week04 - Math 23 Sections 110-113 B Dodson Week 4 Homework...

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Math 23Sections 110-113B. DodsonWeek 4 Homework:13.1, 13.2 vector functions, derivatives13.3 arc length, curvature13.4 velocity, accelerationProblem 13.2.9:Find the derivative of the vector functionr(t) =< t2,1-t,t >.Solution:We just take the derivative of the components,r(t) =<(t2),(1-t),(t)>=<2t,-1,12t>,where (t12) =12t-12.Week 4 Homework:13.3 arc length,curvature (1st Wed)13.4 velocity, accelerationProblem 13.2.17:Ifr(t) =<6t5,4t3,2t >,find the unit tangent vectorTwhent= 1.
2.Solution:First,r(t) =<30t4,12t2,2>= 2<15t3,6t2,1>,andr(1) = 2<15,6,1>,so the length|r(1)|= 2225 + 36 + 1 = 2252,soT(1) =1252<15,6,1> .Note that findingT(t) first, beforesettingt= 1,makes the problem harder; whilesettingt= 1 inr(t) before differentiatingchanges the derivative to 0 : the abovesteps are in exactly the correct order to get theright answer with least computation.Week 4 Homework:13.3 curvature (2nd Wed)Problem 13.3.16Use formula (9) to find the curvature ofr(t) =< t2,2t,lnt >.

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