Engineering Calculus Notes 338

Engineering Calculus Notes 338 - x which is the same as...

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326 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION The idea is this: suppose the function f ( −→ x ) when restricted to the level set L ( g,c ) has a local maximum at −→ x 0 : this means that, while it might be possible to Fnd nearby points where the function takes values higher than f ( −→ x 0 ), they cannot lie on the level set. Thus, we are interested in Fnding those points for which the function has a local maximum along any curve through the point which lies in the level set . Suppose that −→ p ( t ) is such a curve; that is, we are assuming that g ( −→ p ( t )) = c for all t , and that −→ p (0) = −→ x 0 . in order for f ( −→ p ( t )) to have a local maximum at t = 0, the derivative must vanish—that is, 0 = d dt v v v v t =0 [ f ( −→ p ( t ))] = −→ f ( −→ x 0 ) · −→ v where −→ v = ˙ −→ p (0) is the velocity vector of the curve as it passes −→ x 0 : the velocity must be perpendicular to the gradient of f . This must be true for any curve in the level set as it passes through −→
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Unformatted text preview: x , which is the same as saying that it must be true for any vector in the plane tangent to the level set L ( g,c ) at −→ x : in other words, −→ ∇ f ( −→ x ) must be normal to this tangent plane. But we already know that the gradient of g is normal to this tangent plane; thus the two gradient vectors must point along the same line—they must be linearly dependent! This proves (see ±igure 3.24 ) Proposition 3.6.11 (Lagrange Multipliers) . If −→ x is a local extreme point of the restriction of the function f ( −→ x ) to the level set L ( g,c ) of the function g ( −→ x ) , and c is a regular value of g . Then −→ ∇ f ( −→ x ) and −→ ∇ g ( −→ x ) must be linearly dependent: −→ ∇ f ( −→ x ) = λ −→ ∇ g ( −→ x ) (3.27) for some real number λ ....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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