Instructions.

A 3D view of Example III. The surface \(z = \frac{x^2}{2} + \frac{y^2}{2}\) is shown with the level curve \(f = 1\text{,}\) the circle \(x^2 + y^2 = 2\text{,}\) passing through the point \((1,1,1)\text{;}\) its dashed projection lies in the \(xy\)-plane. Drag the slider to rotate the direction \(\mathbf u = \langle\cos\varphi, \sin\varphi\rangle\) at \(P_0(1,1)\text{:}\) the vertical plane in the direction \(\mathbf u\) cuts the surface in a curve whose tangent line at \((1,1,1)\) has slope \(D_{\mathbf u} f = \cos\varphi + \sin\varphi\text{.}\) Use the buttons to snap to the answers of Example III, and note that when \(D_{\mathbf u} f = 0\text{,}\) the vector \(\mathbf u\) is tangent to the level curve. Drag the figure to view it from a different angle, or press the “Rotate” button.