Find the best approximation to the first derivative of f ( x ) based on the values of

Karina Trujillo

Karina Trujillo

Answered question

2022-06-14

Find the best approximation to the first derivative of f ( x ) based on the values of f ( x ) , f ( x + h ) , f ( x + 2 h ). What is the accuracy of this approximation?

Answer & Explanation

Kaydence Washington

Kaydence Washington

Beginner2022-06-15Added 32 answers

f ( x ) f ( x + h ) f ( x ) h for small h but that may be closer to f ( x + h 2 ).
So if you have a reasonable approximation to both f ( x + 3 h 2 ) and f ( x + h 2 ) and if you think you function is fairly smooth (in the sense of an almost constant second derivative in the range [ x , x + 2 h ] then you might reduce the earlier estimate by half the difference between those two, giving
f ( x ) f ( x + h 2 ) 1 2 ( f ( x + 3 h 2 ) f ( x + h 2 ) )
and
f ( x + h ) f ( x ) h 1 2 ( f ( x + 2 h ) f ( x + h ) h f ( x + h ) f ( x ) h ) = 4 f ( x + h ) 3 f ( x ) f ( x + 2 h ) 2 h .
In fact this is the only estimate which gives correct results for quadratic functions and any h.
minwaardekn

minwaardekn

Beginner2022-06-16Added 6 answers

From the Taylor series, f ( x + h ) = f ( x ) + h f ( x ) + h 2 f ( x ) / 2 + O ( h 3 ) and f ( x h ) = f ( x ) h f ( x ) + h 2 f ( x ) / 2 + O ( h 3 )
The only combination which gets rid of the f ( x ) term is f ( x + h ) f ( x h ) 2 h = f ( x ) + O ( h 2 )

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