Errors and Approximations, Maxima and Minima, Lagrange’s Method

  • Errors and Approximations

If {f} is a function of {x,y,z}, then the error in {f} is

{df = \frac {\partial f}{\partial x} dx + \frac {\partial f}{\partial y} dy + \frac {\partial f}{\partial z} dz}

However, this is an approximation.

Note: If necessary, we take log on both sides.

  • Maxima and Minima (Functions of Several Variables)

We have studied the process of finding extreme values of {y=f(x)}. Now, we will see the same for {f = f(x,y)}.

First, we equate {\frac {\partial f}{\partial x}} and {\frac {\partial f}{\partial x}} to {0}. We get those pairs {(x,y)}, which can either be maxima, minima or saddle points.

We then find {f_{xx} = r}, {f_{xy} = s} and {f_{yy}=t} for these pairs.

I) {rt > s^2}, {r < 0} – Gives maximum value

II) {rt > s^2}, {r > 0} – Gives minimum value

III) {rt < s^2}Saddle point

IV) {rt = s^2} – No conclusion possible

  • Lagrange’s Method of Undetermined Multipliers

This method is used when the maximum and minimum of a function are to be obtained under certain constraints (like the optimization problems). We form a linear relation between the function and the constraint using a Lagrange multiplier and differentiate it partially w.r.t. the variables. We then eliminate the variables and parameter and get an equation. The roots of that equation are the extreme values.

Advertisements
Posted in M I

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s