Module 5 : Linear and Quadratic Approximations, Error Estimates, Taylor's Theorem, Newton and Picard Methods
Lecture 14 : Taylor's Theorem [Section 14.1]
14.1
Taylor 's Theorem and its applications
In previous section we used , the tangent line, a polynomial of degree one in to approximate a given
function for near . One can try to approximate the function by a higher degree polynomial,
hoping that the polynomial of higher degree will give a better approximation to for near . To analyze
this, we need a generalization of the extended mean value theorem:
14.1.1
Theorem (Taylor's Theorem):
Then, .
for some
The above expression is also known as the Taylor 's formula for around .
Proof:
We assume the proof. The interested reader may refer a book on advanced calculus.
, the tangent line, a polynomial of degree one in 
to approximate a given
function 
for 
near 
. One can try to approximate the function 
by a higher degree polynomial,
hoping that the polynomial of higher degree will give a better approximation to 
for 
near 
. To analyze
this, we need a generalization of the extended mean value theorem: 
. 
around 
. 
be as in theorem 14.1.1. 
