Find the curvature K of the curve at the point r(t)=e^tcos tcdot i+e^tsin tcdot j+e^tk,P(1,0,1)

illusiia 2020-12-28 Answered
Find the curvature K of the curve at the point r(t)=etcosti+etsintj+etk,P(1,0,1)
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Bella
Answered 2020-12-29 Author has 81 answers
r(t)=etcosti+etsintj+etk
Differentiate
r(t)=et(costsint)i+et(sint+cost)j+etk
Find Magnitude
|r(t)|=et(costsint)2+(sint+cost)2+1
|r(t)|=et(cos2tsin2t2costsint)2+(sin2t+cos2t+2costsint)2+1
|r(t)|=et3
Recall that:T(t)=r(t)|r(t)|
ThereforeT(t)=(et(costsint)i+et(sint+cost)j+etk)(et3)
T(t)=13[(costsint)i+(sint+cost)j+k]
DifferentiateT(t)=13[(sintcost)i+(costsint)j+0k]
Find Magnitude|T(t)|=13(sintcost)2+(costsint)2+0)
|T(t)|=13(sin2t+cos2t+2sintcost)+(cos2t+sin2t2costsint)+0)
|T(t)|=23
Recall thatk(t)=23(et3)=2(3et)
Result
k(t)=2(3et)
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