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math10c fall 2009 practice midterm1

# math10c fall 2009 practice midterm1 - Problem 1[8 points...

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Problem 1. [ 8 points. ] Consider the vectors v = i + 2 j , w = 3 i - j . (i) [3 points] Draw the two vectors v and w in standard position, and draw their sum v + w . (ii) [2 points] Compute the vector 2 v + 3 w . (iii) [3 points] Find the unit vector u in the direction of w . Answer: (i) The vector v + w in standard position joins the origin to the vertex of the parallel- ogram spanned by v and w . (ii) We have 2 v + 3 w = 2 ( i + 2 j ) + 3 (3 i - j ) = 11 i + j . (iii) We have u = w || w || = 3 i - j p 3 2 + ( - 1) 2 = 3 10 i - 1 10 j . Problem 2. [10 points.] Consider the function f ( x, y ) = ( x - 1) 2 + ( y - 1) 2 . (i) [5 points] Draw the contour diagram for f ( x, y ) and clearly label the level curves. Show the con- tours for at least three levels. (ii) [5 points] Draw the graph of z = f ( x, y ) . Answer: (i) The contour diagram consists in circles centered at (1 , 1) of radius c where c is the level. For instance, for levels c = 1 , c = 4 , c = 9 , we draw three circles of center (1 , 1) and radii 1 , 2 and 3 respectively. (ii) The graph z = f ( x, y ) is a paraboloid whose lowest point is (1 , 1 , 0) .

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