1211s5 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 1st Semester Solution to Assignment 5 1(a We

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 1st Semester: Solution to Assignment 5 1. (a) We define D ( a,b ) = ∑ n i =1 ‡ y i- ( a x i + b ) · 2 (b) The required system of equations is ∂D ∂a = ∂D ∂b = 0, which is    ∑ 2 ‡ y i- ( a x i + b ) ·‡- 1 x i · = 0 ∑ 2 ‡ y i- ( a x i + b ) · (- 1) = 0 = ⇒    ‡ ∑ 1 x i 2 · a + ‡ ∑ 1 x i · b = ∑ y i x i ‡ ∑ 1 x i · a + nb = ∑ y i = ⇒ ( a = n ∑ y i /x i- ( ∑ 1 /x i )( ∑ y i ) n ( ∑ 1 /x i 2 )- ( ∑ 1 /x i ) 2 b = ( ∑ 1 /x i 2 )( ∑ y i )- ( ∑ 1 /x i )( ∑ y i /x i ) n ( ∑ 1 /x i 2 )- ( ∑ 1 /x i ) 2 Also, D aa = 2 ∑ 1 /x i 2 ,D ab = D ba = 2 ∑ 1 /x i ,D bb = 2 n . At critical point ( a ,b ) ,D aa = 2 ∑ 1 /x i 2 > 0, and D aa D bb- D ab D ba = 4 ‡ n ∑ 1 /x i 2- ( ∑ 1 /x i ) 2 · = 2 ∑ ≤ i ≤ m ≤ j ≤ n (1 /x i- 1 /x j ) 2 > 0 hence D ( a,b ) has a local minimum at ( a ,b ). Also, D ( a,b ) is a paraboloid function, then we obtain that ( a ,b ) is a global minimum point for D ( a,b )....
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This note was uploaded on 05/04/2011 for the course MATH 1211 taught by Professor Wang during the Spring '11 term at HKU.

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1211s5 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 1st Semester Solution to Assignment 5 1(a We

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