lecture09

lecture09 - MATH 100 Lecture 9 Level Curves Proposition for...

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2006 Fall MATH 100 Lecture 9 1 MATH 100 Lecture 9 Level Curves () ( ) () () ( ) () ( ) () ( ) () () 0 ' , ' , ' , ' , , then , , curve level the zed parameteri : Proof , at curve level of ctor tangent ve , , , curve level for the : n Propositio 0 0 0 0 0 0 = = + = = = = = t y t x y x f t y y x f t x y x f C t y t x f t y y t x x y x y x f y x f C y x f y x
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2006 Fall MATH 100 Lecture 9 2 MATH 100 Lecture 9 Level Curves () () () 4 3, at ector gradient v and 4 3, through curve level find , , : Ex n explanatio Geometric 2 2 y x y x f + = 22 2 2 Sol: + = 3 4 25 , 2 2 3,4 6 8 xy f x i y j f ij += ∇= + + G G G G
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2006 Fall MATH 100 Lecture 9 3 MATH 100 Lecture 9 Level Curves () ( ) 22 Ex: , 2000 2 4 ,at 20,5,1000 , whether the function is increasing along 1 west 1,0 or northeast 1,1 ? 2 fx y x y =− = ( ) ()( ) ( ) ) ( ) Sol: 4 , 8 20,5 4 20 , 8 5 1,0 80, 40 1,0 80 0 decreasing 1 4 20 , 8 5 2 w ne , y x y Df ∇= −= −⋅ = < 1,1 40 increasing 2 =>
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2006 Fall MATH 100 Lecture 9 4 MATH 100 Lecture 9 Level Curves Along what direction there is no decreasing or increasing? () 2 , 1 or 2 , 1 then , 2 or 2 take 2 1 0 40 80 0 5 , 20 , Let : Ans 1 2 1 2 1 2 1 = = = = = = u u u u u u u f u u u G G G
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lecture09 - MATH 100 Lecture 9 Level Curves Proposition for...

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