Transcript Section 12.11 Applications of Taylor Polynomials

Section 12.11
Applications of Taylor
Polynomials
TAYLOR POLYNOMIAL OF
ORDER n
The Taylor polynomial of order n based at a,
Tn(x), for the function f is the nth partial sum of
the Taylor series at a for f. Thus,
f (a)
2

Tn ( x)  f (a)  f (a)( x  a) 
( x  a)  
2!
(n)
f (a)

( x  a) n
n!
MACLAURIN POLYNOMIALS
When a = 0 in the Taylor Polynomial of order n,
we call it the Maclaurin polynomial of order
n. That is, the Maclaurin polynomial of order n
is the nth partial sum of the Maclaurin series for
a function f.
(n)


f (0) 2
f (0) n
Tn ( x)  f (0)  f (0) x 
x 
x
2!
n!
TAYLOR’S THEOREM WITH
REMAINDER
Let f be a function whose (n + 1)st derivative f (n + 1) (x)
exists for each x in an open interval I containing a. Then,
for each x in I,
f (a)
f ( x)  f (a)  f (a)( x  a) 
( x  a) 2  
2!
f ( n ) (a)

( x  a) n  Rn ( x)
n!
whose remainder term (or error) Rn(x) satisfies the following
equation
f ( n1) (c)
Rn ( x) 
( x  a) n1
(n  1)!
and c is some number between x and a.
USEFUL TOOLS FOR
BOUNDING |Rn(x)|
It is usually impossible to get an exact value for
Rn(x). So, we usually bound |Rn(x)|. Our
primary tools are:
1. The triangle inequality. |a ± b| ≤ |a| + |b|
2. The fact that a fraction gets larger as its
denominator gets smaller.
3. The fact that a fraction gets larger as its
numerator gets larger.