Lista 1

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INSTITUTO FEDERAL CATARINENSE
Professora: Katielle de M. Bilhan
Disciplina: Ca´lculo I
LISTA DE EXERCI´CIOS 1
1. Marque verdadeiro ou falso, justificando as alternativas falsas:
( ) −2 ∈ [−2, 2]
( ) −2 ∈ (−2, 2]
( ) 3 ∈ (−2, 2)
( )
√
7 /∈ [−2, 2]
( ) pi ∈ [−3; 3, 14]
( ) −7 ∈ [−2,∞]
( ) 12 /∈ [0, 1)
( ) 2 ⊂ [−3; 3, 14]
( ) [0, 1] ⊂ [−2,∞]
( ) 12 ⊂ [0, 1)
2. Determinar todos os intervalos de nu´meros reais que satisfac¸am as desigualdades abaixo.
(a) 3− x < 5 + 3x (b) 2x− 5 < 1
3
+
3x
4
+
1− x
3
(c) 2 > −3− 3x ≥ −7
(d)
5
x
<
3
4
(e) x2 ≤ 9 (f) x2 − 3x+ 2 > 0
(g) 1− x− 2x2 ≥ 0 (h) x3 + 1 > x2 + x (i) 2
x− 2 ≤
x+ 2
x− 2 ≤ 1
(j) x4 ≥ x2 (k) x
x− 3 < 4 (l)
1
2x− 3
4 + x
> 1
(m)
3
x− 5 ≤ 2 (n) x
3 − 3x+ 2 ≤ 0 (o) x2 + x− 2 ≥ 0.
(p)
x+ 2
10
− 1 ≤ 1− x
4
(q)
1
2
(x+
1
3
)− 1 > −1
3
(x− 1
2
) (r)
x+ 1
2
< 5 + x ≤ 2x− 1
4
(s)
{
3−x
2 < 1 +
5−2x
3
2x−3
4 <
x+5
6
(t)
{
4x−5
3 − 3x+15 < −x
2+7x
5 − 3x2 > 0
(u) 4 ≥ 2x− 5
3
≥ −2
3. Resolver as inequac¸o˜es modulares em R.
(a) |x+ 12| < 7 (b) |3x− 4| ≤ 12 (c) |2x− 5| > 3
(d) 1 < |x+ 2| < 4 (e) |14x− 6| ≤ 4 (f) |6x| > 2
(g) | − x+ 12| ≥ 3 (h) |3x| > |5− 2x| (i) 1|x+1||x−3| ≥ 15
(j)
∣∣∣7−2x5+3x ∣∣∣ ≤ 12 (k) 1 < |x+ 2| < 4 (l) |x|+ 1 < x
GABARITO
2.
1
(a) (−1
2
,∞+) (b) (−∞, 68
19
) (c) (−5
3
, 4
3
] (d) (−∞, 0) ∪ [ 20
3
,∞+) (e) [−3, 3]
(f) (−∞, 1) ∪ (2,∞+) (g) [−1, 1
2
] (h)(−1, 1) ∪ (1,∞+) (i) (−∞, 0) (j) (−∞,−1] ∪ [1,∞+)
(k) (−∞, 3) ∪ (4,∞+) (l) (−14,−4) (m) (−∞, 5) ∪ [ 13
2
,∞+) (n) (−∞,−2] (o) (−∞,−2] ∪ [1,∞+)
(p) (−∞, 3] (q) ( 6
5
,∞+) (r) ∅ (s) (−∞, 19
9
) (t) (−∞, 14
13
)
(u) [− 1
2
, 17
2
]
3.
(a) (−19,−5) (b) [−8/3, 16/3] (c) (−∞, 1) ∪ (4,+∞) (d) (−1, 2) (e) [1/7, 5/7]
(f) (−∞,−1/3) ∪ (3,+∞) (g) (−∞,−1/4] ∪ [15,+∞) (h) (−∞,−5) ∪ (1,+∞) (i) [−2,−1) ∪ (1, 3) ∪ (3, 4] (j) [9/7, 19]
(k) (−6,−3) ∪ (−1, 2) (l) ∅
2