m2b-sol-20C-sp2006

# m2b-sol-20C-sp2006 - Print Name TA Name Math 20C Midterm...

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Print Name: Section Number: TA Name: Section Time: Math 20C Midterm Exam 2. May 26, 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. (a) (5 points) Find and sketch the domain of the function f ( x, t ) = ln(2 x + 3 t ). (b) (5 points) Find all possible constants c such that the function f ( x, t ) above is solution of the wave equation, f tt - c 2 f xx = 0. (a) The argument in the ln function must be positive. then, the domain is D = ( x, t ) IR 2 : 2 x + 3 t > 0 . x -2 3 t t = -(2/3) x (b) f t = 3 2 x + 3 t , f x = 2 2 x + 3 t , f tt = - 9 (2 x + 3 t ) 2 , f xx = - 4 (2 x + 3 t ) 2 , 0 = f tt - cf xx = - 9 (2 x + 3 t ) 2 + c 2 4 (2 x + 3 t ) 2 = 1 (2 x + 3 t ) 2 ( - 9 + 4 c 2 ) 4 c 2 = 9 , c = ± 3 2 .

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2. (a) (5 points) Find the direction in which f ( x, y ) increases the most rapidly, and the directions in which f ( x, y ) decreases the most rapidly at P 0 , and also find the
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