Calc1231W9T2 - Taylor polynomial of a function

Tuesday, 18 September 2018

10:07 PM

Machine generated alternative text:
If the first n derivatives of a function f all exist at = a then the Taylor polynomial pn of degree n about a is 
f(2) (a) 
pn(x) _ f(O) (a) + f(l) _ a) + 
2 
2! 
Note- f (n) (a) denotes the nth derivative of f evaluated at x = a, with f (0) (a) f(a) 
and f@) = sin(x) Then 
Suppose a 
2 
. f(l) ( # ) 
. f(2) ( # ) 
• f(3) ( # ) 
Since 
sin@), the pattern repeats itself after n

 

Machine generated alternative text:
The Taylor polynomial of degree 5 for f@) = sin(x) about — is equal to 
P5(c) — 
The Taylor polynomials pn@) for n = 1, 2, 3, 4 and 5 are pictured in the GeoGebra app below Which looks like the best approximation to sin@) 
T ne Taylor polynomial ot degree 5 
Note: the Maple syntax for T is pi. 
near

 

Machine generated alternative text:
to find a rational approximation to In 
Now use the Taylor polynomial of degree 3 for sin@) 
to find a rational approximation to sin 
Use the Taylor polynomial of degree 2 for ln@) near x 
—1 
—o 
near x — 
P'2(x) — 
10 
In 
9 
1 
1 
17/162 
1 
485/4374 
3

 

Machine generated alternative text:
The Taylor polynomial of degree 4 for f@) = In(l + x) about = 0 exists, and is equal to 
The first few Taylor polynomials for In(l + x) near x = O are shown in the GeoGebra app below; they are reasonable approximations to In(l + x) provided we choose < 
P30@) 
n — 3D

 

Machine generated alternative text:
The Taylor polynomialw of degrees 1, • 
• , 5 for f(x) 
ln@) about x 
1 are shown in the GeoGebra app below, along with their polynomial equations. 
f@) 
+ 24 . 
3! 
These Taylor polynomial have an incremental nature. For example, the degree 6 Taylor polynomial for f(x) about = 1 is

 

 

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