Section 10.4
Taylor Polynomials
Approximating Functions by Polynomials
n
Tn ( x)  
i 0
T0 ( x) 
f
f (i ) (a)
( x  a )i
i!
(0)
(a)
( x  a)0  f  a 
0!
T1(x) = f(a) + f (a)(x – a)
Tn(x)
f  x  e
x
x2
T2  1  x 
T1  1  x
2
x 2 x3
T3  1  x  
2 6
Remainder in a Taylor Polynomial
Let Tn be the Taylor polynomial of order n for f .
The remainder in using Tn to approximate f is
Rn  f  x   Tn
Taylor’s Inequality
 x   M for x  a  d ,
then Rn  x  satisfies the inequality
If f
 n 1
M
n 1
Rn  x  
xa
for x  a  d .
 n  1!
 a  x  a n1


 n  1!
f
 n 1
How large should we take n to be in order to achieve a
desired accuracy?
| Rn (x) | = | f (x) – Tn (x)|
There are three possible methods for estimating the size
of the error:
1. If a graphing device is available, we can use it to
graph | Rn (x) | and thereby estimate the error.
2. If the series happens to be an alternating series, we
can use the Alternating Series Estimation Theorem.
3. In all cases we can use Taylor’s Inequality which
says that if
then
(a) Approximate the function
Taylor polynomial of degree 2 at a = 8.
by a
( (b) How accurate is this approximation when
7  x  9?
What is the maximum error possible
in using the approximation
3
5
x
x
when –0.3 ≤ x ≤ 0.3?
sin x  x  
3! 5!
Use this approximation to find sin 12°
correct to six decimal places.
| R6 ( x) || sin x  ( x  x 
1
6
3
1
120
5
x )|
What if we want to approximate sin 72° instead of sin 12°?
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.
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.
The polynomial is often a Taylor polynomial that
has been modified so that the error is spread more
evenly throughout an interval.
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.