I'm learning about linear approximations, where a function $f(x,y) \approx L(x,y)$ if both are evaluated at the same point $(a,b)$ but $L(x,y)$ becomes more and more error prone when moving away from $(a,b)$.

I also know that the tangent plane in $(a,b)$ is $$L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)$$

where $dz=f_x(a,b)(x-a)+f_y(a,b)(y-b)$ is the distance between $f(a,b)$ and the tangent plane, and where $\Delta z=f(a+(x-a), b+(y-b))-f(a,b)$ is the distance between $f(a,b)$ and $f(a+(x-a), b+(y-b))$.

However, I'm a bit lost when I'm asked to estimate the error in some exercises. I'm not sure whether I should use $dz$ or $\Delta z$. Some exercises I have are:

1 $\bullet$ Find an estimation of error when $\cos(0.1) \sin(-0.02)$ is approximated by a linearization of $f(x,y)=\cos(y)+\sin(y.x)$ at $P=(0,0)$.

2 $\bullet$ Find $a$ and $b$ $\in R$ so that linearization of $f(x,y)=ay+\cos(xy)+\sin(bx)$ at $P=(0,0)$ is $P(x,y)=1+2x-y$. Provide an error estimation using this $P$ to approximate $f(0.1, -0.2)$

Not sure which one I'm asked to use in those two problems...


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