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Exam2.YellowSol

# Exam2.YellowSol - yellow MATH 32A EXAM 2 LAST NAME FIRST...

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yellow MATH 32A EXAM 2 NOVEMBER 13, 2009 LAST NAME FIRST NAME ID NO. WRITE CLEARLY AND LEGIBLY. TO RECEIVE CREDIT, YOU MUST EXPLAIN YOUR WORK AND CIRCLE YOUR ANSWERS . PLEASE DO NOT WRITE BELOW THIS LINE SCORES 1( 2 0p o i n t s ) 4( 2 o i n t s ) 2( 2 o i n t s ) 5( 2 o i n t s ) 3( 2 o i n t s ) 6( 2 o i n t s ) TOTAL

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2 PROBLEM 1 Find the curvature κ (9) for the parametric curve (called a clothoid): x ( t )= Z t 0 sin u 3 3 du, y ( t Z t 0 cos u 3 3 du 0.5 1 Clothoid Curve Solution: We have x 0 ( t )=s in u 3 3 x 0 ( t u 2 cos u 3 3 y 0 ( t )=cos u 3 3 y 0 ( t u 2 sin u 3 3 The curvature is κ ( t | x 0 ( t ) y 0 ( t ) x 0 ( t ) y 0 ( t ) | ( x 0 ( t ) 2 + y 0 ( t ) 2 ) 3 2 x 0 ( t ) y 0 ( t ) x 0 ( t ) y 0 ( t u 3 3 ± u 2 sin u 3 3 ² cos u 3 3 ± u 2 cos u 3 3 ² = u 2 ± cos 2 u 3 3 +sin 2 u 3 3 ² = u 2 ( x 0 ( t ) 2 + y 0 ( t ) 2 ) 3 2 =(s 2 u 3 3 +cos 2 u 3 3 ) 3 2 =1 Therefore κ ( t |− u 2 | / 1= u 2 and κ (9) = 81.
3 PROBLEM 2 Consider the helix r ( t )= h cos t, sin t, t i . (A) Find the vectors T and N at r ( π 2 ). (B) A runner runs along the helix. When he passes the point r ( π 2 ), his speed is 4 m/s and he is accelerating at a rate of 1 3 m/s 2 . Find his acceleration vector a at this moment. Note: the runner’s acceleration

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Exam2.YellowSol - yellow MATH 32A EXAM 2 LAST NAME FIRST...

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