Tangent plane

The formula comes from requiring that the plane \(z=f(a,b)+m(x-a)+n(y-b)\) pass through the point of tangency and that its slices by \(y=b\) and by \(x=a\) be the tangent lines of those curves. This forces \(m=f_x(a,b)\) and \(n=f_y(a,b)\). For example, for \(z=x^2+y^2\) at \((1,2)\) we have \(f_x=2\), \(f_y=4\) and the plane is \(z=5+2(x-1)+4(y-2)=2x+4y-5\).

The normal vector can be read from the equation: writing it as \(f_x\,x+f_y\,y-z=\text{constant}\), the normal is \(\langle f_x,f_y,-1\rangle\), with a minus sign in the third component. The plane is horizontal exactly where \(f_x=f_y=0\), the candidates for maxima, minima and saddle points.

The most immediate use is the linearization \(L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\), which approximates \(f\) near \((a,b)\): with \(f=\sqrt{x^2+y^2}\) at \((3,4)\), it gives \(f(3.03,3.96)\approx4.986\), against the exact value \(4.9862\ldots\). The error grows away from the point of tangency, and the approximation is good only if \(f\) is differentiable there.

The existence of the partials is not enough for the plane to be a good approximation: that requirement is formulated by the concept of differentiability, which comes next and which justifies calling this plane "tangent".