Lista de exs 3 (gabarito)

Prévia do material em texto

Universidade Federal do Rio de Janeiro
Cálculo II - 2013/2 - Lista 3 - Gabarito
Professora Talita Mello
1. (a) (0, 3)
(b) (−2, 2)
(c) (−1,√3)
(d) (−4, 0)
2. (a) (
√
2, pi4 )
(b) (4, −pi6 )
(c) (2, −pi3 )
(d) (
√
13, arctan(−23 ))
3. Grá�co
4. Grá�co
5. (a) (0, 3, 1)
(b) (3, 0,−6)
(c) (2,−2√3, 5)
6. (a) (
√
2, −pi4 , 4)
(b) (2, 4pi3 , 2)
(c) (3
√
2, pi4 ,−2)
7. (a) (0, 0, 1)
(b) (
√
3
4 ,
1
4 ,
√
3
2 )
(c) (
√
2
2 ,
√
6
2 ,
√
2)
8. (a) (3, pi, pi2 )
(b) (2, 0, pi3 )
(c) (2, pi4 ,
pi
4 )
9. Grá�co
10. x = 0 ou x = 6
11. (a) pruv =
11
√
13
13 (2, 3)
(b) pruv =
3
√
5
5 (4, 2, 0)
12. Demonstração
13.
√
6
6 (2, 1,−1) e −
√
6
6 (2, 1,−1).
1
14. 4
15. 19
16. Sim
17. (a) y = x2 − 3
(b) y = −2x+ 9, −1 ≤ x ≤ 2
(c) y = (1−x)
2
4 + 4, −5 ≤ x ≤ 1
18. (a) 1 ≤ t ≤ 5
(b) t > 0, t 6= 1
19. (a) (1, 0, 0)
(b) (1, 12 , 3)
20. Demonstração
21. Demonstração
22. (a) (2 cos t, 2 sin t, 4 sin t cos t)
(b) (t, 12 t
2 − 12 , 12 t2 + 12 )
(c) (
√
1
2 t− 2, 12 t− 2, 14 t2 − 4), t > 4.
23. Demonstração
24. Demonstração
25. Demonstração
26. (a) σ′(t) = (2t, 1, 1
2
√
t
), t > 0
(b) σ′(t) = 4e4tk
(c) σ′(t) = (−3 sin 3t, 1, 3 cos 3)
27. (a) σ′(1) = (30, 12, 2)
(b) σ′(1) = ( 12 ,−1, 12 )
(c) σ′(pi6 ) = (1,
√
3, −32 )
(d) σ′(pi2 ) = (−2epi,−epi, 2epi)
28. σ′(t) = (1, 2t, 3t2), T (1) =
√
14(1, 2, 3), A(t) = (0, 2, 6) e V (t)×A(t) = (6t2, 6t, 2).
29. σ′(t) = (2e2t,−2e−2t, 2te2t + e2t), T (0) = √13(2,−2, 1), A(0) = (4, 4, 4) e V (t)A˙(t) = 12e4t−
8e−4t + 12te4t + 8t2e4t.
30. (a) (1 + 5t, 1 + 4t, 1 + 3t)
(b) (−1, 1, 1 + t)
2
(c) (−pi2 t,
1
4 + t, 1 + 4t)
(d) (pit, 1 + 12 t,−1)
31. (a) ( 12 ,
1
3 ,
1
4 )
(b) ( 103 ,
−124
5 ,
−4
3 )
32. r(t) = ( t
3
3 , t
4 + 1, −t
3
3 )
33. Demonstração
34. (a) 20
√
29
(b)
pi2
2
√
5
(c)
1
4 (e
2 + 1)
(d) 2e− 12e3 − 53
35. (a) σ(s) = ( s
√
2
2 sin(ln(
s
√
2
2 )),
s
√
2
2 cos(ln(
s
√
2
2 )))
(b) σ(s) = (1 + s
√
30
15 , 3 +
s
√
30
30 ,
−s√30
6 )
(c) σ(s) = (3 sin( s
√
13
13 ),
4s
√
13
13 , 3 cos(
s
√
13
13 ))
3