1 / 4

Implicit Functions

Implicit Functions. The equation y = mx + b is an explicit function y is considered the dependent variable x is the independent variable The equation f( x , y , m , b ) = 0 is an implicit function

liz
Download Presentation

Implicit Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Implicit Functions • The equation y = mx + b is an explicit function • y is considered the dependent variable • x is the independent variable • The equation f(x,y,m,b) = 0 is an implicit function • The relationships between the variables and the parameters are implicitly present in the equation but not explicitly stated

  2. Derivatives from Implicit Functions • In many circumstances, it will be helpful to compute derivatives directly from implicit functions • If f(x,y)=0, then its total differential of f1dx + f2dy = 0 • Thus,

  3. Production Possibility Frontier • Earlier example: 2x2 + y2 = 225 • Can be rewritten: f(x,y) = 2x2 + y2 - 225 = 0 • Then, the opportunity cost trade-off between x and y is Because fx = 4x and fy = 2y

  4. Implicit Function Theorem • It may not always be possible to solve implicit functions of the form g(x,y)=0 for unique explicit functions of the form y = f(x) • Mathematicians have derived the necessary conditions • In many economic applications, these conditions are the same as the second-order conditions for a maximum (or minimum)

More Related