
So, we can use the Alternating Series Estimation Theorem.
APPROXIMATING FUNCTIONS Example 2 a
3 5 7
sin3! 5! 7!
x x xx x
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The error in approximating sin x by
the first three terms of its Maclaurin series
is at most
If –0.3 ≤ x ≤ 0.3, then |x| ≤ 0.3 So, the error is smaller than
APPROXIMATING FUNCTIONS
7 7| |
7! 5040
x x
Example 2 a
78(0.3)
4.3 105040
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To find sin 12°, we first convert to radian
measure.
Correct to six decimal places, sin 12° ≈ 0.207912
APPROXIMATING FUNCTIONS Example 2 a
3 5
12sin12 sin
180
1 1sin
15 15 15 3! 15 5!
0.20791169
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The error will be smaller than 0.00005
if:
Solving this inequality for x, we get:
The given approximation is accurate to within 0.00005 when |x| < 0.82
APPROXIMATING FUNCTIONS Example 2 b
7| |0.00005
5040
x
7 1/ 7| | 0.252 or | | (0.252) 0.821x x
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What if we use Taylor’s Inequality
to solve Example 2?
APPROXIMATING FUNCTIONS
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Since f (7)(x) = –cos x, we have
|f (7)(x)| ≤ 1, and so
Thus, we get the same estimates as with the Alternating Series Estimation Theorem.
APPROXIMATING FUNCTIONS
76
1| ( ) | | |
7!R x x
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What about graphical
methods?
APPROXIMATING FUNCTIONS
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The figure shows the graph of
APPROXIMATING FUNCTIONS
3 51 16 6 120| ( ) | | sin ( ) |R x x x x x
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We see that |R6(x)| < 4.3 x 10-8
when |x| ≤ 0.3
This is the same estimate that we obtained in Example 2.
APPROXIMATING FUNCTIONS
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For part b, we want |R6(x)| < 0.00005
So, we graph both y = |R6(x)| and y = 0.00005, as follows.
APPROXIMATING FUNCTIONS
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By placing the cursor on the right intersection
point, we find that the inequality is satisfied
when |x| < 0.82
Again, this is the same estimate that we obtained in the solution to Example 2.
APPROXIMATING FUNCTIONS
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If we had been asked to approximate sin 72°
instead of sin 12° in Example 2, it would have
been wise to use the Taylor polynomials at
a = π/3 (instead of a = 0).
They are better approximations to sin x for values of x close to π/3.
APPROXIMATING FUNCTIONS
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Notice that 72° is close to 60° (or π/3
radians).
The derivatives of sin x are easy to compute at π/3.
APPROXIMATING FUNCTIONS
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The Maclaurin polynomial approximations
to the sine curve are graphed
in the following figure.
APPROXIMATING FUNCTIONS
3
1 3
3 5 3 5 7
5 7
( ) ( )3!
( ) ( )3! 5! 3! 5! 7!
xT x x T x x
x x x x xT x x T x x
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APPROXIMATING FUNCTIONS3
1 3
3 5 3 5 7
5 7
( ) ( )3!
( ) ( )3! 5! 3! 5! 7!
xT x x T x x
x x x x xT x x T x x
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You can see that as n increases, Tn(x)
is a good approximation to sin x on a larger
and larger interval.
APPROXIMATING FUNCTIONS
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One use of the type of calculation
done in Examples 1 and 2 occurs in
calculators and computers.
APPROXIMATING FUNCTIONS
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For instance, a polynomial approximation
is calculated (in many machines) when:
You press the sin or ex key on your calculator.
A computer programmer uses a subroutine for a trigonometric or exponential or Bessel function.
APPROXIMATING FUNCTIONS
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The polynomial is often a Taylor
polynomial that has been modified so
that the error is spread more evenly
throughout an interval.
APPROXIMATING FUNCTIONS
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Taylor polynomials
are also used frequently
in physics.
APPLICATIONS TO PHYSICS
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To gain insight into an equation, a physicist
often simplifies a function by considering only
the first two or three terms in its Taylor series.
That is, the physicist uses a Taylor polynomial as an approximation to the function.
Then, Taylor’s Inequality can be used to gauge the accuracy of the approximation.
APPLICATIONS TO PHYSICS
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The following example shows
one way in which this idea is used
in special relativity.
APPLICATIONS TO PHYSICS
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In Einstein’s theory of special relativity,
the mass of an object moving with velocity v
is
where:
m0 is the mass of the object when at rest.
c is the speed of light.
SPECIAL RELATIVITY Example 3
0
2 21 /
mm
v c
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The kinetic energy of the object is
the difference between its total energy
and its energy at rest:
K = mc2 – m0c2
SPECIAL RELATIVITY Example 3
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a. Show that, when v is very small compared
with c, this expression for K agrees with
classical Newtonian physics: K = ½m0v2
b. Use Taylor’s Inequality to estimate
the difference in these expressions for K
when |v| ≤ 100 ms.
SPECIAL RELATIVITY Example 3
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Using the expressions given for K and m,
we get:
SPECIAL RELATIVITY Example 3 a
2 20
220
02 2
1/ 222
0 2
1 /
1 1
K mc m c
m cm c
v c
vm c
c
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With x = –v2/c2, the Maclaurin series
for (1 + x) –1/2 is most easily computed as
a binomial series with k = –½.
Notice that |x| < 1 because v < c.
SPECIAL RELATIVITY Example 3 a
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Therefore, we have:
SPECIAL RELATIVITY
312 21/ 2 21
2
3 512 2 2 3
2 33 512 8 16
(1 ) 12!
...3!
1 ...
x x x
x
x x x
Example 3 a
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Also, we have:
SPECIAL RELATIVITY
2 4 62
0 2 4 6
2 4 62
0 2 4 6
1 3 51 ... 1
2 8 16
1 3 5...
2 8 16
v v vK m c
c c c
v v vm c
c c c
Example 3 a
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If v is much smaller than c, then all terms
after the first are very small when compared
with the first term.
If we omit them, we get:
SPECIAL RELATIVITY
22 21
0 022
1
2
vK m c m v
c
Example 3 a
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Let:
x = –v2/c2
f(x) = m0c2[(1 + x) –1/2 – 1]
M is a number such that |f”(x)| ≤ M
SPECIAL RELATIVITY Example 3 b
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Then, we can use Taylor’s Inequality
to write:
SPECIAL RELATIVITY Example 3 b
21| ( ) |
2!
MR x x
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We have and
we are given that |v| ≤ 100 m/s.
Thus,
SPECIAL RELATIVITY
2 5/ 2304'''( ) (1 )f x m c x
2 20 0
2 2 5/ 2 2 2 5/ 2
3 3| ''( ) | ( )
4(1 ) 4(1 100 / )
m c m cf x M
v c c
Example 3 b
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Thus, with c = 3 x 108 m/s,
So, when |v| ≤ 100 m/s, the magnitude of the error in using the Newtonian expression for kinetic energy is at most (4.2 x 10-10)m0.
SPECIAL RELATIVITY
2 4100
1 02 2 5/ 2 4
31 100| ( ) | (4.17 10 )
2 4(1 100 / )
m cR x m
c c
Example 3 b
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SPECIAL RELATIVITY
The upper curve in the figure is the graph
of the expression for the kinetic energy K
of an object with velocity in special relativity.
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SPECIAL RELATIVITY
The lower curve shows the function
used for K in classical Newtonian physics.
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SPECIAL RELATIVITY
When v is much smaller than the speed
of light, the curves are practically identical.
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Another application to physics
occurs in optics.
OPTICS
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This figure is adapted from Optics,
4th ed., by Eugene Hecht.
OPTICS
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It depicts a wave from the point source S
meeting a spherical interface of radius R
centered at C. The ray SA is refracted toward P.
OPTICS
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Using Fermat’s principle that light travels so
as to minimize the time taken, Hecht derives
the equation
where:
n1 and n2 are indexes of refraction.
ℓo, ℓi, so, and si are the distances indicated in the figure.
OPTICS
2 11 2 1 i o
o i i o
n s n sn n
R
Equation 1
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By the Law of Cosines, applied to
triangles ACS and ACP, we have:
OPTICS Equation 2
2 2
2 2
( ) 2 ( )cos
( ) 2 ( ) cos
o o o
i i i
R s R R s R
R s R R s R
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As Equation 1 is cumbersome to work with,
Gauss, in 1841, simplified it by using the linear
approximation cos ø ≈ 1 for small values of ø.
This amounts to using the Taylor polynomial of degree 1.
OPTICS
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Then, Equation 1 becomes the following
simpler equation:
OPTICS
2 2 11
o i
n n nns s R
Equation 3
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The resulting optical theory is known as
Gaussian optics, or first-order optics.
It has become the basic theoretical tool used to design lenses.
GAUSSIAN OPTICS
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A more accurate theory is obtained
by approximating cos ø by its Taylor
polynomial of degree 3.
This is the same as the Taylor polynomial of degree 2.
OPTICS
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This takes into account rays for which ø
is not so small—rays that strike the surface
at greater distances h above the axis.
OPTICS
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We use this approximation to derive the more
accurate equation
OPTICS
21
2 2
22 1 1 21 1 1 1
2 2
o i
o o i i
nn
s s
n n n nh
R s s R s R s
Equation 4
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The resulting optical theory is known
as third-order optics.
THIRD-ORDER OPTICS