orlovskihmw

2022-07-11

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

Sariah Glover

Expert

Let $F:x↦ax+b$b be the linear approximation of $f$ around $0$, and $G:x↦cx+d$ be the one of $g$. Then we have $FG:x↦\left(ax+b\right)\left(cx+d\right)=ac{x}^{2}+\left(bc+ad\right)x+bd$ which is clearly not linear.
But actually the linear approximation of $fg$ is $x↦\left(bc+ad\right)x+bd$ 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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