MT2 w soln

# MT2 w soln - Midterm 2 The exam starts at 4:10pm and lasts...

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Midterm 2 The exam starts at 4:10pm and lasts for 50 minutes. You turn in your work exactly at 5pm. You cannot use calculators, any books or other supplementary materials. However, you are allowed to have a formula sheet the size of which is a quarter of a regular sheet of paper. You can write on both sides of it. If you need to ask a question, please raise your hand and I will address your question. If cheating is noticed draconian measures will be taken. Please write legibly. Sketches should be done in a clear and neat way. The main features of graphs should be speciFed. Your explanations should be brief and clear. Please don’t forget to write your name above each of the problems. Good luck! 1

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Your Name Problem 1 The surface S is given in Cartesian coordinates by x 3 + y 3 + z 3 + 1 = ( x + y + z + 1) 3 . Find the equation of the tangent plane at the point (2 , - 2 , - 1). Solution Let F ( x, y, z ) = x 3 + y 3 + z 3 - ( x + y + z +1) 3 +1 then the surface S is given by F ( x, y, z ) = 0. We have at the point (2 , - 2 , - 1) F x = 3 x 2 - 3( x + y + z + 1) 2 = 12 , F y = 3 y 2 - 3( x + y + z + 1) 2 = 12 , F z = 3 z 2 - 3( x + y + z + 1) 2 = 3 . The equation of the tangent plane at the point (2
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## This note was uploaded on 03/09/2010 for the course PHYSICS 7A taught by Professor Lanzara during the Fall '08 term at Berkeley.

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MT2 w soln - Midterm 2 The exam starts at 4:10pm and lasts...

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