If f ( x ) = g ( x ) h ( x )does the...

orlovskihmw

orlovskihmw

Answered

2022-07-11

If f ( x ) = g ( x ) h ( x )
does the linear approximation of f ( x ) equals the linear approximation of g ( x ) times the linear approximation of h ( x )?
is it true for quadratic approximations as well?

Answer & Explanation

Sariah Glover

Sariah Glover

Expert

2022-07-12Added 16 answers

Let F : x a x + bb be the linear approximation of f around 0, and G : x c x + d be the one of g. Then we have F G : x ( a x + b ) ( c x + d ) = a c x 2 + ( b c + a d ) x + b d which is clearly not linear.
But actually the linear approximation of f g is x ( b c + a d ) x + b d since you just need to get rid of the x 2 term.
For quadratic approximations, it's the same : you can multiply the two approximations, but then you need to get rid of all the terms with x k where k > 4.
except for some particular cases, it is false.

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