Linearization of function of 2 variables: Matlab example (symbolic toolbox), Taylor series revision

Antonio Sala, UPV

Difficulty: ** ,       Relevance: ,      Duration: 16:57

*Enlace a Spanish version

Summary:

This video develops a Matlab example of the linearization of a function of 2 variables $f(x_1,x_2)=x_1^2∕2+x_2^2+\cos <!--FUNCTION\; APPLICATION-->(1.1x_1+e^{-x_2})+0.25$ at an (arbitrarily chosen) operating point.

You can consult material dedicated to the theory in the video [lint3], with a “geometric” (tangent subspace) approach, that you may wish to watch alongside this video, to complement the Taylor series approach that we discuss here.

The function is first defined and represented; later on, a theoretical review of the multivariable Taylor series and the Jacobian and Hessian definitions is made, and it is applied to the actual example under study, to linearize it using the jacobian command.

The final part of the video superimposes the (3D) graphics of the original function and its linearized approximation, verifying that it is indeed the tangent plane in absolute coordinates.

The optional video [linmiso2EN] will evaluate the accuracy of the approximation at a specific numeric point away from the tangency point, and also plot the approximation error (with a contour plot), discussing the relationship with the Hessian.

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.