atividade calculo I

˜
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ATIVIDADE DE REVISAO - CALCULO 1 (2014/1)
EXERC´
ICIOS
Prof. Juliana Dias
1. Determine o valor de x que satisfaz as seguintes inequa¸oes: c˜ (a)

2x − 1
≤0
x+1

(b)

(c)

3x − 2
≤0
2−x

(d) x(2x − 1) ≥ 0

(e) (2x − 3)(x2 + 1) ≤ 0

(f )

1−x
≥0
3−x

x−3
>0
x2 + 1

2. Verifique as seguintes identidades:
(a) x2 − a2 = (x − a)(x + a)
(b) x3 − a3 = (x − a)(x2 + ax + a2 )
(c) x4 − a4 = (x − a)(x3 + ax2 + a2 x + a3 )
(d) x5 − a5 = (x − a)(x4 + ax3 + a2 x2 + a3 x + a4 )
Obs: xn − an = (x − a)(xn−1 + axn−2 + a2 xn−3 + . . . + an−2 x + an−1 ), sendo n = 0 um n´mero natural. u 3. Utilizando quando necess´rio o exerc´ anterior, simplifique as express˜es: a ıcio o (a)

x2 − 1 x−1 (b)

x3 − 8 x2 − 4

(c)

4x2 − 9 x−1 (d)

(x + h)3 − x3 h (e)

(x + h)2 − (x − h)2 h (f )

1 x+h − h 1 x 4. Considere um polinˆmio do segundo grau na forma: p(x) = ax2 + bx + c, sendo o a = 0, b e c s˜o reais dados. a (a) Verifique que ax2 + bx + c = a

x+

b
2a

2

−

∆
, sendo ∆ = b2 − 4ac
4a2

C´lculo I a Prof. Juliana Dias

(b) Verifique se ∆ ≥ 0, as ra´ do polinˆmio p(x) = ax2 + bx + c s˜o dadas pela ızes o a f´rmula: o √
−b ± ∆ x= 2a

5. Resolva as seguintes inequa¸oes: c˜ (a) x2 − 3x + 2 < 0

(b) x2 − 5x + 6 ≥ 0

(c) x2 − 3x > 0

(d) 4x2 − 4x + 1 < 0

6. Resolva as equa¸oes e inequa¸˜es: c˜ co
(a) |x| ≤ 1

(b) |x − 2| = −1

(c) |2x − 1| < x

(d) |x − 3| < x + 1

(e) |2x2 − 1| < 1

(f ) |3x − 1| <

1
3

7. Verifique as seguintes identidades sendo x > 0 e y > 0.
√

√ x− y
√
√
(b) x − y = 4 x − 4 y
(a) x − y =

√
√
x+ y
√
4 x3 + 4 x2 y +

4

xy 2 +

4

y3

8. Determine o dom´ ınio e esboce o gr´fico das seguintes fun¸oes: a c˜

(a) f (x) = 3x

(b) g(x) = −x

(c) h(x) = −x + 1

(d) t(x) = 3


 2x se x ≤ −1
(e) f (x) =
 −x + 1 se x > −1 x2 − 1
(g) h(x) = x−1 (f ) h(s) = |s − 1|
(h) f (x) =

x2