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Unformatted text preview: Math 241 Fall 2010 Final Exam Solutions THIS IS A ROUGH DRAFT; IF YOU FIND AN ERROR, TELL ME SO THAT I CAN FIX IT FOR YOUR CLASSMATES! email@example.com 1. There are lots of ways that we could complete this problem, but the fastest is a 3-step process: i) Find a normal vector, n = a i + b j + c k ii) Find a base point ( x ,y ,z ) iii) Plug into the standard point-slope-style formula a ( x- x ) + b ( y- y ) + c ( z- z ) = 0 To find n , well cross two vectors in the plane. We have lots of choices, but two example vectors in the plane are (1- 3) i + (1- (- 1)) j + (0- (- 2)) k which points from the first point to the second point, and (0- 3) i + (1- (- 1)) j + (2- (- 2)) k which points from the first point to the third point. Simplifying and crossing, we get (- 2 i + 2 j + 2 k ) (- 3 i + 2 j + 4 k ) = 4 i + 2 j + 2 k . Taking the second point as our base point (arbitrarily), we have 4( x- 1) + 2( y- 1) + 2( z- 0) = 0 which in slope-intercept-style looks like 2 x + y + z = 3 (either form is fine, or any equation which is equivalent to these). 2. (a) If we start with the unit circle centered at the origin and stretch it out to be twice as wide and thrice as tall, we have the given ellipse. Thus a quick-and-dirty parametrization in stretched polar coordinates is r ( t ) = 2cos( t ) i + 3sin( t ) j defined on the interval 0 t < 2 , for example. This is by no means a unique solution. (b) We have v ( t ) = r ( t ) =- 2sin( t ) i + 3cos( t ) j and a ( t ) = r 00 ( t ) = v ( t ) =- 2cos( t ) i- 3sin( t ) j so that ( t ) = k v a k k v k 3 = k (6sin 2 ( t ) + 6cos 2 ( t )) k k p (4sin 2 ( t ) + 9cos 2 ( t ) 3 = 6 (4 + 5cos 2 ( t )) 3 / 2 . (c) The fraction above is at its maximum when cos 2 ( t ) is as small as possible, that is when cos 2 ( t ) = 0 i.e. t = (2 n +1) 2 for integers n . Conversely, the fraction is at its minimum when the denominator is as big as possible, that is when cos 2 ( t ) = 1, i.e. t = n for integers n ....
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This note was uploaded on 02/27/2012 for the course MATH 2 taught by Professor Staff during the Spring '08 term at Maryland.
- Spring '08