lecture13-09

lecture13-09 - Math 18.02(Spring 2009 Lecture 13 Lagrange...

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Unformatted text preview: Math 18.02 (Spring 2009): Lecture 13 Lagrange Multipliers March 5 Last time: Vector fields, Gradient, directional derivatives. Today: Tangent plane to (level) surfaces. Lagrange multipliers. Reading Material: From Simmons : 19.5 and 19.8. 2 Recall: Gradient and level curves and surfaces Given a function z = f ( x, y ) the ~ O f satisfies remarkable proprieties: • (1) The direction of ~ O f is the direction of the max increase of f • (2) For any P = ( x , y ) the ~ O f ( x , y ) ⊥ the level curve of f ( x, y ) through P . • (3) ~ O f points ”uphill” • (4) | ~ O f | measures the ”steepness” of the landscape. Remark. One should remark that proprieties (1) and (2) above are also valid for functions of three variables w = f ( x, y, z ) . In particular (2) reads as follow • (2)’ For any P = ( x , y , z ) the ~ O f ( x , y , z ) ⊥ to the level surface of f ( x, y, z ) through P . Propriety (2) above is particularly useful in order to find tangent lines along level curves and (2)’ to find tangent planes along level surfaces. We will use (2)’ in the next example: 1 Exercise 1. Consider the ellipsoid 4 = 2 x 2 + y 2 + z 2 . Compute the tangent plane to this ellipsoid at the point P = (1 , 1 , 1) . Solution: Remark. We already know how to compute the tangent plane to the graph of a function z = f ( x, y ) at a point P = ( x , y , z ) , where z = f ( x , y ) : We will now find the same equation by using (2)’. We first observe that if we define the func- tion g ( x, y, z ) = f ( x, y )- z , then the graph of the function f is also the level surface S = { ( x, y, x ) /g ( x, y, z ) = 0 } and the point P belongs to this level surface. By (2)’ we know that ~ O g ( P ) is a normal vector to the tangent plane we are looking for. We have ~ O g = f x ˆ i + f y ˆ j- ˆ k hence the equation of the plane is f x ( x , y )( x- x ) + f y ( x , y )( y- y )- ( z- f ( x , y )) = 0 , which coincide with the equation of the tangent plane that we already know....
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This note was uploaded on 03/08/2010 for the course MATH 18.022 taught by Professor Hartleyrogers during the Spring '06 term at MIT.

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lecture13-09 - Math 18.02(Spring 2009 Lecture 13 Lagrange...

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