Baixe o app para aproveitar ainda mais
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
Compartilhar