Example 4-5-1 - Maple Help



Chapter 4: Partial Differentiation





Example 4.5.1



Let  and let P be the point $\left(1,2\right)$.

 a) Obtain $\nabla f$ at P.
 b) Graph the surface $z=f\left(x,y\right)$.
 c) On the same set of axes, graph the level curve through P, and $\nabla f$ at P.
 d) At P, show that $\nabla f$ is orthogonal to a vector tangent to the level curve through P.
 e) At P, obtain $\mathrm{ψ}=\left(\nabla f\right)·\mathbf{u}$, the directional derivative of $f$ in the direction  . Show that $\mathrm{ψ}$ is a maximum when u is along $\nabla f\left(\mathrm{P}\right)$ and that this maximum is $∥\nabla f\left(\mathrm{P}\right)∥$.