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Contents >> Applied Mathematics >> Numerical Methods >> Numerical Differentiaon >> Example

Numerical differentiaon - Example

Example [1]

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

Values of function у (х)

x

y

Δy

Δ2y

Δ3y

Δ4y

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, х0 = 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 functionat 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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Last updated: April 30, 2015.