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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.

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  • $\begingroup$ There’s a version of Taylor’s theorem for multivariable functions with an associated exact bound for the error. However, this is a fairly simple function and you’re only looking at the first-order approximation, so it shouldn’t be too hard to work out that error with a bit of algebra and trig identities. $\endgroup$ – amd Oct 12 '18 at 0:18

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