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f2sol-20C-sp2006

# f2sol-20C-sp2006 - Print Name TA Name Math 20C Final Exam...

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Print Name: Section Number: TA Name: Section Time: Math 20C Final Exam. June 15, 2006 No calculators or any other devices are allowed on this exam. Write your solutions clearly and legibly; no credit will be given for illegible solutions. Read each question carefully. If any question is not clear, ask for clarification. Answer each question completely, and show all your work. 1. (10 points) Find the plane through the point P 0 = (2 , - 1 , 1) which is perpendicular to the planes 2 x - y - z = 3 and x + 2 y + z = 2. The plane is determined by its normal vector n and a point. We choose the point to be P 0 = (2 , - 1 , 1). The normal vector can be computed as n = n 1 × n 2 , n 1 = h 2 , - 1 , - 1 i , n 1 = h 1 , 2 , 1 i . where n 1 and n 2 are the normal vectors to the planes 2 x - y - z = 3 and x + 2 y + z = 2, respectively. Then, n = i j k 2 - 1 - 1 1 2 1 = h ( - 1 + 2) , - (2 + 1) , (4 + 1) i n = h 1 , - 3 , 5 i . Then, the equation of the plane is ( x - 2) - 3( y + 1) + 5( z - 1) = 0 x - 3 y + 5 z = 10 .

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2. (8 points) Decide whether the lim ( x,y ) (0 , 0) y 4 - x 2 y 4 + x 2 exists. Give reasons your answer. Consider the path given by the line x = 0, then lim (0 ,y ) (0 , 0) y 4 - x 2 y 4 + x 2 = lim y 0 y 4 + y 4 = lim y 0 1 = 1 .
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