Trabajo colaborativo calculo 3 Spira Mirabilis

Spira Mirabilis

La  espiral  logaritmica,  llamada  la  spira  mirabilis  o  eadem  mutata  resugno  es  una  curva  param´etrica  de  la forma

c(t) = (aebtcos(t), aebtsen(t))

Desarrollo

1. Muestre que la magnitud de la curva, ||c(t)|| es ||c(t)|| = aebt

||c(t)|| = .(aebtcos(t))2 + (aebtsen(t))2[pic 1]

||c(t)|| = .a2e2btcos2(t) + a2e2btsen2(t)[pic 2]

||c(t)|| = .a2e2bt(cos2(t) + sen2(t))[pic 3]

||c(t)|| = .a2e2bt(cos2(t) + sen2(t))[pic 4][pic 5][pic 6]

||c(t)|| = √a2e2bt[pic 7]

[pic 8]

1. Muestre que el vector tangente a la curva es

cj(t) = (aebt(bcos(t) − sen(t)))i + (aebt(bsen(t) + cos(t)))j

cj(t) =

d(aebtcos(t))

i +[pic 9]

dt

d(aebtsen(t))

j[pic 10]

dt

cj(t) = (abebtcos(t) − aebtsen(t))i + (abebtsen(t) + aebtcos(t))j

[pic 11]

1. Muestre que la rapidez de la curva esta dada por la expresi´on s(t) = aebt√b2 + 1[pic 12]

s(t) = ||cj(t)||

s(t) = .(aebt(bcos(t) − sen(t)))2 + (aebt(bsen(t) + cos(t)))2[pic 13]

s(t) = .a2e2bt(bcos(t) − sen(t))2 + (bsen(t) + cos(t))2[pic 14]

s(t) = aebt√b2cos2(t) − 2bcos(t)sen(t) + sen2(t) + b2sen2(t) + 2bsen(t)cos(t) + cos2(t) s(t) = aebt.b2(cos2(t) + sen2(t)) −2bcos(t)sen(t) + 2bsen(t)cos(t) + cos2(t) + sen2(t)[pic 15][pic 16][pic 17][pic 18][pic 19]

s        ˛1¸        x

s        ˛0¸        x

s(t) = aebt√b2 + 1[pic 20]

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