C LculoI Lista2

C´ alculo Diferencial e Integral I - Engenharia de Produ¸c˜ ao Professor Em´ılio
Lista de Exerc´ıcios 2
1. Calcule
√
(a) f (0), f (2), f ( 2), onde f (x) =

x x2 − 1

Resp.: 0, 2/3,

√

2

f (a + b) − f (a − b)
, onde f (x) = x2 e ab = 0 Resp.: 4 ab f (a + b) − f (a − b)
, onde f (x) = 3x + 1 e ab = 0 Resp.:
(c)
ab
(b)

6/a

f (x + h) − f (x) h (a) f (x) = 2x + 1 Resp.: 2

(b) f (x) = −2x + 4

Resp.: −2

(c) f (x) = x2

Resp.: 2x + h

(d) f (x) = −x2 + 5

Resp.: −2x + h

(e) f (x) = x3

Resp.: 3x2 + 3x + h2

(f) f (x) = 1/x

3. Simplifique

Resp.: −1/(x2 + xh)

3. Determine o dom´ınio das seguintes fun¸c˜oes. x x−1
Resp.: Df = R − {−1, 1}
(b) f (x) =
Resp.: Df = (−∞, −1) ∪ [1, ∞)
(a) f (x) = 2 x −1 x+1 √
2x − 1
(c) f (x) =
Resp.: Df = (1/3, 1/2]
(d) f (x) = 5 − 2x2 Resp.: Df = [− 5/2, 5/2]
1 − 3x
√
√ x+1 (e) f (x) = x− 5 − 2x Resp.: Df = [0, 5/2]
(f) f (x) = 2
Resp.: Df = {x ∈ R|x = 0, x = 1} x +x
√
(g) f (x) = x2 + 9 Resp.: Df = R
(h) f (x) = 1 − (x + 2)2 Resp.: Df = [−3, −1]
√
(i) f (x) = x − x Resp.: Df = [1, +∞)
√
1
√
4. Verifique que 1 + x2 − |x| =
. Conclua que a` medida que |x| cresce
2
|x|
+
1
+
x
√
1 + x2 − |x| se aproxima de 0.
√
5. Dˆe o dom´ınio e esboce o gr´afico de f (x) = x2 − 1. Resp.: Df = (−∞, −1] ∪ [1, +∞)
6. Considere a fun¸c˜ao f (x) = m´ax{x, 1/x}.
(a) Calcule f (2), f (−1) e f (1/2).

Resp.: 2, −1, 2

(b) Determine o dom´ınio de f e esboce seu gr´afico.

1

Resp.: Df = R∗

7. Considere a fun¸c˜ao f (x) = m´ax{n ∈ Z | n ≤ x}. (Fun¸c˜ao maior inteiro.)
(a) Calcule f (1/2), f (1), f (5/4) e f (−7/2).

Resp.: 0, 1, 1, −4

(b) Determine o dom´ınio de f e esboce seu gr´afico.
8. Esboce o gr´afico das seguintes fun¸co˜es.

 x se x ≤ 2
(a) f (x) =
 3 se x > 2

Resp.: Df = R

(b) f (x) =




2x

se x ≤ −1

 −x + 1 se x > −1

(c) g(x) = |x − 1|

(d) g(x) =

|x| x (g) y(x) = −x3

 x2 se x ≤ 1
(i) t(x) =
 2 − (x − 2)2 se x > 1

x2 − 1 x−1 (f) h(x) = |x| − 1

(e) h(x) =

(h) y(x) = x4

2