Let f ( x ) = a r c t a n ( x )....

glitinosim3

glitinosim3

Answered

2022-07-04

Let f ( x ) = a r c t a n ( x ). Use the derivative approximation:
f ( x ) = 8 f ( x + h ) 8 f ( x h ) f ( x + 2 h ) + f ( x 2 h ) 12 h to approximate f ( 1 4 π ) using h 1 = 2 , 4 , 8 . Try to take h small enough that the rounding error effect begins to dominate the mathematical error. For what value of h does this begin to occur?

Answer & Explanation

Tristin Case

Tristin Case

Expert

2022-07-05Added 15 answers

Partial answer
To get an idea how good the approximation is, you can calculate the result for the special functions f ( x ) = 1 , x , x 2 , x 3 . Here, we get the exact result for f ( x ) = 1 , x , x 2 , x 3 , x 4 , so the formula is exact upto degree 4 (Polynomials upto degree 4 are differentiated exactly by the formula)
For f ( x ) = x 5 , you get f ( x ) = 5 x 4 4 h 4 , so you have an error of 4 h 4

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