Ficha 2 - Fun ções Exponencial e Logar ítmica PDF

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Escola Superior de Tecnologia e Gest˜ao Departamento de Matem´atica Matem´atica Geral B 1.o Semestre

2020/2021

Fun¸c˜oes Exponencial e Logar´ıtmica Fun¸c˜ ao Exponencial 1. Complete os espa¸cos com a informa¸c˜ao solicitada. Fun¸c˜ ao exponencial: x 7→ ax , onde a ´e uma constante positiva diferente de 1. a>1

0 0 ⇔ x ∈ • f (x) = 1 ⇔ x =

e f (x) < 0 ⇔ x ∈ ; lim

x→−∞

f (x) =

2. Escreva como potˆencia de expoente natural.  −2  −3 3 1 (a) 3−1 (b) (c) 4 2 3. Escreva na forma de raiz. (a)

1 32

(b)

3 24

 − 1 2 1 (c) 2

4. Escreva como potˆencia de base natural. s  r −2 1 1 3 (a) (b) 4 3

e lim f (x) = x→+∞

; .

5. Escreva: (a) 16 como potˆencia de base 4; 1 ; 2

(b) 8 como potˆencia de base

(c) 1 como potˆencia de base −2; (d) 0.01 como potˆencia de base 10; 6. Sabendo que 4x = 6, determine: (a) 4x+1

(b) 4x−2

(c) 42x

(d)

4−x

(e) 2x

7. Resolva as seguintes equa¸c˜oes. (a)

2x+1 = 16

(d) 4x+1 − 4x = 24

(b) 8x+1 = 4x

(e)

(c) 5 × 5x =

√ 5

2x+3 − x21+x = 0

Fun¸c˜ ao logar´ıtmica x 7→ loga x, onde a > 1 constante.

8. Complete os espa¸cos com a informa¸c˜ao solicitada. Para f (x) = loga x, com a > 1, tem-se: • Df =

e Df′ =

• f (x) > 0 ⇔ x ∈ f (x) < 0 ⇔ x ∈

; e

;

• f ´e estritamente crescente em estritamente decrescente em • f (x) = 0 ⇔ x = • lim+ f (x) = x→0

9. Calcule o valor de: (a) log2 8

(b) log2 64

(c) log5 25

(d) log4 16

(e) log3 27

(f) log7 1

  (g) ln e5

(h) log 10000

(i) log 0.01

e ;

; e lim

x→+∞

f (x) =

.

10. Escreva as express˜oes seguintes na forma de logaritmo. (a) log3 6 + log 3 2

(b) log2 9 − log 2 3 log 5 + log 3 (e) 2

(d) 1 + log 3 4

(c) 5 × log 2

11. Determine o valor exato de: (a) 2log2 (7)   (d) log4 210 (g) log4

 √ 3 2

(b) 51+log 5 3   1 (e) ln e3

(c) 32 log3 4

(h) log4 8 + log 4 2

(i) log

(f) log√2 1 √  √  20 − log 2

12. Determine o dom´ınio de cada uma das fun¸c˜oes f definidas por: (a) f (x) = log3 (x + 2)   (b) f (x) = log3 x2 + 2

(c) f (x) = log3 (|x − 3|)   (d) f (x) = log5 4 − x2

(e) f (x) = ln (x + 2) + ln (x + 1)   x+1 (f) f (x) = ln x−2

13. Resolva, em R, as seguintes equa¸c˜oes: (a) 2x+1 + 1 = 9

(b) 3 − e2 x = 1

(c) 2x = −4

(d) log2 x = 3

(e) ln(x) = −1

(f) ln(x + 1) = 2

(g) 2 ln(x + 2) + 2 = 3

(h) ln(x) + ln(3) = ln(9)

(i) ln2 (x) − 3 ln(x) = 0

(j) ln(x2 + 1) ln(3 x − 4) = 0

(k) log3 (x + 2) = 1 − log 3 x

(l) log5 (x + 3) = log5 (x − 1) + log 5 (2).

14. Considere a fun¸c˜ao h, real de vari´avel real, definida por:    − ln(2 − x) + 2 x se x < 1 . h (x) =   e1−x + 1 se x ≥ 1 x (a) Calcule:

i. h (0);

ii. h (2);

iii. h (1) .

(b) Determine o valor dos seguintes limites:

i. lim h (x); x→+∞

ii. lim h (x). x→−∞

(c) Estude a continuidade da fun¸c˜ao h em x = 1.

15. Considere as fun¸c˜oes reais de vari´avel real definidas por f (x) = 3−2 ex/2 e g(x) = 3+log2 (x−1). Caracterize as fun¸c˜oes inversas de f e de g (dom´ınio, contradom´ınio e express˜ao anal´ıtica).

Solu¸co ˜es 1. Df = R; D′f = R+ ; f (x) > 0 ⇐⇒ x ∈ R; f (x) = 1 ⇐⇒ x = 0; lim f (x) = 0; x→−∞

lim f (x) = +∞

x→+∞

2. -

 1 1 (a) 3

 2 4 (b) 3

(c) 23

3. (a)

√ 3

√ 4 8

(b)

√ 2

(c)

4. 2

1

(a) 3− 2

(b) 4 3

5. (a) 42

(b)

 −3 1 2

(a) 24

(b)

3 8

0

(d) 10−2

(c) (−2)

6. (c) 36

1 6

(d)

7. (a)

x=3

(d) x =

3 2

(b) x = −3 (e)

(c) x = −

(e)

√ 6

1 2

x=4

8. Df = R+ ; D′f = R; f (x) > 0 ⇐⇒ x ∈ ]1, +∞[ e f (x) < 0 ⇐⇒ x ∈ ]0, 1[ ; f ´e estritamente crescente em R+ ; f (x) = 0 ⇐⇒ x = 1; lim+ f (x) = −∞; lim f (x) = +∞ x→+∞

x→0

9. (a) 3

(b) 6

(c) 2

(d) 2

(e) 3

(f) 0

(g) 5

(h) 4

(i) − 2

(a) log3 12

(b) log 2 3

10. (c) log 32

(d) log3 12

√ (e) log(3 5)

11. (a) 7

(b) 15

(c)

16

(d)

5

(f)

(g)

1 6

(h)

2

(i)

1 2

0

(e) −3

12. (a) ]−2, +∞[ (b) R

(c) R\ {3}

(d) ]−2, 2[ (e) ]−1, +∞[ (f) ]−∞, −1[ ∪ ]2, +∞[

13.

14.

 √  (b) CS = ln 2 ;   (f) CS = e2 − 1 ;

(a) CS = {2};   1 ; (e) CS = e   (i) CS = 1, e3 ;

  5 (j) CS = ; 3

  1 ; 2

(a)

i. h (0) = ln

(b)

i. lim h (x) = 0;

ii. h (2) =

(c) CS = ∅; √ (g) CS = { e − 2};

(d) CS = {8};

(k) CS = {1};

(l) CS = {5}.

1 1 + ; e 2

iii. h (1) = e0 + 1 = 2.

ii. lim h (x) = −∞.

x→+∞

x→−∞

(c) A fun¸c˜ao h ´e cont´ınua em x = 1. 15. f −1 : ] − ∞, 3[ → x

7→

;

R 2 ln



3−x 2



(h) CS = {3};

g −1 : R

→

]1, +∞[

x

7→

1 + 2x−3...