Problems for Edward F. Redish |
For what values of θ is an approximation of O(θ) good for these functions? an approximation of O(θ^{3})?
Solution
To get the Taylor series for a function we need its derivatives. These are:
so the Taylor series for the two functions are:
The accuracy of the one-term and two-term Taylor series approximations are shown in the figures below.
so the Taylor series for the two functions are:
The constant approximation for cos is within 5% out to about 0.3 rad (almost 20^{o}!) and within 10% out to about 0.5 rad.
The two term series for cos is within 1% to about 0.65 rad and within 10% out to about 1.0 rad.
The three term series for cos is within 1% out to almost 1.2 radians.
The linear term for tan is within 1% to about 0.65 rad and within 10% out to about 0.55 rad.
The two term series for tan is within 1% to about 0.53 rad and within 10% out to about 0.9 rad.
University of Maryland | Physics Department | Physics 374 Home |
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Last revision 16. September, 2005.