1698 - 1 MATH 203 Final Exam May 23, 2007 PART I: Answer...

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1 MATH 203 Final Exam May 23, 2007 PART I: Answer all parts of Questions 1-8 (points as indicated). Question 1 (8 points) Given the two lines x - 2 3 = y 4 = z + 1 5 and x + 2 6 = y - 1 8 = z 10 (a) Are they parallel? Explain. (b) Find an equation of the plane containing the two lines. Question 2 (8 points) (a) Find an equation of the tangent plane at (2 , 1 , 3) to the surface z = xy + y . (b) Find the parametric equations of the normal line; that is, the line perpendicular to the tangent plane, passing through (2 , 1 , 3). Question 3 (8 points) Consider the function T ( x,y,z ) = 9 1 + x 2 + y 2 - 3 z . (a) Find the rate of change of T at the point (1 , 1 , 1) in the direction towards the point (11 , 21 , 6). (b) In what direction does T increase most rapidly at the point (1 , 1 , 1)? (c) At what rate is T increasing in the direction given by the answer to part (b)? Question 4 (8 points) Let f ( x,y,z ) = x yz 2 . Use linear approximation (differentials) to approximate f ( . 98 , 1
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1698 - 1 MATH 203 Final Exam May 23, 2007 PART I: Answer...

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