Listas de Cálculo Diferencial e Integral 1 - 2015.1 UERJ física

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IME 508 - CDI I T12 (2015-1)
UFRRJ - UNIVERSIDADE ESTADUAL DO RIO DE JANEIRO
IME - INSTITUTO DE MATEMA´TICA E ESTATI´STICA
ANMAT - DEPARTAMENTO DE ANA´LISE MATEMA´TICA
Prof.a Cristiane Oliveira de Faria
LISTA 1 DE CA´LCULO DIFERENCIAL E INTEGRAL I
Revisa˜o: inequac¸o˜es, raizes e mo´dulos
1. Escreva usando a notac¸a˜o cient´ıfica:
(a) 100.000.000 (c) (0, 00000009)2 (e) 2× 1016 − 0, 3× 1015
(b) 0, 0000021× 8.000 (d) √0, 00000002× 0, 0002 (f)
(
1.600× 20.000
4.000
)1/3
2. Simplifique:
(a)
(62)3/4 × 2−1/2
33/2
(b)
(
√
2)7/2 × 4√2
21−pi
(c)
3
√
3 × 62−
√
3
21−
√
3
(d)
22x × 10x
20x
(e)
(2pi)1−x × 4x
pi × 2x−4 (f)
x2(x+1)
(2x)x + x2x
3. Resolva as inequac¸o˜es:
(a) 3− 4x ≥ x+ 2 (b) 2x2 − x− 10 > 0 (c) −3x2 + 7x− 6 > 0
(d) x2 + x < x3 + 1 (e)
2x− 1
1− x ≤ 0 (f)
x
2x− 3 ≤ 3
(g) x+
1
x
≥ 2 (h) 1
x− 2 ≥
5
2x− 1 (i)
x+ 2
2− x <
x2 − 4
2x− 1
4. Resolva para x e represente a soluc¸a˜o na reta nume´rica
(a) |x− 3| = 6 (b) |3x− 1| = |x− 5| (c) |2x+ 3| = 2x+ 3
(d) |3 + 2x| ≤ 2 (e) |2x+ 5| > 3 (f) |3− 4x| > x+ 2
(g)
x+ 1
|x2 − 1| ≥ x−
1
x− 1 (h)
x+ 1
|x| +
1
|x2 − x| ≥
|x|
3
RESPOSTAS
1.(a) 108 (b) 168× 10−4 (c) 81× 10−16 (d) 2× 10−6 (e) 19, 7× 1015 (f) 20
2.(a) 2 (b) 21+pi (c) 18 (d) 2x (e) 25 × pi−x (f) x
x+2
2x + xx
3.(a) (−∞, 1/5] (b) (−∞,−2) ∪ (5/2,∞) (c) ∅
(d) (−1, 1) ∪ (1,∞) (e) (−∞, 1/2] ∪ (1,∞) (f) (−∞, 3/2) ∪ [5/3,∞)
(g) (0,∞) (h)(−∞, 1/2) ∪ (2, 3] (i) [−2, 1−√6] ∪ (1/2, 2) ∪ [1 +√6,∞)
4.(a) {−3, 9} (b) {−2, 3/2} (c) x ≥ −3/2
(d) [−5/2,−1/2] (e) (−∞,−4) ∪ (−1,∞) (f) (−∞, 1/5) ∪ (5/3,∞)
(g)(−∞,−1) ∪ (−1, 0] ∪ (1, 2] (h)[−1.1, 0) ∪ (0, 1] ∪ (1, 4]