Taylor Polynomials and Approximations
Polynomial functions can be used to approximate other elementary functions such as sin
x
,
x
e
,
and ln
x
.
On your calculator, graph:
1
sin
y
x
3
5
2
3!
5!
x
x
y
x
and compare.
_______________________________________________________________________________
Definition of an nth-degree Taylor polynomial
:
If
f
has
n
derivatives at
x
=
c
, then the polynomial
2
...
2!
!
n
n
n
f
c
f
c
P
x
f
c
f
c
x
c
x
c
x
c
n
is called the
n
th-degree Taylor polynomial for
f
at
c
, named after Brook Taylor, an English
mathematician.
________________________________________________________________________________
If
c
= 0, then
2
0
0
0
0
...
2!
!
n
n
n
f
f
P
x
f
f
x
x
x
n
is called the
n
th-degree Maclaurin
polynomial for
f
, named after another English mathematician, Colin Maclaurin.
________________________________________________________________________________
Ex
. Find the Maclaurin polynomial of degree
n
= 4 for
cos
f
x
x
, and then use
4
P
x
to
approximate the value of
cos 0.8, cos 1.1, and cos 4.
Check your answers.
What do you notice
about your approximations?

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- Summer '16
- Taylor Series, Taylor Polynomials, Brook Taylor, sin x