math 105 Prelim1 +key 2007

# Math 105 Prelim1 - Prelim 1 Solutions Problem 1(Total 10 points Give the least square line that has the best fit to the following data points(0

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Prelim 1 Solutions Problem 1 (Total 10 points) Give the least square line that has the best ﬁt to the following data points: (0 , 2), (1 , - 2), (2 , - 2) and (3 , 2). Hint: Recall that the slope and y -intercept of the best ﬁt line are given by the formulas nb + ± n X i =1 x i ! m = n X i =1 y i and ± n X i =1 x i ! b + ± n X i =1 x 2 i ! m = n X i =1 x i y i . Solution: We begin by computing the sums required for the equations given in the hint: n X i =1 x i = 6 n X i =1 y i = 0 n X i =1 x 2 i = 14 n X i =1 x i y i = 0 Plugging these values into the equations gives (4) b + (6) m = 0 and (6) b + (14) m = 0 Solving these equations by any method yields the unique solutions m = 0 and b = 0. Therefore the best ﬁt line is y = 0, a horizontal line through the origin. Problem 2 (Total 16 points) Draw Venn diagrams that satisfy the following requierments. Let A U , B U , and C U and (a) ( 5 points) A B 6 = , C B and C A 6 = . (b) ( 5 points) A B 6 = and C A B. (c) ( 6 points) A B , C B 6 = , and C A 0 6 = . Solution: (Other satisfactory solutions exist. ..) 1

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Problem 3 (Total 16 points) (1) Give the equations for the lines that go through the following pairs of points. Write all equations in point slope form. (a) ( 3 points) ( - 1 , 3) and (2 , 4). (b) ( 3 points) ( - 6 , - 2) and (9 , 3). (c) ( 3 points) (1 , 2) and ( - 4 , 2). (2)
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## This note was uploaded on 04/21/2008 for the course MATH 1105 taught by Professor Hurtado during the Fall '07 term at Cornell University (Engineering School).

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Math 105 Prelim1 - Prelim 1 Solutions Problem 1(Total 10 points Give the least square line that has the best fit to the following data points(0

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