Exercicios Calculo (169)

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2.8 4ª Prova – 19 de Maio de 2018 
 
𝑄𝑢𝑒𝑠𝑡ã𝑜 1. 
 
𝑎) 𝑆𝑒𝑗𝑎 𝑓: [0,2018] → ℝ 𝑐𝑜𝑛𝑡í𝑛𝑢𝑎 𝑒𝑚 [0,2018] 𝑒 𝑑𝑒𝑟𝑖𝑣á𝑣𝑒𝑙 𝑒𝑚 (0,2018) 𝑠𝑎𝑡𝑖𝑠𝑓𝑎𝑧𝑒𝑛𝑑𝑜 
𝑓 ′(𝑥)𝑓 ′(2018 − 𝑥) ≤ 0 ∀𝑥 ∈ (0,2018).𝑆𝑎𝑏𝑒𝑛𝑑𝑜 𝑞𝑢𝑒 𝑒𝑥𝑖𝑠𝑡𝑒 𝑛𝑜 𝑚á𝑥𝑖𝑚𝑜 𝑢𝑚 𝑐 𝑡𝑎𝑙 𝑞𝑢𝑒 
𝑓 ′(𝑐) = 0 𝑒 𝑞𝑢𝑒 𝑓 ′′(𝑐) > 0, 𝑠𝑒 𝑓 ′(𝑐) = 0 𝑒𝑛𝑐𝑜𝑛𝑡𝑟𝑒 𝑜𝑠 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙𝑜𝑠 𝑜𝑛𝑑𝑒 𝑓 é 𝑐𝑟𝑒𝑠𝑐𝑒𝑛𝑡𝑒 
𝑒 𝑑𝑒𝑐𝑟𝑒𝑠𝑐𝑒𝑛𝑡𝑒. 
 
𝑏) 𝑆𝑒𝑗𝑎 𝑓(𝑥) = ln(1 − ln 𝑥) . 𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑜 𝑑𝑜𝑚í𝑛𝑖𝑜, 𝑜𝑠 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙𝑜𝑠 𝑑𝑒 𝑐𝑟𝑒𝑠𝑐𝑖𝑚𝑒𝑛𝑡𝑜 𝑒 
𝑑𝑒𝑐𝑟𝑒𝑠𝑐𝑖𝑚𝑒𝑛𝑡𝑜 , 𝑎 𝑐𝑜𝑛𝑐𝑎𝑣𝑖𝑑𝑎𝑑𝑒 𝑒 𝑜𝑠 𝑝𝑜𝑛𝑡𝑜𝑠 𝑑𝑒 𝑖𝑛𝑓𝑙𝑒𝑥ã𝑜 𝑑𝑒 𝑓. 
 
𝑄𝑢𝑒𝑠𝑡ã𝑜 2. 
 
𝑎) 𝐶𝑎𝑙𝑐𝑢𝑙𝑒 lim
𝑥→0+
sen𝑥
√1− cos𝑥
. 
 
𝑏) 𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒, 𝑠𝑒 𝑒𝑥𝑖𝑠𝑡𝑖𝑟,𝑜 𝑝𝑜𝑛𝑡𝑜 𝑑𝑒 𝑖𝑛𝑓𝑙𝑒𝑥ã𝑜 𝑑𝑎 𝑓𝑢𝑛çã𝑜 𝑓(𝑥) = 𝑥|𝑥|. 
 
𝑄𝑢𝑒𝑠𝑡ã𝑜 3. 
 
𝑎) 𝑆𝑒𝑗𝑎 𝑓 ′′(𝑥) = −4 cos(2𝑥) , 𝑐𝑜𝑚 𝑓(0) = 1 𝑒 𝑓 ′(0) = 0. 𝐴𝑠 𝑟𝑒𝑡𝑎𝑠 𝑦 =
1
2
 𝑒 𝑦 = −
1
2
, 
𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑎𝑚 𝑜 𝑔𝑟á𝑓𝑖𝑐𝑜 𝑑𝑒 𝑓 𝑒𝑚 𝑝𝑜𝑛𝑡𝑜𝑠 𝑞𝑢𝑒 𝑠ã𝑜 𝑜𝑠 𝑣é𝑟𝑡𝑖𝑐𝑒𝑠 𝑑𝑒 𝑢𝑚 𝑝𝑜𝑙í𝑔𝑜𝑛𝑜. 
𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒𝑎 á𝑟𝑒𝑎 𝑑𝑒𝑠𝑠𝑒 𝑝𝑜𝑙í𝑔𝑜𝑛𝑜, 𝑐𝑜𝑚 𝑥 ∈ [0,𝜋]. 
 
𝑏) 𝑆𝑒𝑗𝑎𝑚 𝑓, 𝑔 𝑓𝑢𝑛çõ𝑒𝑠 𝑑𝑖𝑓𝑒𝑟𝑒𝑛𝑐𝑖á𝑣𝑒𝑖𝑠, 𝑡𝑎𝑖𝑠 𝑞𝑢𝑒 lim
𝑥→0
𝑓 ′(𝑥) = lim
𝑥→0
𝑔′(𝑥) = 1 𝑒 
lim
𝑥→0
𝑓(𝑥) = lim
𝑥→0
𝑔(𝑥) = 0. 𝐶𝑎𝑙𝑐𝑢𝑙𝑒 lim
𝑥→1
𝑓(𝑥2 −1)
𝑔(𝑥3 − 1)
. 
 
𝑄𝑢𝑒𝑠𝑡ã𝑜 4. 𝑃𝑟𝑒𝑡𝑒𝑛𝑑𝑒-𝑠𝑒 𝑒𝑠𝑡𝑒𝑛𝑑𝑒𝑟 𝑢𝑚 𝑐𝑎𝑏𝑜 𝑑𝑒 𝑢𝑚𝑎 𝑢𝑠𝑖𝑛𝑎 𝑑𝑒 𝑓𝑜𝑟ç𝑎 à 𝑚𝑎𝑟𝑔𝑒𝑚 𝑑𝑒 
𝑢𝑚 𝑟𝑖𝑜 𝑑𝑒 9𝑚 𝑑𝑒 𝑙𝑎𝑟𝑔𝑢𝑟𝑎 𝑎𝑡é 𝑢𝑚𝑎 𝑓á𝑏𝑖𝑐𝑎 𝑠𝑖𝑡𝑢𝑎𝑑𝑎 𝑑𝑜 𝑜𝑢𝑡𝑟𝑜 𝑙𝑎𝑑𝑜 𝑑𝑜 𝑟𝑖𝑜, 30𝑚 𝑑𝑜 
𝑟𝑖𝑜 𝑎𝑏𝑎𝑖𝑥𝑜. 𝑂 𝑐𝑢𝑠𝑡𝑜 𝑝𝑎𝑟𝑎 𝑒𝑠𝑡𝑒𝑛𝑑𝑒𝑟 𝑢𝑚 𝑐𝑎𝑏𝑜 𝑝𝑒𝑙𝑜 𝑟𝑖𝑜 é 𝑑𝑒 𝑅$5,00 𝑜 𝑚𝑒𝑡𝑟𝑜,𝑒𝑛𝑞𝑢𝑎𝑛𝑡𝑜 
𝑞𝑢𝑒 𝑝𝑎𝑟𝑎 𝑒𝑠𝑡𝑒𝑛𝑑ê-𝑙𝑜 𝑝𝑜𝑟 𝑡𝑒𝑟𝑟𝑎 𝑐𝑢𝑠𝑡𝑎 𝑅$4,00 𝑜 𝑚𝑒𝑡𝑟𝑜.𝑄𝑢𝑎𝑙 𝑜 𝑝𝑒𝑟𝑐𝑢𝑟𝑠𝑜 𝑚𝑎𝑖𝑠 
𝑒𝑐𝑜𝑛ô𝑚𝑖𝑐𝑜 𝑝𝑜𝑠𝑠í𝑣𝑒𝑙? 
 
𝑄𝑢𝑒𝑠𝑡ã𝑜 5. 
 
𝑆𝑒𝑗𝑎 𝑓(𝑥) =
(𝑥 − 1)3
(𝑥 + 1)2
. 𝑆𝑎𝑏𝑒𝑛𝑑𝑜 𝑞𝑢𝑒 𝑓 ′(𝑥) =
(𝑥 − 1)2(𝑥 + 5)
(𝑥 + 1)3
 𝑒 𝑓 ′′(𝑥) =
24(𝑥 − 1)
(𝑥 + 1)4
 
𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒: 
 
(𝑖) 𝑂𝑠 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙𝑜𝑠 𝑑𝑒 𝑐𝑟𝑒𝑠𝑐𝑖𝑚𝑒𝑛𝑡𝑜 𝑒 𝑑𝑒𝑐𝑟𝑒𝑠𝑐𝑖𝑚𝑒𝑛𝑡𝑜; 
(𝑖𝑖) 𝑂𝑠 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙𝑜𝑠 𝑜𝑛𝑑𝑒 𝑎 𝑐𝑜𝑛𝑐𝑎𝑣𝑖𝑑𝑎𝑑𝑒 é 𝑣𝑜𝑙𝑡𝑎𝑑𝑎 𝑝𝑎𝑟𝑎 𝑐𝑖𝑚𝑎 𝑜𝑢 𝑝𝑎𝑟𝑎 𝑏𝑎𝑖𝑥𝑜; 
(𝑖𝑖𝑖) 𝐴𝑠 𝑎𝑠𝑠í𝑛𝑡𝑜𝑡𝑎𝑠, 𝑠𝑒 𝑒𝑥𝑖𝑠𝑡𝑖𝑟𝑒𝑚; 
(𝑖𝑣) 𝑂𝑠 𝑝𝑜𝑛𝑡𝑜𝑠 𝑑𝑒 𝑚á𝑥𝑖𝑚𝑜 𝑒 𝑑𝑒 𝑚í𝑛𝑖𝑚𝑜 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑜𝑠, 𝑠𝑒 𝑒𝑥𝑖𝑠𝑡𝑖𝑟𝑒𝑚; 
(𝑣) 𝑂𝑠 𝑝𝑜𝑛𝑡𝑜𝑠 𝑑𝑒 𝑖𝑛𝑓𝑙𝑒𝑥ã𝑜,𝑠𝑒 𝑒𝑥𝑖𝑠𝑡𝑖𝑟𝑒𝑚; 
(𝑣𝑖) 𝑂 𝑔𝑟á𝑓𝑖𝑐𝑜 𝑑𝑒 𝑓(𝑥).