This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 18.02 (Spring 2009): Lecture 10 Second derivative test. Global max/min: Boundaries and infinity. Level curves and lever surfaces February 26 Last time: Max/Min problems. Least squares approximation. Today: Second derivative test. Global max/min: Boundaries and infinity. Level curves and level surfaces. Reading Material: From Simmos 19.1, 19.7. From the Lecture Notes SD. 2 Second derivatives Given a function z = f ( x,y ) we denoted the first partial derivatives of f as f x = ∂f ∂x and f y = ∂f ∂y . Then it is simple to define the second partial derivatives: f xx = 2 nd derivative w.r.t. x = ∂ ∂f ∂x ∂x = ∂ 2 f ∂x 2 f xy = derivative w.r.t. y of derivative w.r.t. x = ( f x ) y = ∂ ∂f ∂x ∂y = ∂ 2 f ∂y dx f yy = 2 nd derivative w.r.t. y = ∂ ∂f ∂y ∂y = ∂ 2 f ∂y 2 . Remark. There is a very surprising theorem: if all second derivatives are continuous then f xy = f yx , and this in fact is going to be the case for all functions we will be dealing with in this class! 1 3 2 nd derivative test Last time we learned that for a function z = f ( x,y ) local max/min occur on critical pints , that is on points ( x ,y ) such that f x ( x ,y ) = 0 f y ( x ,y ) = 0 We also learned that not all critical points are either a local max or min, take for example the point (0 , 0) critical point for the function z = x 2 y 2 which is a saddle point. Question: Is there a way to recognize if a critical point is a max a min or a saddle? The answer to this question is a “partial” YES . We first have to define the Discriminant D : Definition 1. Given a function z = f ( x,y ) we define D = f xx f yy ( f xy ) 2 = f xx f xy f yx f yy note f xy = f yx ....
View
Full
Document
This note was uploaded on 03/01/2009 for the course 18 18.02 taught by Professor Auroux during the Spring '08 term at MIT.
 Spring '08
 Auroux
 The Land

Click to edit the document details