Lista de exercícios 26 02

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UNIVERSIDADE FEDERAL DO CARIRI
CURSO: Engenharia Civil
Lista de exerc´ıcios
Professor: Valdineˆs Leite ∗
February 26, 2018
1 Func¸o˜es Vetoriais: Curvas
1. Desenhe a imagem.
(a) f(t) = (sin t, t)
(b) f(t) = (t, 2− t, 2t)
(c) f(t) = (1, cos t, sin t)
(d) f(t) = (1, t)
(e) f(t) = (t, t+ 1)
(f) f(t) = (2t− 1, t+ 2)
(g) f(t) = (t, t3)
(h) f(t) = (t2, t)
(i) f(t) = (cos t, 2 sin t)
(j) f(t) = (sin t, sin2 t)
(k) f(t) = (sin t, sin t)
(l) f(t) = (
√
2 cos t, 2 sin t)
(m) f(t) = (sin t, t)
(n) f(t) = (t2, t4)
2. Desenhe a imagem.
(a) f(t) = (1, t, 1), t ∈ R
(b) f(t) = (1, 1, t), t ≥ 0
(c) f(t) = (t, t, 1), t ≥ 0
(d) f(t) = (1, 0, t), t ∈ R
(e) f(t) = (cos t, sin t, 2)
(f) f(t) = (t, t, 1/t), t > 0
(g) f(t) = (t, t, 1 + sin t), t ≥ 0
(h) f(t) = (sin t, sin t, t), t ≥ 0
(i) f(t) = (cos t, sin t, e−t), t ≥ 0
(j) f(t) = (e−t cos t, e−t sin t, e−t), t ≥ 0
(k) f(t) = (sin t, sin t,
√
2 cos t), 0 ≤ t ≤ 2pi
(l) f(t) = (1+sin t, 1+sin t, cos t), −pi/2 ≤ t ≤ pi/2
3. Sejam F (t) = (t, sin t, 2) e G(t) = (3, t, t2). Cal-
cule
(a) 〈F (t), G(t)〉
(b) e−tF (t)
(c) F (t)− 2G(t)
(d) F (t)×G(t)
4. Calcule os limites.
(a) limt→0
(
e−3t, t
2
sin2 t
, cos 2t
)
(b) limt→1
(
t2−1
t−1 ,
√
t+ 8, sinpitln t
)
(c) limt→∞
(
te−t, t
3+t
2t3−1 , t sin
1
t
)
5. Calcule F ′ e F ′′.
(a) F (t) = (3t2, e−t, ln(t2 + 1))
(b) F (t) = (
3
√
t2, cos t2, 3t)
(c) F (t) = (sin 5t, cos 4t,−e−2t)
6. Determine a equac¸a˜o da reta tangente a` tra-
jeto´ria da func¸a˜o dada, no ponto dado.
(a) F (t) = (cos t, sin t) e F (pi/3)
(b) F (t) = (t2, t) e F (1)
(c) F (t) = (1/t, 1/t, t2 sin t) e F (2)
∗Universidade Federal do Cariri, Juazeiro do Norte, CE, BR (Email: valdines.leite@ufca.edu.br).
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