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)|
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(3)(0)||= - 1|
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 p1(x), p3(x), and p5(x).
We can write the entire Taylor series as