discursiva 01

Prévia do material em texto

1. Determine o plano tangente ao gráfico z = x² + y² , no ponto (1,2,3) 
𝑑𝑧
𝑑𝑥
= 2𝑥 𝑒 
𝑑𝑧
𝑑𝑦
 = 2𝑦 ∴ 𝑧𝑥 = 2 𝑒 𝑧𝑦 = 4 
 
𝑧 − 3 = 2(𝑥 − 1) + 4(𝑦 − 2) ∴ 𝑧 − 3 = 2𝑥 + 4𝑦 − 3 ∴ 𝑧 = 2𝑥 + 4𝑦 
 
 
2. Determine as derivadas de primeira ordem da função: 
𝑓(𝑥, 𝑦) = 𝑧 = 𝑥𝑦 − 4𝑥𝑦³ 
𝑧𝑥 = 𝑦.𝑥𝑦−1−4𝑦3 𝑒 𝑍𝑦 = 12𝑦²𝑥 
 
3. Calculando a integral a seguir, obtém-se: 
∫ ∫ 2𝑥 + 2𝑦 𝑑𝑦𝑑𝑥
𝑥
𝑥²
4
2
 
 
 
4. Determine e classifique os extremos da função Z = x²y + 24x² - 4y 
 
𝑑𝑧
𝑑𝑥
= 2𝑥𝑦 + 48𝑥 𝑒 
𝑑𝑧
𝑑𝑦
= 𝑥2 − 4 
{
2𝑥𝑦 + 48𝑥
𝑥2 − 4
 ∴ 𝑥 = ±2 , 𝑦 = −24 𝑙𝑜𝑔𝑜 𝑃1(2, −21) 𝑒 𝑃2(−2, −21) 𝑝𝑜𝑛𝑡𝑜𝑠 𝑐𝑟𝑖𝑡𝑖𝑐𝑜𝑠 
 
𝑑
𝑑𝑥𝑥
= 2𝑦 + 48 , 
𝑑
𝑑𝑥𝑦
= 2𝑥 , 
𝑑
𝑑𝑦𝑦
= 0 , 
𝑑
𝑑𝑦𝑥
= 2𝑥 
𝐻 = 
𝑧𝑥𝑥 𝑧𝑥𝑦
𝑧𝑦𝑥 𝑧𝑦𝑦
 ∴ 
2𝑦 + 48 2𝑥
2𝑥 0
→ 𝐻 = 0 − 4𝑥2 
|𝑝1( 2,−21) 𝐻 = −16 ∴ 𝐻 < 0 ∴ 𝑃1(2, −21) é 𝑝𝑜𝑛𝑡𝑜 𝑑𝑒 𝑠𝑒𝑙𝑎 
|𝑝2(−2,−21) 𝐻 = −16 ∴ 𝐻 < 0 ∴ 𝑃2(2, −21) é 𝑝𝑜𝑛𝑡𝑜 𝑑𝑒 𝑠𝑒𝑙𝑎 
 
 
 
5. Calcule lim
𝑥 →1
𝑦→1
(
𝑥2−𝑦²
𝑥3−𝑥³𝑦
) 
 
6. Considere as funções vetoriais: 
 
𝑓(𝑡) = (4𝑡 + 2)𝑖 + (𝑡2 + 7)𝑗 − (𝑡 + 5)𝑘 
𝑓(𝑔) = (2𝑡 + 2)𝑖 − (5𝑡4 + 2𝑡2)𝑗 + (𝑡 + 9)𝑘 
Qual o resultado do produto escalar f(t) e f(g)? 
 
𝑓(𝑡) × 𝑓(𝑔) = ( (4𝑡 + 2)𝑖 + (𝑡2 + 7)𝑗 − (𝑡 + 5)𝑘 ) × ((2𝑡 + 2)𝑖 − (5𝑡4 + 2𝑡2)𝑗 + (𝑡 + 9)𝑘) 
= ((8𝑡2 + 8𝑡 + 4𝑡 + 4)𝑖 + (−5𝑡6 − 2𝑡3 − 35𝑡5 − 14𝑡2)𝑗 + (−𝑡2 − 9𝑡 − 5𝑡 − 45)𝑘) 
= −7𝑡2 − 𝑡 − 41 − 5𝑡6 − 2𝑡3 − 35𝑡5 ∴ 
= −5𝑡6 − 35𝑡5 − 2𝑡3 − 7𝑡2 − 𝑡 − 41 
 
 
 
 
 
 
7. 
 
𝑑𝑖𝑣𝑓 = 25𝑥4𝑦3𝑖 − 24𝑥𝑦2𝑧
5
𝑗 + 32𝑧 𝑘 
 
 
8. 
 
𝑑𝑧
𝑑𝑥
= 2𝑥 + 3 𝑒 
𝑑𝑧
𝑑𝑦
= 4𝑦 − 4 
{
2𝑥 + 3
4𝑦 − 4
 ∴ 𝑥 = −
3
2
 , 𝑦 = 1 𝑙𝑜𝑔𝑜 𝑃1 (−
3
2
 ,1) 𝑝𝑜𝑛𝑡𝑜𝑠 𝑐𝑟𝑖𝑡𝑖𝑐𝑜𝑠 
 
𝑑
𝑑𝑥𝑥
= 2, 
𝑑
𝑑𝑥𝑦
= 0 , 
𝑑
𝑑𝑦𝑦
= 4 , 
𝑑
𝑑𝑦𝑥
= 0 
𝐻 = 
𝑧𝑥𝑥 𝑧𝑥𝑦
𝑧𝑦𝑥 𝑧𝑦𝑦
 ∴ 
2 0
0 4
→ 𝐻 = 8 
|𝑝
1( −
3
2
 ,1) 
 𝐻 = 8 ∴ 𝐻 = 8 ∴ 𝑃1 (−
3
2
 ,1) 
 
 
 
9. 
 
 
10. 
 
 
 
11. 
 
𝑊𝑥 = 30𝑥
5 + 2𝑥𝑦2 ∴ 𝑊𝑥𝑥 = 150𝑥
4 + 2𝑦² 
𝑊𝑦 = 20𝑦
3 + 2𝑦𝑥² ∴ 𝑊𝑥𝑥 = 60𝑦
2 + 2𝑥² 
 
 
12. 
 
 
 
𝜕𝑧
𝜕𝑦
= 𝑤𝑦 = −
𝐹𝑦
𝐹𝑧
= −
5𝑧 + 4𝑥
3𝑧2𝑥3 + 5𝑦
 
 
 
 
 
13. 
 
𝐺𝑓 = 8𝑥 𝑖 + 8𝑧 𝑗 + (8𝑦 + 3)𝑘 ∴ 𝑃(2,2,2) → 8 × 2 𝑖 + 8 × 2 𝑗 + (8 × 2 + 5)𝑘 
𝐺𝑓 = 16𝑖 + 16𝑗 + 21𝑘 ∴ 𝑢 =
1𝑖+2𝑗+2𝑘 
√12+22+2²
 → 
1𝑖+2𝑗+2𝑘 
3
 
(16𝑖 + 16𝑗 + 21𝑘) × 
1𝑖 + 2𝑗 + 2𝑘 
3
 ∴ 
16 + 32 + 42
3
 
= 
90
3
 
∆
→ 𝑓𝑢 = 30 
 
 
 
14. 
 
𝑧 = 𝑥
1
2 × 𝑦
1
2 ∴ 
𝑑𝑧
𝑑𝑥
= 
1
2
 𝑥−
1
2 𝑦
1
2 → 
1
2
√
𝑥
𝑦
 ∴ 𝑍𝑥𝑝 = 
1
2
 
𝑧 = 𝑥
1
2 × 𝑦
1
2 ∴ 
𝑑𝑧
𝑑𝑥
= 
1
2
 𝑦−
1
2 𝑥
1
2 → 
1
2
√
𝑥
𝑦
 ∴ 𝑍𝑦𝑝 = 
1
2
 
 
𝑧 − 4 = 
1
2
 . (𝑥 − 2) + 
1
2
. (𝑦 − 2) ∴ 2𝑧 − 8 = 𝑥 + 𝑦 − 4 → 2𝑧 = 𝑥 + 𝑦 + 4 
 
 
15. 
 
 
𝑑𝑤
𝑑𝑥
 × 
𝑑𝑥
𝑑𝑡
+
𝑑𝑤
𝑑𝑦
 × 
𝑑𝑦
𝑑𝑡
+
𝑑𝑤
𝑑𝑧
 × 
𝑑𝑧
𝑑𝑡
 
(12𝑥 × 1) + (4𝑦 × 2) + (10𝑧 × (−𝑢)) ∴ 12𝑥 + 8𝑦 − 10𝑧𝑢 
12𝑡 + 16𝑡 − 10𝑢(𝑡 − 𝑢) ∴ 10𝑢2 − 10𝑢𝑡 + 28𝑡