I've got a problem which asks me to find linear approximation of multi-variable function and its maximum error.
Here's the problem :
By about how much will
$$g(x,y,z)=x+x\cos(z)-y\sin(z)+y$$ change if the point $P(x, y, z)$ moves from $(2, -1, 0)$ a distance of $ds = 0.2$ unit toward the point $(0, 1, 2)$? Also find an upper bound for the error of it.
I know I should find its linearization $L(x,y)$ at $(2, -1, 0)$ and plug its $x$-increment and $y$-increment in $L(x,y)$.
But I don't know how to derive an upper bound for it.
In one-variable calculus, I remember I dealt this kind of problem with Taylor's inequality, but this is not one-variable situation and I feel I'm stuck.