Spring 2007 - Hall's Class - Practice Exam 1

# Spring 2007 - Hall's Class - Practice Exam 1 - Math 20C...

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Unformatted text preview: Math 20C Practice Midterm I Solutions 1. (a) (5 points) Find the center and radius of the sphere x 2 + 2 x + y 2- 4 y + z 2 = 8 . We need to rewrite this equation in the form ( x- x ) 2 + ( y- y ) 2 + ( z- z ) 2 = r 2 , a sphere of radius r and center ( x ,y ,z ). We do this by completing the square of x 2 + 2 x and y 2- 4 y : x 2 + 2 x = x 2 + 2 x + 1- 1 = ( x + 1) 2- 1 y 2- 4 y = y 2- 4 y + 4- 4 = ( y- 2) 2- 4 Substituting these into the original equation, we get: ( x + 1) 2- 1 + ( y- 2) 2- 4 + z 2 = 8 , i.e., ( x + 1) 2 + ( y- 2) 2 + z 2 = 13 . Thus our sphere has radius √ 13 and center (-1, 2, 0). (b) (5 points) Find and describe the intersection of the sphere with the xz-plane. The xz-plane consists of all points with y-coordinate 0. Setting y = 0 gives ( x + 1) 2 + (0- 2) 2 + z 2 = 13 = ⇒ ( x + 1) 2 + z 2 = 9 . This is a circle in the xz-plane of radius 3 and center (-1, 0, 0). (c) (10 points) Write a parametrization of the form x = f ( t ) ,y = g ( t ) ,z = h ( t ) ,a ≤ t ≤ b for the curve in part (b). Let x =- 1 + 3cos( t ) ,y = 0 ,z = 3sin( t ) ,a = 0 ,b = 2 π. 1 Then ( x + 1) 2 + z 2 = (3cos( t )) 2 + (3sin( t )) 2 = 9(sin( t ) 2 + cos( t ) 2 ) = 9 Thus our answer checks out....
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Spring 2007 - Hall's Class - Practice Exam 1 - Math 20C...

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