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It is instructive to compute the Taylor series for several of the elementary functions. We take up this task in the present section.
We compute the first few derivatives of f (x) = sin(x) :
|f (2)(x)||= - sin(x)|
|f (3)(x)||= - cos(x)|
|f (4)(x)||= sin(x)|
At this point, we have arrived back at sin(x) , and the pattern will repeat itself. Evaluating the derivatives at 0 , we find that
|f (0)||= 0|
|f (2)(0)||= 0|
|f (3)(0)||= - 1|
|f (4)(0)||= 0|
and so on. This allows us to write down the first few terms of the Taylor series for f (x) = sin(x) at 0 :
|x - + - + ...|
Below, we plot the graph of sin(x) , together with the graphs of its Taylor polynomials p 1(x) , p 3(x) , and p 5(x) .
We can write the entire Taylor series as
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