Mat Sol Revisao

  • November 2019
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Resolu¸ c˜ ao 3.1 a) Df = R e f −1 (x) = ln( x5 ) − 2 3x b) Df = R \ {−3} e f −1 (x) = − x−2

c) Df = R \ {1} e f −1 (x) = 3.2

x+5 x−1

P = 15 e Q = 30.

3.3 a) S = 34 b) P = 12 3.4

Existe lucro quando 7 < Q < 25.

3.5 a) +∞

b) +∞

c) −∞

d) − 23

e) 32

f) −∞

g)

1 6

h)

1 2

j) e1/3

i) 2 3.6

p=2

3.7

p=4

3.8

X = A − 2B T

3.9

X=A

3.10

a) rA = 3 

13

 AB =  −14

15 −7

b) rB = 3  −3 −8  6 9 

−58 −138 −6 3.11

a=3

3.12

a=6

20

1

3.13 a) x = 3, y = −1 e z = 1 b) x = 4, y = 2 − t e z = 2, com um parˆametro livre t ∈ R, c) Sistema Imposs´ıvel 3.14 a)

a 6= −3 : Sistema Poss´ıvel e Determinado a = −3 e b 6= a = −3 e b =

b)

3 ıvel 2 : Sistema Imposs´ 3 ıvel e Indeterminado 2 : Sistema Poss´

a 6= −10 : Sistema Poss´ıvel e Determinado a = −10 e b 6= −4 : Sistema Imposs´ıvel a = −10 e b = −4: Sistema Poss´ıvel e Indeterminado

3.15

A−1

 −3  2 =  6 4

3.16

1

2



 −3 −8  −2 −5

A inversa da matriz dos coeficientes do sistema AX = B ´e, 

−1

A



9 17 −19    = −4 −7 8   −5 −9

10

e a solu¸c˜ ao ´e ent˜ ao,       x 2 −3       −1 −1      X = y  = A B = A 1 =  1   z

2

1

3.17 a) det A = −40 b) det B = 0 c) det C = 30 As linhas e colunas de A e C s˜ao linearmente independentes porque o determinante ´e diferente de zero. As linhas e colunas de B s˜ao linearmente dependentes porque o determinante ´e nulo.

2

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