Seção 13 3 R

Prévia do material em texto

2  SEÇÃO 13.3 COMPRIMENTO DE ARCO E CURVATURA
 1. 13 (b− a)
 2. 3 e2pi − 1
 3. 8
 4. 3+ 22 ln 2+ 3
 5. ln pi
2
4 + 2+
pi
2 +
pi
4
pi 2
4 + 2− ln 2
 6. r (t (s)) = 12 s + 1 sen ln
1
2 s + 1 i
+ cos ln 12 s + 1 j
 7. r (t (s)) = 1+ 230 s i + 3+
1
30 s j −
5
30 s k
 8. r (t (s)) = cos3 12 cos
− 1 1 − 45 s i
+ sen3 12 cos
− 1 1 − 45 s j + 1 −
4
5 s k
 9. (a) 15 4 cos 4t, 3,−4 sen 4t , sen 4t, 0,− cos 4t
(b) 1625
 10. (a)
1
1+ t2
1, 2t, t2 ,
1
2 (1+ t2 )
−2t, 2 1 − t2 , 2t
(b) 1
3 2 (1+ t2 )2
 11. (a) 12 − 2 sen t, cos t, cos t ,
1
2 − 2 cos t, − sen t, − sen t
(b) 12
 12. (a)
1
t2 + 2
t2 , 2t, 2 , 1
t2 + 2
2t, 2 − t2 ,−2t
(b) 2
(t2 + 2)2
 13. (a)
1
e2 t + 1
2e2 t , e2 t ,−1 ,
1
e2 t + 1
1 − e2 t , 2et , 2et
(b) 2e
2 t
(e2 t + 1)2
 14. (a)
1
2t2 + 1
2t, 2t2 , 1 , 1
2t2 + 1
1 − 2t2 , 2t, −2t
(b) 2
(2t2 + 1)2
 15. 
2
(4t2 + 1)3/ 2
 16. 
3
(1 + 18t2 )3/ 2
 17. 1+ 4t
2 + t4
6 (t4 + t2 + 1)3/ 2
 18. 6
2 (2t2 − 4t + 5)3/ 2
 19. 2
(1 + cos2 t)3/ 2
 20. 24
 21. 
2
(4x + 1)3/ 2
 22. |sen x |
(1 + cos2 x )3/2
 23. 
|x |
(x 2 + 1)3/2
 24. 
6
|t | (9t2 + 4)3/2
 25. 
2+ t2
(1 + t2 )3/2
13.3 RESPOSTAS Revisão técnica: Ricardo Miranda Martins – IMECC – Unicamp