\[ \newcommand{\vect}[1]{\mathbf{\mathrm{#1}}} \]

Key terms and concepts for sections 12.6-12.7

Tangent plane (tangentplan). For a level surface (nivåflate) \(f=c\): \[ f_x(x_0,y_0,z_0)(x-x_0) + f_y(x_0,y_0,z_0)(y-y_0) + f_z(x_0,y_0,z_0)(z-z_0)=0. \]

A tangent plane is the linearization (linearisering) of a function near a given point \((x_0,y_0)\): \[ f(x,y) \approx f(x_0,y_0) + f_x(x_0,y_0)(x-x_0) + f_y(x_0,y_0)(y-y_0). \]

Local extrema: Local maxmima and local minima. No points nearby give higher (respectively lower) function values. If \(f\) has a local extreme value at \((x_0,y_0)\), then \(\nabla f(x_0,y_0)=\vect{0}\) if the gradient exists.

A critical point (kritisk punkt) of a function is an interior point where both partials are zero, or one or both do not exist.

A saddle point (sadelpunkt) of a function is a critical point where nomatter how close you get to the point, there are always points of higher function value and points of lower function value nearby. The graph has the shape of a saddle (locally).

The second derivative test for local extreme values (andrederiverttesten for lokale ekstremverdier) can often tell if a point is a local minimum, a local maximum or a saddle.

When looking for global maxima/minima on closed and bounded (also called compact) domains, we:

  1. List the local maximum (minimum) values in the interior of the domain.
  2. List the local maximum (minimum) values ion the boundary of the domain.
  3. The maximum (minimum) of the values in the two lists is what we want.
2013-03-08, spreeman