Lect28 - Optimization Which point on the line y = 1 - 2 x...

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92.131 Lecture 28 1 of 13 Ronald Brent © 2010 All rights reserved. x -2 -1 1 2 -2 -1 0 1 2 y Optimization Which point on the line x y 2 1 = is closest to the origin?
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92.131 Lecture 28 2 of 13 Ronald Brent © 2010 All rights reserved. Recall the distance between a point ( x , y ) and (0, 0) is 2 2 y x d + = . If we require the point to be on the line, then the distance is 1 4 5 ) 2 1 ( ) ( 2 2 2 + = + = x x x x x d To find the absolute minimum of this function we first find critical points 1 4 5 2 5 ) ( 2 + = x x x x d which equals 0 at 5 2 = x . The point is 5 1 , 5 2 . And the distance is 5 1 ) 5 / 2 ( = d .
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92.131 Lecture 28 3 of 13 Ronald Brent © 2010 All rights reserved. The method for solving optimization problems is as follows*: 1. Understand the Problem. What is being asked? Can you envision the problem? What is the numerical information being given? What are the units? 2. Build a Mathematical Model This will involve drawing pictures, labeling variables and using either algebraic, geometric, or trigonometric laws to write down equations relating the variables. a. One equation will be the function you wish to maximize or minimize. This is called the objective function . It will depend upon one or more variables. b. One equation will relate the variables in the problem. This is called the constraint equation . c. Using the constraint and the objective, many times the objective function can be reduced to a function of only one variable.
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92.131 Lecture 28 4 of 13 Ronald Brent © 2010 All rights reserved. 3. What is the domain of the objective function? This needs to be determined from all the given data and the constraints. 4.
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This note was uploaded on 02/13/2012 for the course MATH 92.131 taught by Professor Staff during the Fall '09 term at UMass Lowell.

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Lect28 - Optimization Which point on the line y = 1 - 2 x...

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