Linearization of an implicitly defined function $f(x,y,z)=e^{xz}y^2+\sin(y)\cos(z)+x^2 z$
Find equation of tangent plane at $(0,\pi,0)$ and use it to approximate $f(0.1,\pi,0.1)$. Find equation of normal to tangent plane.
My attempt: I found that tangent plane is $(2\pi-1)(y-\pi)=0$ or $y=\pi$ (all partial derivatives except $y$ equal 0).
I don't know how to find linear approximation from it.
What is the equation for linear approximation?
Possibly it is $L(x,y,z)=f(0,\pi,0)+(2\pi-1)(y-\pi)$

This is not an implicit function. An implicit function is like $0=e^{xz}y^2+\sin(y)\cos(z)+x^2 z$. Instead, your function is a 3D explicit function. 
You are correct on your linear approximation $L(x,y,z)=f(0,\pi,0)+(2\pi-1)(y-\pi)$ and that is exactly the tangent plane $w=f(0,\pi,0)+(2\pi-1)(y-\pi)$.
