Exercicios Calculo (188)

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𝑏) 𝑈𝑚 𝑚𝑜𝑛𝑔𝑒 𝑡𝑖𝑏𝑒𝑡𝑎𝑛𝑜 𝑑𝑒𝑖𝑥𝑎 𝑜 𝑚𝑜𝑛𝑎𝑠𝑡é𝑟𝑖𝑜 à𝑠 7 ℎ𝑜𝑟𝑎𝑠 𝑑𝑎 𝑚𝑎𝑛ℎã 𝑒 𝑠𝑒𝑔𝑢𝑒 𝑠𝑢𝑎 𝑐𝑎𝑚𝑖𝑛ℎ𝑎𝑑𝑎 
𝑢𝑠𝑢𝑎𝑙 𝑝𝑎𝑟𝑎 𝑜 𝑡𝑜𝑝𝑜 𝑑𝑎 𝑚𝑜𝑛𝑡𝑎𝑛ℎ𝑎,𝑐ℎ𝑒𝑔𝑎𝑛𝑑𝑜 𝑙á à𝑠 7 ℎ𝑜𝑟𝑎𝑠 𝑑𝑎 𝑛𝑜𝑖𝑡𝑒.𝑁𝑎 𝑚𝑎𝑛ℎã 𝑠𝑒𝑔𝑢𝑖𝑛𝑡𝑒, 
𝑒𝑙𝑒 𝑝𝑎𝑟𝑡𝑒 𝑑𝑜 𝑡𝑜𝑝𝑜 à𝑠 7 ℎ𝑜𝑟𝑎𝑠 𝑑𝑎 𝑚𝑎𝑛ℎã,𝑝𝑒𝑔𝑎 𝑜 𝑚𝑒𝑠𝑚𝑜 𝑐𝑎𝑚𝑖𝑛ℎ𝑜 𝑑𝑒 𝑣𝑜𝑙𝑡𝑎 𝑒 𝑐ℎ𝑒𝑔𝑎 𝑎𝑜 
𝑚𝑜𝑛𝑎𝑠𝑡é𝑟𝑖𝑜 à𝑠 7 ℎ𝑜𝑟𝑎𝑠 𝑑𝑎 𝑛𝑜𝑖𝑡𝑒. 𝑈𝑠𝑒 𝑜 𝑇𝑒𝑜𝑟𝑒𝑚𝑎 𝑑𝑜 𝑉𝑎𝑙𝑜𝑟 𝐼𝑛𝑡𝑒𝑟𝑚𝑒𝑑𝑖á𝑟𝑖𝑜 𝑝𝑎𝑟𝑎 𝑚𝑜𝑠𝑡𝑟𝑎𝑟 
𝑞𝑢𝑒 𝑒𝑥𝑖𝑠𝑡𝑒 𝑢𝑚 𝑝𝑜𝑛𝑡𝑜 𝑛𝑜 𝑐𝑎𝑚𝑖𝑛ℎ𝑜 𝑞𝑢𝑒 𝑜 𝑚𝑜𝑛𝑔𝑒 𝑣𝑎𝑖 𝑐𝑟𝑢𝑧𝑎𝑟 𝑒𝑥𝑎𝑡𝑎𝑚𝑒𝑛𝑡𝑒 𝑛𝑎 𝑚𝑒𝑠𝑚𝑎 ℎ𝑜𝑟𝑎 
𝑑𝑜 𝑑𝑖𝑎 𝑒𝑚 𝑎𝑚𝑏𝑎𝑠 𝑎𝑠 𝑐𝑎𝑚𝑖𝑛ℎ𝑎𝑑𝑎𝑠. 
 
𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟𝑒 𝑓(𝑡) 𝑎 𝑓𝑢𝑛çã𝑜 𝑞𝑢𝑒 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑎 𝑜 𝑝𝑒𝑟𝑐𝑢𝑟𝑠𝑜 𝑑𝑜 𝑚𝑜𝑛𝑔𝑒 𝑎 𝑝𝑎𝑟𝑡𝑖𝑟 𝑑𝑜 
𝑚𝑜𝑛𝑎𝑠𝑡é𝑟𝑖𝑜 𝑎𝑡é 𝑜 𝑝𝑜𝑛𝑡𝑜 𝑑𝑎 𝑚𝑜𝑛𝑡𝑎𝑛ℎ𝑎, 𝑒 𝑔(𝑡) 𝑎 𝑓𝑢𝑛çã𝑜 𝑞𝑢𝑒 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑎 𝑜 
𝑐𝑎𝑚𝑖𝑛ℎ𝑜 𝑑𝑒 𝑣𝑜𝑙𝑡𝑎 𝑑𝑜 𝑚𝑜𝑛𝑔𝑒 𝑎𝑜 𝑚𝑜𝑛𝑎𝑠𝑡é𝑟𝑖𝑜.𝑁𝑎 𝑎𝑢𝑠ê𝑛𝑐𝑖𝑎 𝑑𝑒 𝑝𝑎𝑟𝑎𝑑𝑎𝑠 𝑎𝑜 𝑙𝑜𝑛𝑔𝑜 
𝑑𝑜 𝑝𝑒𝑟𝑐𝑢𝑟𝑠𝑜 𝑛𝑜𝑠 𝑑𝑜𝑖𝑠 𝑑𝑖𝑎𝑠, 𝑐𝑜𝑛𝑠𝑖𝑑𝑒𝑟𝑎𝑚𝑜𝑠 𝑓 𝑒 𝑔 𝑐𝑜𝑛𝑡í𝑛𝑢𝑎𝑠 𝑛𝑜 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙𝑜 𝑓𝑒𝑐ℎ𝑎𝑑𝑜 
𝑑𝑒 7 ℎ𝑜𝑟𝑎𝑠 𝑑𝑎 𝑚𝑎𝑛ℎã (7ℎ) 𝑎𝑡é à𝑠 7 ℎ𝑜𝑟𝑎𝑠 𝑑𝑎 𝑛𝑜𝑖𝑡𝑒 (19ℎ). 
 
𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟𝑒 𝑞𝑢𝑒 𝑜 𝑚𝑜𝑛𝑎𝑠𝑡é𝑟𝑖𝑜 𝑒𝑛𝑐𝑜𝑛𝑡𝑟𝑎-𝑠𝑒 𝑛𝑜 𝑛í𝑣𝑒𝑙 𝑑𝑎 𝑏𝑎𝑠𝑒 𝑑𝑎 𝑚𝑜𝑛𝑡𝑎𝑛ℎ𝑎 (ℎ = 0) 
𝑒 𝑞𝑢𝑒 𝑛𝑜 𝑡𝑜𝑝𝑜 𝑑𝑎 𝑚𝑜𝑛𝑡𝑎𝑛ℎ𝑎 ℎ = ℎ𝑚𝑜𝑛𝑡𝑎𝑛ℎ𝑎 , 𝐶𝑜𝑚 𝑖𝑠𝑠𝑜 𝑓(7) = 𝑔(19) = 0 𝑒, 
𝑓(19) = 𝑔(7) = ℎ𝑚𝑜𝑛𝑡𝑎𝑛ℎ𝑎 . 𝑆𝑒𝑛𝑑𝑜 ℎ(𝑡) = 𝑓(𝑡) − 𝑔(𝑡), 𝑜𝑛𝑑𝑒 ℎ é 𝑐𝑜𝑛𝑡í𝑛𝑢𝑎 𝑒𝑚 
[7,19], 𝑡𝑒𝑚𝑜𝑠: 
 
ℎ(7) = 𝑓(7) − 𝑔(7) = 0− ℎ𝑚𝑜𝑛𝑡𝑎𝑛ℎ𝑎 = −ℎ𝑚𝑜𝑛𝑡𝑎𝑛ℎ𝑎 ; ℎ(7) 0 
 
𝐶𝑜𝑚𝑜 ℎ é 𝑢𝑚𝑎 𝑓𝑢𝑛çã𝑜 𝑐𝑜𝑛𝑡í𝑛𝑢𝑎 𝑛𝑜 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙𝑜 𝑓𝑒𝑐ℎ𝑎𝑑𝑜 [7,19] 𝑒 0 é 𝑢𝑚 𝑛ú𝑚𝑒𝑟𝑜 
𝑒𝑛𝑡𝑟𝑒 ℎ(7) 𝑒 ℎ(19),𝑒𝑛𝑡ã𝑜, 𝑝𝑒𝑙𝑜 𝑇𝑒𝑜𝑟𝑒𝑚𝑎 𝑑𝑜 𝑉𝑎𝑙𝑜𝑟 𝐼𝑛𝑡𝑒𝑟𝑚𝑒𝑑𝑖á𝑟𝑖𝑜,𝑒𝑥𝑖𝑠𝑡𝑒 𝑎𝑙𝑔𝑢𝑚 
𝑛ú𝑚𝑒𝑟𝑜 𝑡 ∈ (7,19) 𝑡𝑎𝑙 𝑞𝑢𝑒 ℎ(𝑡) = 0 ⟹ 𝑓(𝑡) = 𝑔(𝑡). 𝑂𝑢 𝑠𝑒𝑗𝑎,𝑒𝑥𝑖𝑠𝑡𝑒 𝑢𝑚 𝑝𝑜𝑛𝑡𝑜 
𝑛𝑜 𝑐𝑎𝑚𝑖𝑛ℎ𝑜 𝑞𝑢𝑒 𝑜 𝑚𝑜𝑛𝑔𝑒 𝑣𝑎𝑖 𝑐𝑟𝑢𝑧𝑎𝑟 𝑒𝑥𝑎𝑡𝑎𝑚𝑒𝑛𝑡𝑒 𝑛𝑎 𝑚𝑒𝑠𝑚𝑎 ℎ𝑜𝑟𝑎 𝑑𝑜 𝑑𝑖𝑎𝑠 𝑒𝑚 
𝑎𝑚𝑏𝑎𝑠 𝑎𝑠 𝑐𝑎𝑚𝑖𝑛ℎ𝑎𝑑𝑎𝑠. 
 
𝑸𝒖𝒆𝒔𝒕ã𝒐 𝟑. 
 
𝑎) 𝑆𝑒𝑛𝑑𝑜 
𝑓(𝑥) = {
3𝑥 + 6𝑎, 𝑠𝑒 𝑥