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Several proof and application of Taylor formula
Abstract:Taylor formula is an important formula in higher mathematics, it
plays a very important role in theoretical and methodological.In the
elementary function, polynomial is the most simple function, because the
operations of polynomial function only add, subtract, multiply three
operations.Obviously,the study of functional state and function value of
approximate calculation of great significance, If we can replace
polynomial function approximation for rational fraction function,
especially the unreasonable function and elementary transcendental
function to the thought of a "approximation",while error can meet the
requirements.
The introduction :As everybody knows, polynomial function is one of
the functions whose structure are very simple and convenient to make
calculation,it is an important content to use polynomial approximation
function of approximate calculation and theoretical analysis .As we know
it is very convenient to use
polynomial approximation
f (x)? f ?(x0)(x ? x0) replacing the function value f (x) and (x ? x0) its
nearby, but its accuracy often can not meet our actual demand, it request
-s us to find another polynomial ,the famous mathematician Taylor puts
forward the famous theorem,it is a good way to solve the polynomial
approximation by replacing some of the more complex functions such a
complicated problem with Taylor formula
With more than Palin type of Taylor formula .
Theorem 1:If the function f in the point xo until N derivative,there are
f ?x? ? f ?x0?? f ??x0??x ? x0?? f ???x0??x ? x0?2 ??? f ?x0??x ? x0?n ?o ?x ? x0??n?
?n?
?
?
2!
n!
Prove:Assuming-that
Tn ? f ?x0?? f ??x0??x ? x0?? f ???x0??x ? x0?2 ??? f ?x0??x ? x0?n
?n?
(1)
2!
n!
Rn ? f ?x??Tn ?x?
Qn(x) ??x ? x0?n
Rn ?x? ? 0
As long as lim
,
By the relation( 1) we can get
Q ?x?
x?x
0
n
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