1211a5 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 1st Semester: Assignment 5 Due date: 2010 November 19 (Friday), 5:00 pm. 1. Suppose that you are given n pairs of data ( x 1 ,y 1 ) , ( x 2 ,y 2 ) ,..., ( x n ,y n ) and you seek to fit a function of the form y = a x + b to these data. (a) Use the method of least squares as outlined in § 4.4 to construct a function D ( a,b ) that gives the sum of the squares of the distances between observed and predicted y-values of the data. (b) Show that the “best fit” curve of the form y = a x + b should have a = n ∑ y i /x i- ( ∑ 1 /x i )( ∑ y i ) n ∑ 1 /x 2 i- ( ∑ 1 /x i ) 2 , and b = ( ∑ 1 /x 2 i )( ∑ y i )- ( ∑ 1 /x i )( ∑ y i /x i ) n ∑ 1 /x 2 i- ( ∑ 1 /x i ) 2 . (All sums are from i = 1 to n .) (c) Find the curve of the form y = a x + b that best fits the following data: (1 , 0) , (2 ,- 1) , ( 1 2 , 1 ) , and ( 3 ,- 1 2 ) ....
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