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.

Posted in M I

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