c3-t2-a

# Obtain an arc length parameterization for the curve r

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Obtain an arc-length parameterization for the curve r ( t )=<t , 3cos(2t), 3sin(2t) > in terms of the initial point (0, 3, 0) which is the terminal point of r (0) in standard position. Rather than overloading the symbol r , write this new parameterization as R (s). How are R and r related? First, since r (0) = <0, 3, 0>, the (signed) distance along the curve from the point given by r (0) on the graph to that given by r (t) is . s ϕ ( t ) t 0 r ( u ) du t 0 <1, 6 sin(2 u ),6cos(2 u )> du t 0 37 1/2 du 37 1/2 ( t 0) 37 1/2 t . Solving for t in terms of s yields t 37 1/2 s so that ϕ 1 ( t ) 37 1/2 t . Thus, R ( s ) r ( ϕ 1 ( s )) r (37 1/2 s ) < s 37 1/2 , 3cos 2 s 37 1/2 , 3sin 2 s 37 1/2 >. Evidently, R (s) = r ( ϕ -1 (s)), or equivalently, r (t) = R ( ϕ (t)). ______________________________________________________________________ 4. (10 pts.) A particle moves smoothly in such a way that at a particular t i m et=0 ,w e have v ( 0 )=<1,2> a n d a ( 0 )=<3,0> . I fw e write a (0) in terms of T (0) and N (0), then a ( 0 )=a T (0) T ( 0 )+a N (0) N (0), where (a) T (0)

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TEST2/MAC2313 Page 3 of 5 ______________________________________________________________________ 5. (10 pts.) (a) Find the limit. lim t 1 3 t 2 , ln( t ) t 2 1 ,sin( π 2 t ) 3, 1 2 ,1 The only silliness is the middle limit. Either recognize a loggy derivative limit as part of the mess or use l’Hopital’s Rule there. (b) Find parametric equations for the line tangent to the graph of r ( t )=( 2-l n ( t )) i + t 2 j at the point where t 0 =1 . Since r ( t )=<- t -1 ,2 t >, r ( 1 )=<2- ln(1),1>=<2,1> ,a
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Obtain an arc length parameterization for the curve r t t...

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