Spring 2007 - Linshaw's Class - Quiz 2

Spring 2007 - Linshaw's Class - Quiz 2 - , where C 1 is a...

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Name: PID: TA: Sec. No: Sec. Time: Math 20C. Quiz 2 April 20, 2007 1. Find parametric equations for the tangent line to the curve given by r ( t ) = t 5 i + t 4 j + t 3 k , at the point (1 , 1 , 1). Solution: Note that r (1) = (1 , 1 , 1). We have r ( t ) = 5 t 4 i + 4 t 3 j + 3 t 2 k , so r (1) = 5 i + 4 j + 3 k . The tangent line passes through (1 , 1 , 1) and is parallel to the vector 5 i + 4 j + 3 k . We can parametrize this line as follows: x = 1 + 5 t, y = 1 + 4 t, x = 1 + 3 t. 2. Find the position function r ( t ) of a particle whose acceleration is a ( t ) = t i + t 2 j + cos 2 t k , which satis±es the initial conditions v (0) = i + k , and r (0) = j . Solution: Integrating a ( t ) yields v ( t ) = t 2 2 i + t 3 3 j + 1 2 sin 2 t k + C 1
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Unformatted text preview: , where C 1 is a constant vector. Using the rst initial condition yields v (0) = C 1 = i + k , so v ( t ) = ( t 2 2 + 1) i + t 3 3 j + ( 1 2 sin 2 t + 1) k . Integrating v ( t ) yields r ( t ) = ( t 3 6 + t ) i + t 4 12 j + (-1 4 cos 2 t + t ) k + C 2 , where C 2 is another constant vector. Using the second initial condition, we have r (0) =-1 4 k + C 2 = j , so C 2 = j + 1 4 k . Finally, we obtain r ( t ) = ( t 3 6 + t ) i + ( t 4 12 + 1) j + (-1 4 cos 2 t + t + 1 4 ) k ....
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