a curve is described by r(t) = <e^(t) sint , e^(t) cost , 0> Its graph over the

Question

a curve is described by r(t) = <e^(t) sint , e^(t) cost , 0>

Its graph over the interval [0]<_ t <_ [(3pi) / 2] with pts spiraling out from (0,1) plotted for t valuess of 0 , (pi)/3 , (2pi)/3 , (pi) , (4pi)/3

1) find the arclength function s(t) where we measure the curve starting from (0,1) at t=0. use the arclength function to find the length of the curve

2) solve the arclength function for "t" in terms of "s" and use this formula to parametrize "r" in terms of the arclength "s" ;ie find r(s)

**please help explain and show work on how to get to answers, to understand****

Explanation / Answer

first of all, notice e^(-t) appears in every coordinate. if we have u = e^(-t), then what we have is:

r(t) = (ucos(t), usin(t),u)

now if u was constant, this would just be a circle. and if u was t, this would be an ever-expanding helix.

but u = e^(-t), which starts out as 1 (for t = 0), and then gradually decays asymptotically to 0.

so our helix, instead of widening, shrinks from a radius of 1 down to nothing, and eventually tries to converge to a single point (think of something like a tornado).

as t approaches infinity, r(t) ---> (0,0,0).

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