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Function that has derivatives at point a can be presented by polynom with coefficients that are calculated by Taylor formula. For details visit this page for example.
With the help of LeoStatistic we can investigate how good is approximation with Taylor polynomials for sin(x) near the x=0.
In tab "Data" click in "Insert" button. Insert parameter as argument with arithmetic progression from 0 to 9.99 with step 0.01.
Add others as arguments by formulas:
Result presentation has to be like this:
Obviously the large grade of Taylor polynom the longer from initial point x=0 they can approximate the sin(x). What is really amazing is that if to use approximation scheme of LeoStatistic and one can find that formula:
s = +0.806542*x +0.504436*x^2 -0.591369*x^3 +0.15451*x^4 -0.0156156*x^5 +0.000548931*x^6
will practically exact fit all displayed domain:
This circumstance has not to surprised us too much. At first approximation polynom in above has one more extra coefficient that is had happened more important then a highest grade of its term. Other important distinction from Taylor polynomial presentation is that just approximation of the data in interval 0 - 10 not guarantee a fit data outside the interval. If you look at behavior of the fitting curve outside of interval special in negative region of argument.
As for Taylor polynomial approximation it is symmetrically good matcher from basic point. In our example area of fit with Taylor polynom 11th grade is about from -4.5 till 4.5. Just about the same by all range as our approximating formula that covers dominion from 0 to 10.
Screenshots of the LeoStatistic software:
click on picture to enlarge
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