Atv( 9b)

Prévia do material em texto

9. b) ∫ [(6 − 𝑡)𝑖 + 3√𝑡 𝑗 +
4
𝑡2
 𝑘] 𝑑𝑡
2
1
 
 
∫ 6𝑡 −
𝑡2
2
 𝑖
2
1
 + ∫ 2𝑡
3
2
2
1
 j + ∫ −
4
𝑡
2
1
 k 
6.(2) ‒ 6.(1) ‒ (
22
2
−
12
2
) 2√2³ ‒ 2√1³ −
4
2
+
4
1
 
12 ‒ 6 ‒ (2 −
1
2
) 2√2² . 2 ‒ 2 ‒2 + 4 
6 ‒ 
3
2
 4√𝟐 ‒ 2 j 2k 
𝟗
𝟐
 i 
𝟗
𝟐
 𝒊 + (𝟒√𝟐 − 𝟐)𝒋 + 𝟐𝒌 
 
 c) ∫ [ (𝑠𝑒𝑛𝑡)𝑖 + (1 + 𝑐𝑜𝑠𝑡)𝑗 + (𝑠𝑒𝑐²𝑡)𝑘 ] 𝑑𝑡
𝜋
4
−
𝜋
4
 
 ∫ −𝑐𝑜𝑠𝑡
𝜋
4
−
𝜋
4
 i +∫ (𝑡 − 𝑠𝑒𝑛𝑡)𝑗
𝜋
4
−
𝜋
4
 + ∫ 𝑡𝑔𝑡 𝑘 
𝜋
4
−
𝜋
4
 
 ‒cos(45) ‒ [‒cos(‒45)] │ 𝜋
4
−
𝜋
4
− [𝑠𝑒𝑛(45) − 𝑠𝑒𝑛(−45)]│tg(45) ‒ tg(‒45) 
 −
√2
2
+
√2
2
 │ 
2𝜋
4
+ 
2√2
2
 │ 1 ‒ (‒1) 
 (0)i │ 
𝜋+2√2
2
 j │ 2k 
 
𝝅 + 𝟐√𝟐
𝟐
 𝒋 + 𝟐𝒌 
 
 d) ∫ [
1
𝑡
 𝑖 +
1
5−𝑡
 𝑗 +
1
2𝑡
 𝑘] 𝑑𝑡
4
1
 
 
∫ 𝑙𝑛│𝑡│𝑖 + ∫ −𝑙𝑛│5 − 𝑡│𝑗 + ∫ 
1
2
𝑙𝑛│𝑡│𝑘
4
1
4
1
4
1
 
 ln│4│ ‒ ln│1│ ‒ln│5 ‒ 4│+ ln│5 ‒ 1│ 1
2
𝑙𝑛│4│ −
1
2
𝑙𝑛│1│ 
 ln│4│ ‒ 0 ‒ln│1│ + ln│4│ 
𝑙𝑛│4│
2
 
 ln│4│i 0 + ln│4│ ln│2│k 
 ln│4│j 
 𝒍𝒏│𝟒│𝒊 + 𝒍𝒏│𝟒│𝒋 + 𝒍𝒏│𝟐│𝒌 
10. 𝑇 =
𝑣(𝑡)
│𝑣(𝑡)│
 
 𝐾 = 
1
│𝑣(𝑡)│
 . │𝑇´│ 
 𝑁 = 
𝑇´
│𝑇´│
 
Identidades trigonometricas: 
sen² + cos² = 1 
sec = 
1
𝑐𝑜𝑠
 
𝑑(𝑠𝑒𝑐²𝑡)
𝑑𝑡
= 𝑠𝑒𝑐𝑡. 𝑡𝑔𝑡 
 
a. r (t) = ( 3sent)i + ( 3cost)j + 4tk 
 v(t) = (3cost)i + (‒3sent)j + 4k 
 │v(t)│ = √(3𝑐𝑜𝑠𝑡)2 + (−3𝑠𝑒𝑛𝑡)2 + 4² 
 │v(t)│ = √9𝑐𝑜𝑠²𝑡 + 9𝑠𝑒𝑛²𝑡 + 16 
 │v(t)│ = √9(𝑐𝑜𝑠2𝑡 + 𝑠𝑒𝑛2𝑡) + 16 
 │v(t)│ = √9. (1) + 16 
 │v(t)│ = √25 = 5 
𝑻 = 
𝟑𝒄𝒐𝒔𝒕
𝟐
𝒊 − 
𝟑𝒔𝒆𝒏𝒕
𝟓
𝒋 +
𝟒
𝟓
𝒌 
 
 T ´= 
−3𝑠𝑒𝑛𝑡𝑖−3𝑐𝑜𝑠𝑡𝑗
5
 
 │T ´│ = √
(−3𝑠𝑒𝑛𝑡)2+(−3𝑐𝑜𝑠𝑡)2
(5)2
 
 │T ´│ = √
9𝑠𝑒𝑛𝑡²𝑡 + 9𝑐𝑜𝑠²𝑡
25
 
 │T ´│ = √
9.(𝑠𝑒𝑛2𝑡 +𝑐𝑜𝑠2𝑡) 
25
 
 │T ´│ = √
9.(1)
25
 
 │T ´│ = √
9
25
 = 
3
5
 
 K = 
1
5
 .
3
5
 
𝑲 =
𝟑
𝟐𝟓
 
 𝑁 =
−3𝑠𝑒𝑛𝑡𝑖−3𝑐𝑜𝑠𝑡𝑗
5
3
5
 
 𝑁 =
−3𝑠𝑒𝑛𝑡𝑖−3𝑐𝑜𝑠𝑡𝑗
5
 .
5
3
 
 𝑁 =
3(−𝑠𝑒𝑛𝑡−𝑐𝑜𝑠𝑡)
5
 .
5
3
 
 
𝑵 = (−𝒔𝒆𝒏𝒕)𝒊 − 𝒄𝒐𝒔𝒕𝒋 
 
b. r(t) = (cost + tsent)i + (sent ‒ tcost)j + 3k 
 v(t)= (‒sent+ sent + tcost)i + (cost ‒cost ‒tsent)j 
 v(t) = tcost ‒ tsent 
 │v(t)│ = √(𝑡𝑐𝑜𝑠𝑡)2 + (−𝑡𝑠𝑒𝑛𝑡)2 
 │v(t)│= √𝑡²𝑐𝑜𝑠²𝑡 + 𝑡²𝑠𝑒𝑛²𝑡 
 │v(t)│ = √𝑡²(𝑐𝑜𝑠2𝑡 + 𝑠𝑒𝑛2𝑡) 
 │v(t)│ = √𝑡² = t 
 T = 
𝑡𝑐𝑜𝑠𝑡−𝑡𝑠𝑒𝑛𝑡
𝑡
 
 T = 
𝑡 (𝑐𝑜𝑠𝑡−𝑠𝑒𝑛𝑡)
𝑡
 
𝑻 = (𝒄𝒐𝒔𝒕)𝒊 − (𝒔𝒆𝒏𝒕)𝒋 
T ´= ‒sent i ‒ cost j 
 │T ´│ = √(−𝑠𝑒𝑛𝑡)2 + (−𝑐𝑜𝑠𝑡)² 
 │T ´│ = √𝑠𝑒𝑛²𝑡 + 𝑐𝑜𝑠²𝑡 
 │T ´│ = √1 = 1 
 
K = 
1
𝑇
 . 1 
K = 
𝟏
𝑻
 
 
N = ‒sent i ‒ cost j 
c. r(t) = ln(sec²t)i + (t)j 
 ln(u) = 
𝑢´
𝑢
 
 u= sec²t 
 u ´= sect.tgt 
 
v(t) = 
𝑠𝑒𝑐𝑡.𝑡𝑔𝑡
𝑠𝑒𝑐𝑡
 + j 
 v(t) = (tgt)i + j 
 │v(t)│ = √(𝑡𝑔𝑡)2 + (1)2 
 │v(t)│ = √𝑠𝑒𝑐²𝑡 = sect 
 
T = 
(𝑡𝑔𝑡)𝑖+𝑗
𝑠𝑒𝑐𝑡
 
 
T = 
𝑠𝑒𝑛𝑡
𝑐𝑜𝑠𝑡
1
𝑐𝑜𝑠𝑡
 
 
T = 
𝑠𝑒𝑛𝑡
𝑐𝑜𝑠𝑡
 . 𝑐𝑜𝑠𝑡 
T = sent 
T ´= cost 
│T ´│ = √(𝑐𝑜𝑠𝑡)² 
│T ´│ = √𝑐𝑜𝑠²𝑡 
│T ´│ = cost 
 
K = 
1
𝑠𝑒𝑐𝑡
 . 𝑐𝑜𝑠𝑡 
 
K = 
𝑐𝑜𝑠𝑡
𝑠𝑒𝑐𝑡
 
 
K = 
𝑐𝑜𝑠𝑡
1
𝑐𝑜𝑠𝑡
 
 
K = cost . cost 
K = cos²t 
 
N = 
𝑐𝑜𝑠𝑡
𝑐𝑜𝑠𝑡
 
N = 1