Mat Teste Avaliacao

Curso

Turma

No

A

Nome:

Departamento de Economia e Gest˜ ao Teste de Avalia¸c˜ ao — Matem´ atica I

13/01/2007

Assinale com uma cruz a resposta que considera correcta. O teste tem uma dura¸c˜ao de uma hora Cada pergunta vale dois valores. 1. |4 + 5x| ≥ 3 ⇔

5.

(a)

4 + 5x ≥ 3 ou 4 + 5x ≤ −3

4

(b)

4 + 5x ≥ −3 ou 4 + 5x ≤ 3

3

(c)

4 + 5x ≥ 3 e 4 + 5x ≤ −3

(d)

4 + 5x ≥ −3 e 4 + 5x ≤ 3

2 1 −2

2.

−1

4x3 + 3x2 + x − 3 = x→−∞ 1 + 2x2 lim

(a)

−2

(c) (d)

−∞

3. Dada a f.r.v.r. f (x) = x3 + 4, bijectiva, a sua inversa f −1 (x) ´e definida por, (a)

f −1 (x) =

2

−1

3 2

3x2 lim x→−∞ 2x2 0

(b)

1 0

1 x3 +4

√ 3

(b)

f −1 (x) =

(c)

f −1 (x) =

(d)

Nenhuma das anteriores

√ 3

x3 + 4 −4 + x

4. Considere a fun¸c˜ ao  1 2 2 x − 1, f (x) = −1,

Esta representa¸c˜ao gr´afica corresponde fun¸c˜ao ( 4x2 − 1, se x ≤ 12 (a) f (x) = −x + 12 , se x > 12 ( −4x2 − 1, se x ≤ 21 (b) f (x) = x + 12 , se x > 21 ( 4x2 − 1, se x ≤ 12 (c) f (x) = x + 12 , se x > 12 ( −4x2 − 1, se x ≤ 21 (d) f (x) = −x + 12 , se x > 21

se x ≥ 0 se x < 0

A fun¸c˜ ao f (x),

ln(3 − x) x→2 x

6. Calcule o valor do limite lim

(a)

´e cont´ınua no ponto x = 0.

(a)

+∞

(b)

n˜ ao ´e cont´ınua no ponto x = 0.

(b)

1 2

(c)

n˜ ao ´e cont´ınua no ponto x = −1.

(c)

0

(d)

n˜ ao ´e cont´ınua no ponto x = 12 .

(d)

−∞

a `

x2 − 9 x→−3 x + 3

9. Determine o dom´ınio da fun¸c˜ao ex + 4 f (x) = x+4

7. Calcule o valor do limite lim (a)

−6

(b)

−3

(c)

0

(d)

+∞

(a)

Df = R

(b)

Df = R \ {ln(4)}

(c)

Df = R \ {4}

(d)

Df = R \ {−4}

10. Considere as fun¸c˜oes f (x) = x2 + 3 e g(x) = ex . Temos,

8. Sendo Qs = −15 + 5P e Qd = 25 − 3P os pre¸cos e quantidades de equil´ıbrio s˜ ao,

2

(a)

f (g(x)) = ex

(b)

f (g(x)) = e2x + 3

P = 5, Qs = 10 e Qd = 10

(c)

f (g(x)) = (ex + 3)2

P = 5, Qs = 5 e Qd = 5

(d)

f (g(x)) = e(x ) + 3

(a)

P = 10, Qs = 10 e Qd = 10

(b)

Ps =

(c) (d)

25 3

e Pd =

35 4

Teste de Avalia¸c˜ ao — Matem´ atica I

2

+3

2

13/01/2007

A Departamento de Economia e Gest˜ ao Teste de Avalia¸c˜ ao — Matem´ atica I Solu¸c˜ oes

13/01/2007

1. |4 + 5x| ≥ 3 ⇔

5.

(a)

4 + 5x ≥ 3 ou 4 + 5x ≤ −3

4

(b)

4 + 5x ≥ −3 ou 4 + 5x ≤ 3

3

(c)

4 + 5x ≥ 3 e 4 + 5x ≤ −3

(d)

4 + 5x ≥ −3 e 4 + 5x ≤ 3

2 1 −2

−1

1

2

0

2.

4x3 + 3x2 + x − 3 = x→−∞ 1 + 2x2

−1

lim

(a)

−2

3 2

(c)

3x2 lim x→−∞ 2x2 0

(d)

−∞

(b)

Esta representa¸c˜ao gr´afica corresponde fun¸c˜ao ( 4x2 − 1, se x ≤ 12 (a) f (x) = −x + 12 , se x > 12 ( −4x2 − 1, se x ≤ 21 (b) f (x) = x + 12 , se x > 21 ( 4x2 − 1, se x ≤ 12 (c) f (x) = x + 12 , se x > 12 ( −4x2 − 1, se x ≤ 21 (d) f (x) = −x + 12 , se x > 21

3. Dada a f.r.v.r. f (x) = x3 + 4, bijectiva, a sua inversa f −1 (x) ´e definida por, (a)

f −1 (x) =

(b)

f −1 (x) =

(c)

f −1 (x) =

(d)

1 x3 +4

√ 3 √ 3

x3 + 4 −4 + x

ln(3 − x) x→2 x

6. Calcule o valor do limite lim

Nenhuma das anteriores

4. Considere a fun¸c˜ ao  1 2 2 x − 1, f (x) = −1,

(a)

+∞

(b)

1 2

(c)

0

(d)

−∞

se x ≥ 0 se x < 0 x2 − 9 x→−3 x + 3

7. Calcule o valor do limite lim

A fun¸c˜ ao f (x), (a)

´e cont´ınua no ponto x = 0.

(a)

−6

(b)

n˜ ao ´e cont´ınua no ponto x = 0.

(b)

−3

(c)

n˜ ao ´e cont´ınua no ponto x = −1.

(c)

0

(d)

n˜ ao ´e cont´ınua no ponto x = 12 .

(d)

+∞

1

a `

10. Considere as fun¸c˜oes f (x) = x2 + 3 e g(x) = ex . Temos,

8. Sendo Qs = −15 + 5P e Qd = 25 − 3P os pre¸cos e quantidades de equil´ıbrio s˜ ao, (a)

P = 10, Qs = 10 e Qd = 10 25 3

e Pd =

35 4

2

(a)

f (g(x)) = ex

+3

(b)

f (g(x)) = e2x + 3

(b)

Ps =

(c)

P = 5, Qs = 10 e Qd = 10

(c)

f (g(x)) = (ex + 3)2

(d)

P = 5, Qs = 5 e Qd = 5

(d)

f (g(x)) = e(x ) + 3

2

9. Determine o dom´ınio da fun¸c˜ ao ex + 4 f (x) = x+4 (a)

Df = R

(b)

Df = R \ {ln(4)}

(c)

Df = R \ {4}

(d)

Df = R \ {−4}

Teste de Avalia¸c˜ ao — Matem´ atica I

2

13/01/2007

Curso

Turma

No

B

Nome:

Departamento de Economia e Gest˜ ao Teste de Avalia¸c˜ ao — Matem´ atica I

13/01/2007

Assinale com uma cruz a resposta que considera correcta. O teste tem uma dura¸c˜ao de uma hora Cada pergunta vale dois valores. 1. |2 − 6x| ≤ 4 ⇔

5.

(a)

2 − 6x ≥ 4 ou 2 − 6x ≤ −4

4

(b)

2 − 6x ≥ −4 ou 2 − 6x ≤ 4

3

(c)

2 − 6x ≥ 4 e 2 − 6x ≤ −4

(d)

2 − 6x ≥ −4 e 2 − 6x ≤ 4

2 1

2.

−7x3 + 4x2 + 2x − 1 = x→+∞ 1 − 3x2

−2

−1

lim

(a)

+∞

(b)

4x2 lim x→−∞ −3x2

(c)

7 3

(d)

0

f −1 (x) =

1

2

−1 −2

3. Dada a f.r.v.r. f (x) = x5 − 1, bijectiva, a sua inversa f −1 (x) ´e definida por, (a)

0

1 x5 −1

√ 5

(b)

f −1 (x) =

(c)

f −1 (x) =

(d)

Nenhuma das anteriores

√ 5

x5 − 1 x+1

4. Considere a fun¸c˜ ao  2 x − 2, f (x) = −2,

Esta representa¸c˜ao gr´afica corresponde fun¸c˜ao ( 4x2 − 1, se x ≤ 12 (a) f (x) = −x − 12 , se x > 12 ( −4x2 − 1, se x ≤ 21 (b) f (x) = x − 12 , se x > 21 ( 4x2 − 1, se x ≤ 12 (c) f (x) = x − 12 , se x > 12 ( −4x2 − 1, se x ≤ 21 (d) f (x) = −x − 12 , se x > 21

se x ≥ 0 se x < 0

A fun¸c˜ ao f (x),

ex−2 x→2 x

6. Calcule o valor do limite lim

(a)

n˜ ao ´e cont´ınua no ponto x = 0.

(a)

+∞

(b)

´e cont´ınua no ponto x = 0.

(b)

1 2

(c)

n˜ ao ´e cont´ınua no ponto x = −2.

(c)

0

(d)

n˜ ao ´e cont´ınua no ponto x = 1.

(d)

−∞

a `

x2 − 16 x→4 x − 4

9. Determine o dom´ınio da fun¸c˜ao ex + 5 f (x) = x−5

7. Calcule o valor do limite lim (a)

4

(b)

8

(a)

Df = R

(c)

0

(b)

Df = R \ {ln(5)}

+∞

(c)

Df = R \ {5}

(d)

Df = R \ {−5}

(d)

10. Considere as fun¸c˜oes f (x) = 5x2 e g(x) = ln x. Temos,

8. Sendo Qs = −35 + 4P e Qd = 25 − 2P os pre¸cos e quantidades de equil´ıbrio s˜ ao, (a)

P = 10, Qs = 10 e Qd = 10

(a)

g(f (x)) = ln(5) + ln(x2 )

(b)

P = 10, Qs = 5 e Qd = 5

(b)

g(f (x)) = ln(5 + x2 )

(c)

P = 5, Qs = 5 e Qd = 5

(c)

g(f (x)) = (5 ln x)2

(d)

Ps =

(d)

g(f (x)) = 5 ln(x2 )

35 4

e Pd =

25 2

Teste de Avalia¸c˜ ao — Matem´ atica I

2

13/01/2007

B Departamento de Economia e Gest˜ ao Teste de Avalia¸c˜ ao — Matem´ atica I Solu¸c˜ oes

13/01/2007

1. |2 − 6x| ≤ 4 ⇔

5.

(a)

2 − 6x ≥ 4 ou 2 − 6x ≤ −4

4

(b)

2 − 6x ≥ −4 ou 2 − 6x ≤ 4

3

(c)

2 − 6x ≥ 4 e 2 − 6x ≤ −4

(d)

2 − 6x ≥ −4 e 2 − 6x ≤ 4

2 1

−2

2.

−1

−7x3 + 4x2 + 2x − 1 = x→+∞ 1 − 3x2

0

1

2

−1

lim

(a)

−2

+∞

Esta representa¸c˜ao gr´afica corresponde fun¸c˜ao ( 4x2 − 1, se x ≤ 12 (a) f (x) = −x − 12 , se x > 12 ( −4x2 − 1, se x ≤ 21 (b) f (x) = x − 12 , se x > 21 ( 4x2 − 1, se x ≤ 12 (c) f (x) = x − 12 , se x > 12 ( −4x2 − 1, se x ≤ 21 (d) f (x) = −x − 12 , se x > 21

2

(b)

lim

x→−∞

(c)

7 3

(d)

0

4x −3x2

3. Dada a f.r.v.r. f (x) = x5 − 1, bijectiva, a sua inversa f −1 (x) ´e definida por, (a)

f −1 (x) =

(b)

f −1 (x) =

(c)

f −1 (x) =

(d)

1 x5 −1

√ 5 √ 5

x5 − 1 x+1

ex−2 x→2 x

6. Calcule o valor do limite lim

Nenhuma das anteriores

4. Considere a fun¸c˜ ao  2 x − 2, f (x) = −2,

(a)

+∞

(b)

1 2

(c)

0

(d)

−∞

se x ≥ 0 se x < 0 x2 − 16 x→4 x − 4

7. Calcule o valor do limite lim

A fun¸c˜ ao f (x), (a)

n˜ ao ´e cont´ınua no ponto x = 0.

(a)

4

(b)

´e cont´ınua no ponto x = 0.

(b)

8

(c)

n˜ ao ´e cont´ınua no ponto x = −2.

(c)

0

(d)

n˜ ao ´e cont´ınua no ponto x = 1.

(d)

+∞

1

a `

10. Considere as fun¸c˜oes f (x) = 5x2 e g(x) = ln x. Temos,

8. Sendo Qs = −35 + 4P e Qd = 25 − 2P os pre¸cos e quantidades de equil´ıbrio s˜ ao, (a)

P = 10, Qs = 10 e Qd = 10

(a)

g(f (x)) = ln(5) + ln(x2 )

(b)

P = 10, Qs = 5 e Qd = 5

(b)

g(f (x)) = ln(5 + x2 )

(c)

P = 5, Qs = 5 e Qd = 5

(c)

g(f (x)) = (5 ln x)2

(d)

Ps =

(d)

g(f (x)) = 5 ln(x2 )

35 4

e Pd =

25 2

9. Determine o dom´ınio da fun¸c˜ ao ex + 5 f (x) = x−5 (a)

Df = R

(b)

Df = R \ {ln(5)}

(c)

Df = R \ {5}

(d)

Df = R \ {−5}

Teste de Avalia¸c˜ ao — Matem´ atica I

2

13/01/2007