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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'(x) | = cos(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'(0) | = 1 | ||
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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