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Numerical differentiaon - Example

Example 

Find у ’ (1)  and у ’’ (1) for function у ( х ), given tabular:

Values of function у ( х )

 x y Δy Δ 2 y Δ 3 y Δ 4 y 0.96 0.7825361 – 86029 0.98 0.7739332 – 1326 – 87355 25 1.00 0.7651977 – 1301 1 – 88656 26 1.02 0.7563321 – 1275 – 89931 1.04 0.7473390

S o l u t i o n .  According to the table we’ll receive: h = 0.02, х = 1, х 0 = 1, hence, q = 0. Having made final differences of function у ( х ) and using underlined values, from formula (9) we’ll receive: Similarly, using blacked out values from formula (10) we’ll receive: For comparison with exact values we’ll notice, that tabulated function is Bessel’s function at which the first derivative at х = 1 is equal to – 0.4400506, and the second, accordingly, – 0.325147. From here it is visible, that numerical finding of the second derivative gives greater error, than for the first derivative.

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