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f1sol-20C-fa2005

# f1sol-20C-fa2005 - Print Name Student Number Section Time...

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Print Name: Student Number: Section Time: Math 20C. Final Exam December 8, 2005 Read each question carefully, and answer each question completely. Show all of your work. No credit will be given for unsupported answers. Write your solutions clearly and legibly. No credit will be given for illegible solutions. 1. (8 Pts.) Find the equation of the plane that contains both the point ( - 1 , 0 , 1) and the line x = t , y = - 1 + 2 t , z = 3 t . The plane is determined by a point in the plane and the normal vector. A point in the plane is P 0 = ( - 1 , 0 , 1). To compute the normal vector n , notice that the equation of the line is given by r ( t ) = h 0 , - 1 , 0 i + h 1 , 2 , 3 i t . Denote v = h 1 , 2 , 3 i , and P 1 = r (0) = (0 , - 1 , 0). Then, ~ P 1 P 0 = h- 1 , 1 , 1 , i . Therefore, n = v × ~ P 1 P 0 = i j k 1 2 3 - 1 1 1 = h (2 - 3) , - (1 + 3) , (1 + 2) i , n = h- 1 , - 4 , 3 i . Then, the equation of the plane is - ( x + 1) - 4( y - 0) + 3( z - 1) = 0 , - x - 1 - 4 y + 3 z - 3 = 0 , x + 4 y - 3 z = - 4 . # Score 1 2 3 4 5 6 7 8 9 10 Σ

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2. (8 Pts.) Find the values of the constants a and b such that the function f ( t, x ) = sin( x - at ) + cos( bx + t ) is solution of the wave equation f tt = 4 f xx . f t = - a cos( x - at ) - sin( bx + t ) , f tt = - a 2 sin( x - at ) - cos( bx + t ) , f x = cos( x - at ) - b sin( bx + t ) , f xx = - sin( x - at ) - b 2 cos( bx + t ) , therefore, 0 = f tt - 4 f xx = [ - a 2 sin( x - at
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