Derive the equation of the tangent plane to the surface \(z = f(x,y)\) at a point \(P_0(x_0, y_0, f(x_0,y_0))\) from the slopes \(f_x(x_0,y_0)\) and \(f_y(x_0,y_0)\) of the two tangent lines in the planes \(y = y_0\) and \(x = x_0\text{.}\)
Find the normal vector \(\mathbf n = \nabla w \Big|_{P_0}\) of the tangent plane, where \(w(x,y,z) = f(x,y) - z\text{,}\) and write the parametric equations of the normal line to the surface at \(P_0\text{.}\)
Estimate the change \(\Delta f\) in the value of \(f\) produced by moving a small distance \(ds\) away from \(P_0\) in the direction of a unit vector \(\mathbf u\text{,}\) using the differential \(df = \left(\nabla f \Big|_{P_0} \cdot \mathbf u\right) ds\text{.}\)
Recall that the partial derivative \(f_x(x_0,y_0)\) is the slope of the tangent line to the curve cut from the surface \(z = f(x,y)\) by the plane \(y = y_0\text{,}\) and that \(f_y(x_0,y_0)\) is the slope of the tangent line to the curve cut by the plane \(x = x_0\text{.}\) In this chapter we put these two tangent lines together: the plane containing both of them is the tangent plane to the surface at \(P_0(x_0, y_0, f(x_0,y_0))\text{.}\) We find its equation, use its normal vector to write the equation of the normal line to the surface, estimate the change in \(f\) caused by moving a small distance away from \(P_0\) in a given direction, and finally use the tangent plane to linearize \(f\) near a point.
