Exercicios Calculo (138)

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153 
 
2.4 2ª Prova – 24 de Março de 2018 
 
𝑄𝑢𝑒𝑠𝑡ã𝑜 1. 
 
𝑎)𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑜𝑛𝑑𝑒 𝑎 𝑓𝑢𝑛çã𝑜 𝑓(𝑥) = |𝑥 − 1| + |𝑥 + 2| é 𝑑𝑖𝑓𝑒𝑟𝑒𝑛𝑐𝑖á𝑣𝑒𝑙 𝑒 𝑒𝑛𝑐𝑜𝑛𝑡𝑟𝑒 
𝑓 ′(𝑥). 
 
𝑏)𝑈𝑠𝑒 𝑎 𝑑𝑒𝑓𝑖𝑛𝑖çã𝑜 𝑑𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑑𝑎,𝑝𝑎𝑟𝑎 𝑚𝑜𝑠𝑡𝑟𝑎𝑟 𝑞𝑢𝑒 𝑎 𝑑𝑒𝑟𝑖𝑣𝑎𝑑𝑎 𝑑𝑒 𝑢𝑚𝑎 𝑓𝑢𝑛çã𝑜 𝑝𝑎𝑟 
é 𝑢𝑚𝑎 𝑓𝑢𝑛çã𝑜 í𝑚𝑝𝑎𝑟. 
 
𝑄𝑢𝑒𝑠𝑡ã𝑜 2. 
 
𝑎)𝐴𝑠 𝑝𝑎𝑟á𝑏𝑜𝑙𝑎𝑠 𝑓(𝑥) = 𝑥2 𝑒 𝑔(𝑥) = 𝑎𝑥2 +𝑏𝑥 −
3
4
 𝑠𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑐𝑖𝑎𝑚 𝑛𝑢𝑚 𝑝𝑜𝑛𝑡𝑜 𝑐𝑢𝑗𝑎 
𝑟𝑒𝑡𝑎 𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑒 𝑐𝑜𝑚𝑢𝑚 𝑡𝑒𝑚 𝑐𝑜𝑒𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑒 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑖𝑔𝑢𝑎𝑙 à 1.𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑎 𝑒 𝑏. 
 
𝑏)𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑝𝑎𝑟𝑎 𝑞𝑢𝑎𝑖𝑠 𝑣𝑎𝑙𝑜𝑟𝑒𝑠 𝑑𝑒 𝑥 𝑜 𝑔𝑟á𝑓𝑖𝑐𝑜 𝑑𝑒 𝑓(𝑥) = 𝑥 + 2sen𝑥 𝑡𝑒𝑚 𝑢𝑚𝑎 
𝑟𝑒𝑡𝑎 𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑒 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙. 
 
𝑄𝑢𝑒𝑠𝑡ã𝑜 3. 
 
𝑎)𝐶𝑎𝑙𝑐𝑢𝑙𝑒 𝑜 𝑐𝑜𝑒𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑒 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑑𝑎 𝑟𝑒𝑡𝑎 𝑞𝑢𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑐𝑖𝑎 𝑜 𝑔𝑟á𝑓𝑖𝑐𝑜 𝑑𝑒 𝑓(𝑥) = 2sec 𝑥 , 
𝑛𝑜 𝑝𝑜𝑛𝑡𝑜 𝑑𝑜 𝑔𝑟á𝑓𝑖𝑐𝑜 𝑒𝑚 𝑞𝑢𝑒 𝑥 =
𝜋
3
. 𝐷𝑖𝑣𝑖𝑑𝑖𝑛𝑑𝑜 𝑒𝑠𝑡𝑒 𝑐𝑜𝑒𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑒 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑝𝑜𝑟 8√3 
𝑜𝑏𝑡𝑒𝑟𝑒𝑚𝑜𝑠 𝑜 𝑐𝑜𝑒𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑒 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑑𝑒 𝑢𝑚𝑎 𝑜𝑢𝑡𝑟𝑎 𝑟𝑒𝑡𝑎 𝑎 𝑞𝑢𝑎𝑙 𝑝𝑎𝑠𝑠𝑎 𝑝𝑒𝑙𝑜 𝑝𝑜𝑛𝑡𝑜 (1,0). 
𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑎 á𝑟𝑒𝑎 𝑑𝑜 𝑡𝑟𝑖â𝑛𝑔𝑢𝑙𝑜 𝑐𝑢𝑗𝑜𝑠 𝑣é𝑟𝑡𝑖𝑐𝑒𝑠 𝑠ã𝑜 𝑎 𝑜𝑟𝑖𝑔𝑒𝑚, 𝑜 𝑝𝑜𝑛𝑡𝑜 (1,0) 𝑒 𝑎 
𝑖𝑛𝑡𝑒𝑟𝑠𝑒çã𝑜 𝑑𝑎 𝑠𝑒𝑔𝑢𝑛𝑑𝑎 𝑟𝑒𝑡𝑎 𝑐𝑜𝑚 𝑜 𝑒𝑖𝑥𝑜 𝑑𝑎𝑠 𝑜𝑟𝑑𝑒𝑛𝑎𝑑𝑎𝑠. 
 
𝑏)𝑆𝑒𝑛𝑑𝑜 𝑓(𝑥) = 2|sen𝑥||cos𝑥|, 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑢𝑚𝑎 𝑒𝑞𝑢𝑎çã𝑜 𝑝𝑎𝑟𝑎 𝑎 𝑟𝑒𝑡𝑎 𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑒 𝑎𝑜 
𝑔𝑟á𝑓𝑖𝑐𝑜 𝑑𝑒 𝑓 𝑛𝑜 𝑝𝑜𝑛𝑡𝑜 𝑒𝑚 𝑞𝑢𝑒 𝑥 =
5𝜋
6
. 
 
𝑄𝑢𝑒𝑠𝑡ã𝑜 4. 
 
𝑎)𝑀𝑜𝑠𝑡𝑟𝑒 𝑞𝑢𝑒 𝑎 𝑓𝑢𝑛çã𝑜 𝑦 = 𝑒2𝑥(𝐴. cos3𝑥 +𝐵. sen3𝑥), 𝑠𝑎𝑡𝑖𝑠𝑓𝑎𝑧 𝑎 𝑒𝑞𝑢𝑎çã𝑜 
𝑦′′ − 4𝑦′+ 13𝑦 = 0. 
 
𝑏)𝐶𝑎𝑙𝑐𝑢𝑙𝑒 lim
𝑥→0
2sen(3𝑥) −1
𝑥
. 
 
𝑄𝑢𝑒𝑠𝑡ã𝑜 5. 
 
𝑎)𝐸𝑛𝑐𝑜𝑛𝑡𝑟𝑒 𝑎𝑠 𝑒𝑞𝑢𝑎çõ𝑒𝑠 𝑑𝑒 𝑎𝑚𝑏𝑎𝑠 𝑎𝑠 𝑟𝑒𝑡𝑎𝑠 𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑒𝑠 à 𝑒𝑙𝑖𝑝𝑠𝑒 𝑥2+ 4𝑦2 = 36 
𝑞𝑢𝑒 𝑝𝑎𝑠𝑠𝑎𝑚 𝑝𝑒𝑙𝑜 𝑝𝑜𝑛𝑡𝑜 (12,3). 
 
𝑏)𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑡𝑜𝑑𝑜𝑠 𝑜𝑠 𝑝𝑜𝑛𝑡𝑜𝑠 𝑜𝑛𝑑𝑒 𝑎 𝑟𝑒𝑡𝑎 𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑒 𝑎𝑜 𝑔𝑟á𝑓𝑖𝑐𝑜 𝑑𝑒 
𝑦 = arcsen(
1
2 + 𝑥2
) é ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙.