Taller Resuelto Deri y Opti

1.

𝐴 = 𝜋𝑟2

𝑑𝑟

𝑑𝑡 = 2 𝐼𝑛/𝑠𝑒𝑔[pic 1]

𝑑𝐴

𝑑𝑡 =?        𝑟 = 8𝑖𝑛[pic 2]

𝑑𝐴        𝑑𝑟

𝑑𝑡 = 𝜋2𝑟 ∗ 𝑑𝑡[pic 3][pic 4]

𝑑𝐴[pic 5]

𝑑𝑡

= 2 ∗ 𝜋 ∗ (8𝑖𝑛) ∗ (2 𝐼𝑛/𝑠𝑒𝑔 )

𝑑𝐴 = 32𝜋 = 100.530 𝐼𝑛2/𝑠𝑒𝑔[pic 6]

𝑑𝑡

2.

𝑑𝑣 = 9.5 𝐼𝑛3[pic 7]

𝑑𝑡

𝑑𝑟

𝑑𝑡 = 3 𝐼𝑛[pic 8]

𝑉 = 4 ∗ 𝜋𝑟3[pic 9]

3

𝑑𝑣 = 4 ∗ 𝜋 ∗ 3𝑟2 ∗ 𝑑𝑟[pic 10][pic 11][pic 12]

𝑑𝑡        3        𝑑𝑡

𝑑𝑟 = 4𝜋𝑟2 ∗ 𝑑𝑟[pic 13][pic 14]

𝑑𝑡

𝑑𝑟

= ([pic 15][pic 16]

𝑑𝑣

𝑑𝑡

𝑑𝑡

𝑑𝑟        9.5

) =        =[pic 17][pic 18]

𝑑𝑡

4𝜋𝑟2

𝑑𝑡        4𝜋(3)2

𝑑𝑟        9.5

𝑑𝑡 = 36𝜋 𝐼𝑛/𝑚𝑖𝑛 = 0.0839 𝐼𝑛/𝑚𝑖𝑛[pic 19][pic 20]

3.

1. Y= 3x2*secx

𝑦′ = 𝑑[pic 21]

𝑑𝑥

𝑦′ =  𝑑[pic 22]

𝑑𝑥

(3𝑥2 ∗ sec(𝑥))

(3𝑥2) ∗ sec(𝑥) + 3𝑥2 ∗ 𝑑[pic 23]

𝑑𝑥

(sec(𝑥))

𝑦′ = 3 ∗ 2𝑥 ∗ sec(𝑥) + 3𝑥2 + tan(𝑥) sec(𝑥)[pic 24]

𝑦′ = 3 ∗ 2𝑥 ∗        1[pic 25][pic 26]

+ 3𝑥2 ∗ sin(𝑥) ∗        1

𝑦′ =

6𝑥

[pic 27]

cos(𝑥)

cos(𝑥)

3𝑥2 ∗ sin(𝑥)

+        cos(𝑥)2[pic 28]

cos(𝑥)

cos(𝑥)

𝑦′ =[pic 29]

6𝑥 ∗ cos(𝑥) + 3𝑥2 ∗ sin(𝑥) cos(𝑥)2

1. Y= 2x5 * cosx

𝑦′ = 𝑑[pic 30]

𝑑𝑥

𝑦′ =  𝑑[pic 31]

𝑑𝑥

(2𝑥5 ∗ cos(𝑥))

(2𝑥5) ∗ cos(𝑥) + 2𝑥5 ∗ 𝑑[pic 32]

𝑑𝑥

(cos(𝑥))

𝑦′ = 2 ∗ 5𝑥4 ∗ cos(𝑥) + 2𝑥5 ∗ (−sin(𝑥))

𝑦′ = 10𝑥4 ∗ cos(𝑥) − 2𝑥5 ∗ sin(𝑥)

c) Y= 3(2x4 + x3)5

𝑦′ = 3 ( 𝑑[pic 33]

𝑑𝑥

(2𝑥4 + 𝑥3)5)

𝑦′ = 3 ( 𝑑[pic 34]

𝑑𝑢

(𝑢5) ∗ 𝑑

𝑑𝑥[pic 35]

(2𝑥4 + 𝑥3))

𝑦′ = 3 (5𝑢4 ∗ 𝑑[pic 36]

𝑑𝑥

(2𝑥4 + 𝑥3))

𝑦′ = 3 (5(2𝑥4 + 𝑥3)4 ∗ 𝑑[pic 37]

𝑑𝑥

(2𝑥4 + 𝑥3))

𝑦′ = 15 ((2𝑥4 + 𝑥3)4 ∗ 𝑑[pic 38]

𝑑𝑥

(2𝑥4 + 𝑥3))

𝑦′ = 15(2𝑥4 + 𝑥3)4 ∗ ( 𝑑[pic 39]

𝑑𝑥

(2𝑥4) + 𝑑

𝑑𝑥[pic 40]

(𝑥3))

𝑦′ = 15(2𝑥4 + 𝑥3)4 ∗ (2 𝑑[pic 41]

𝑑𝑥

(𝑥4) + 𝑑

𝑑𝑥[pic 42]

(𝑥3))

𝑦′ = 15(2𝑥4 + 𝑥3)4 ∗ (2 𝑑[pic 43]

𝑑𝑥

(4𝑥3) + 𝑑

𝑑𝑥[pic 44]

(𝑥3))

𝑦′ = 15(2𝑥4 + 𝑥3)4 ∗ (8𝑥3 + 𝑑[pic 45]

𝑑𝑥

(𝑥3))

𝑦′ = 15(2𝑥4 + 𝑥3)4 ∗ (8𝑥3 + 3𝑥2)

[pic 46]

Y= √1−𝑥2

(𝑥+1)3[pic 47][pic 48][pic 49][pic 50]

1

(1−𝑥2)2

[pic 51]

2

(𝑥+1)3[pic 52]

...