
Applications
of differentiation  the graph of a function and its derivatives 
Maclaurin's
formula or Maclaurin's theorem 
The
approximation of the sine function
by polynomial using Taylor's or Maclaurin's formula 
Properties
of the power series expansion of the sine function 






Maclaurin's
formula or Maclaurin's
theorem 
The
formula obtained from Taylor's
formula by setting x_{0}
= 0


that
holds in an open neighborhood of the origin, is called Maclaurin's
formula or Maclaurin's
theorem. 

Consider
the polynomial
f_{n}_{
}(x)
= a_{n}x^{n}
+ a_{n}_{ }_{}_{
1}x^{n }^{}^{
}^{}^{
}^{}^{
}^{1} + · · · + a_{3}x^{3}
+ a_{2}x^{2} + a_{1}x
+ a_{0}, let
evaluate the polynomial and its successive derivatives at the origin, 
f
(0) =
a_{0}, f '(0) = 1· a_{1},
f ''(0) = 1· 2a_{2},
f '''(0) = 1· 2· 3a_{3}, . . .
, f ^{(}^{n}^{)}(0) =
n!a_{n} 
we
get the coefficients, 


Therefore,
the Taylor polynomial of a function f
centered at x_{0}
is the polynomial of degree n
which has the same
derivatives as f
at x_{0},
up to order n. 
If
a
function f
is infinitely differentiable on an interval about a point x_{0}_{ }
or the origin, as are for example e^{x} and
sin
x,
then 
P_{0} (x)
= f (x_{0}), 
P_{1} (x) =
f (x_{0})
+ (x

x_{0})
f ' (x_{0}), 

P_{0},
P_{1},
P_{2},
. . . is a sequence of increasingly approximating polynomials for f. 

The
approximation of the sine function
by polynomial using Taylor's or Maclaurin's formula 
Example: Let
represent the sine function f
(x)
= sin x
by the Taylor polynomial (or power series). 
Solution: The
sine function is the infinitely differentiable function defined
for all real numbers. 
We
use the polynomial with infinitely many terms


to
represent the sine function. We should calculate the function
value f (0),
and some successive derivatives 
of
the sine function, to determine the nth
order derivative expression, therefore


Obtained
values f (0)
and f ^{(n)
}(0)
substituted into Maclaurin's formula with Lagrange
remainder
yield, 


thus, 


Properties
of the power series expansion of the sine function 
The polynomials that describe the sine function
all are symmetric source polynomials of odd degree. Meaning, 
both coordinates
of translations 

are
zero since all even derivatives are 

zero, the infinite polynomial is missing
every next even term, therefore a_{n}_{}_{1}
= 0. 


Notice
that every second polynomial in the above sequence, whose leading term
is negative, represents the variant f
(x)
(or 
f (x))
of the source polynomial of odd degree whose x
or y
variable changed the sign, as are
the graphs, f_{3
}(x)
and f_{7 }(x). 
Observe that graphs
of polynomials
approach closer and closer to the graph of sine function as n
increase. 








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