Exercicios Calculo (132)

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146 
 
2.3 2ª Prova – 23 de Março de 2018 
 
𝑄𝑢𝑒𝑠𝑡ã𝑜 1. 
 
𝑎) 𝑆𝑒 𝑓 𝑓𝑜𝑟 𝑢𝑚𝑎 𝑓𝑢𝑛çã𝑜 𝑑𝑖𝑓𝑒𝑟𝑒𝑛𝑐𝑖á𝑣𝑒𝑙 𝑒 𝑔(𝑥) = 𝑥𝑓(𝑥),𝑢𝑠𝑒 𝑎 𝑑𝑒𝑓𝑖𝑛𝑖çã𝑜 𝑑𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑑𝑎 
𝑝𝑎𝑟𝑎 𝑚𝑜𝑠𝑡𝑟𝑎𝑟 𝑞𝑢𝑒 𝑔′(𝑥) = 𝑥𝑓 ′(𝑥) + 𝑓(𝑥). 
 
𝑏) 𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟𝑒 𝑜𝑠 𝑝𝑜𝑛𝑡𝑜𝑠 𝑑𝑒 𝑎𝑏𝑠𝑐𝑖𝑠𝑠𝑎𝑠 𝑎 𝑒 -𝑎 𝑑𝑜 𝑔𝑟á𝑓𝑖𝑐𝑜 𝑑𝑎 𝑐𝑢𝑟𝑣𝑎 𝑦 = 𝑥2, 𝑜𝑛𝑑𝑒 𝑎 ≠ 0. 
𝐸𝑛𝑐𝑜𝑛𝑡𝑟𝑒 𝑎𝑠 𝑒𝑞𝑢𝑎çõ𝑒𝑠 𝑑𝑎𝑠 𝑟𝑒𝑡𝑎𝑠 𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑒𝑠 𝑎𝑜 𝑔𝑟á𝑓𝑖𝑐𝑜 𝑛𝑒𝑠𝑡𝑒𝑠 𝑝𝑜𝑛𝑡𝑜𝑠 𝑒 𝑒𝑚 𝑠𝑒𝑔𝑢𝑖𝑑𝑎 
𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑜 𝑣𝑎𝑙𝑜𝑟 𝑑𝑒 𝑎 𝑡𝑎𝑙 𝑞𝑢𝑒 𝑎𝑠 𝑟𝑒𝑡𝑎𝑠 𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑒𝑠 𝑛𝑒𝑠𝑡𝑒𝑠 𝑝𝑜𝑛𝑡𝑜𝑠 𝑠𝑒𝑗𝑎𝑚 𝑜𝑟𝑡𝑜𝑔𝑜𝑛𝑎𝑖𝑠. 
 
𝑄𝑢𝑒𝑠𝑡ã𝑜 2. 
 
𝑎) 𝑆𝑒𝑗𝑎 𝑃(𝑥) = 𝑥(𝑥 − 2018)(𝑥 + 2018),𝑚𝑜𝑠𝑡𝑟𝑒 𝑞𝑢𝑒 
𝑃′(𝑥)
𝑃(𝑥)
=
1
𝑥
+
1
𝑥 − 2018
+
1
𝑥 + 2018
. 
𝐶𝑜𝑛𝑐𝑙𝑢𝑎 𝑞𝑢𝑒 𝑃′(1009) = −(1009)2. 
 
𝑏) 𝐸𝑛𝑐𝑜𝑛𝑡𝑟𝑒, 𝑠𝑒 𝑒𝑥𝑖𝑠𝑡𝑖𝑟, 𝑜(𝑠) 𝑝𝑜𝑛𝑡𝑜(𝑠) 𝑑𝑎 𝑐𝑢𝑟𝑣𝑎 𝑦 = 𝑥 + 𝑒𝑥 − 𝑒2𝑥 𝑡𝑎𝑙 𝑞𝑢𝑒 𝑎 𝑟𝑒𝑡𝑎 
𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑒 𝑎𝑜 𝑔𝑟á𝑓𝑖𝑐𝑜 𝑛𝑒𝑠𝑡𝑒(𝑠) 𝑝𝑜𝑛𝑡𝑜(𝑠) 𝑠ã𝑜 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑖𝑠. 
 
𝑄𝑢𝑒𝑠𝑡ã𝑜 3. 
 
𝑎) 𝐻á 𝑢𝑚𝑎 𝑟𝑒𝑡𝑎 𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑒 𝑎𝑜 𝑔𝑟á𝑓𝑖𝑐𝑜 𝑑𝑎 𝑓𝑢𝑛çã𝑜 𝑓(𝑥) = 3𝑥2tg 𝑥 𝑛𝑜 𝑝𝑜𝑛𝑡𝑜 𝑑𝑒 
𝑜𝑟𝑑𝑒𝑛𝑎𝑑𝑎 1.𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑎 á𝑟𝑒𝑎 𝑑𝑜 𝑡𝑟𝑖â𝑛𝑔𝑢𝑙𝑜 𝑐𝑢𝑗𝑜𝑠 𝑣é𝑟𝑡𝑖𝑐𝑒𝑠 𝑠ã𝑜 𝑎 𝑜𝑟𝑖𝑔𝑒𝑚 𝑑𝑜 
𝑠𝑖𝑠𝑡𝑒𝑚𝑎 𝑑𝑒 𝑒𝑖𝑥𝑜𝑠, 𝑜 𝑝𝑜𝑛𝑡𝑜 𝑑𝑒 𝑜𝑟𝑑𝑒𝑛𝑎𝑑𝑎 1 𝑒 𝑎 𝑖𝑛𝑡𝑒𝑟𝑠𝑒çã𝑜 𝑑𝑎 𝑐𝑖𝑡𝑎𝑑𝑎 𝑟𝑒𝑡𝑎 𝑐𝑜𝑚 
𝑜 𝑒𝑖𝑥𝑜 𝑑𝑎𝑠 𝑎𝑏𝑠𝑐𝑖𝑠𝑠𝑎𝑠 . 
 
𝑏)𝐸𝑛𝑐𝑜𝑛𝑡𝑟𝑒 𝑢𝑚𝑎 𝑒𝑞𝑢𝑎çã𝑜 𝑝𝑎𝑟𝑎 𝑎 𝑟𝑒𝑡𝑎 𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑒 𝑎𝑜 𝑔𝑟á𝑓𝑖𝑐𝑜 𝑑𝑒 𝑓(𝑥) = |tg 𝑥| 
𝑛𝑜 𝑝𝑜𝑛𝑡𝑜 𝑑𝑒 𝑎𝑏𝑠𝑐𝑖𝑠𝑠𝑎 𝑥 =
3𝜋
4
. 
 
𝑄𝑢𝑒𝑠𝑡ã𝑜 4. 
 
𝑎)𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑎 𝑑𝑒𝑟𝑖𝑣𝑎𝑑𝑎 𝑑𝑒 𝑦 = sen(√cotg(3𝑥)). 
 
𝑏)𝐶𝑎𝑙𝑐𝑢𝑙𝑒 lim
𝑥→0
sen(3 + 𝑥)2 − sen9
𝑥
. 
 
𝑄𝑢𝑒𝑠𝑡ã𝑜 5. 
 
𝑎)𝑀𝑜𝑠𝑡𝑟𝑒, 𝑢𝑠𝑎𝑛𝑑𝑜 𝑑𝑒𝑟𝑖𝑣𝑎çã𝑜 𝑖𝑚𝑝𝑙í𝑐𝑖𝑡𝑎, 𝑞𝑢𝑒 𝑞𝑢𝑎𝑙𝑞𝑢𝑒𝑟 𝑟𝑒𝑡𝑎 𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑒 𝑒𝑚 𝑢𝑚 𝑝𝑜𝑛𝑡𝑜 𝑃 
𝑎 𝑢𝑚 𝑐í𝑟𝑐𝑢𝑙𝑜 𝑑𝑎𝑑𝑜 𝑝𝑜𝑟 𝑥2 +𝑦2 = 25 é 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑎 𝑟𝑒𝑡𝑎 𝑞𝑢𝑒 𝑐𝑜𝑛𝑡é𝑚 𝑜 𝑝𝑜𝑛𝑡𝑜 𝑃 
𝑒 𝑎 𝑜𝑟𝑖𝑔𝑒𝑚. 
 
𝑏)𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟𝑒 𝑢𝑚 𝑡𝑟𝑖â𝑛𝑔𝑢𝑙𝑜 𝑟𝑒𝑡â𝑛𝑔𝑢𝑙𝑜 𝑐𝑜𝑚 ℎ𝑖𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑎 𝑚𝑒𝑑𝑖𝑛𝑑𝑜 1𝑐𝑚 𝑒 𝑢𝑚 â𝑛𝑔𝑢𝑙𝑜 
𝑎𝑔𝑢𝑑𝑜 𝜃, 𝑐𝑜𝑚 𝑐𝑎𝑡𝑒𝑡𝑜 𝑜𝑝𝑜𝑠𝑡𝑜 ℎ.𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑎 𝑡𝑎𝑥𝑎 𝑑𝑒 𝑣𝑎𝑟𝑖𝑎çã𝑜 𝑑𝑒 𝜃 𝑒𝑚 𝑟𝑒𝑙𝑎çã𝑜 à 
ℎ 𝑞𝑢𝑎𝑛𝑑𝑜 ℎ =
√3
2
𝑐𝑚.