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 x0 = 0     
that holds in an open neighborhood of the origin, is called Maclaurin's formula or Maclaurin's theorem.
Consider the polynomial   fn (x) = anxn + an - 1xn - - 1 + a3x3 + a2x2 + a1x + a0,
let evaluate the polynomial and its successive derivatives at the origin,
f (0) = a0,     f '(0) = 1 a1,     f ''(0) = 1 2a2,     f '''(0) = 1 2 3a3,  . . .  ,  f (n) (0) = n! an  
 we get the coefficients,
Therefore, the Taylor polynomial of a function f centered at x0 is the polynomial of degree n which has the same derivatives as f at x0, up to order n.
If a function f is infinitely differentiable on an interval about a point x0  or the origin, as are for example ex and  sin x, then
                    P0 (x) = f (x0),
                    P1 (x) = f (x0) + (x - x0) f ' (x0),
 
P0, P1, P2, . . . 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  an-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,  f3 (x) and  f7 (x).
Observe that graphs of polynomials approach closer and closer to the graph of sine function as n increase.
Calculus contents D
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