Lista C1 P1(FEI) Novazzi

Prévia do material em texto

CA´LCULO I — LISTA AUXILIAR PARA P1 — PROF. ADILSON NOVAZZI
(1) Determinar o domı´nio, sob a forma de intervalos:
(a) f(x) =
√
1− |x|
ln(2x2 + x)
(b) f(x) =
√
3
x
− 6x
1− x
(c) f(x) =
ln(2− |x+ 4|)√
x2 + 3x
(d) f(x) = ln(6− x− x2) +
√
x+ 2
x− 1
(e) f(x) =
√
−x2 − x+ 6
x− 1 + ln(x+ 5) (f) f(x) =
√
−3 + 5x
2− x
(g) f(x) =
√
9x2 − 12x+ 4
ln(2− x2) (h) f(x) =
√
|3x+ 2| − 8
Respostas:
(a) ]−1,−1/2[ ∪ ]0, 1/2[ ∪ ]1/2, 1] (b) ]−∞,−1] ∪ ]0, 1/2] ∪ ]1,+∞[
(c) ]−6,−3[ (d) ]−3,−2] ∪ ]1, 2[ (e) ]−5,−3] ∪ ]1, 2] (f) [3/4, 2[
(g) ]−√2,√2[− {−1, 1} (h) ]−∞,−10/3] ∪ [2,+∞[
(2) Determinar g(x) nos seguintes casos:
(a) f(x) = 7− 2x e f [g(x)] = 5 + 7x Resp. g(x) = 1− 7x
2
(b) f(x) = 7− 2x e g[f(x)] = 5 + 4x Resp. g(x) = 19− 2x
(c) f(x) = ex e f [g(x)] = x2 + 1 Resp. g(x) = ln(x2 + 1)
(d) f(x) = 3x+ 1, h[g(x)] = 2− 2x e f [h(x)] = 2x− 5 Resp. g(x) = 6− 3x
(e) f(x) = lnx, x > 0 e g[f(x)] = 5x Resp. g(x) = 5ex
(3) Esboc¸ar o gra´fico, indicando domı´nio e imagem:
(a) f(x) = |2x− 3| (b) f(x) = x2 − |3x+ 4| (c) f(x) = x2 − 3|x| − 4
(d) f(x) = |x2 − 3x| − 4 (e) f(x) = |x2 − 3x− 4| (f) f(x) = (2x− 1)|x|
(g) f(x) = (x− 1)|x+ 2| (h) f(x) = | lnx| (i) f(x) = x|x|
(j) f(x) = ln |x| (k) f(x) = e|x| (l) f(x) = sen |x|
(4) Determinar um domı´nio A (o mais amplo poss´ıvel) no qual f seja invert´ıvel, determinar
f−1(x) e esboc¸ar os gra´ficos de f e f−1 (no mesmo par de eixos) indicando domı´nio e
imagem.
(a) f(x) = 3− 2x (b) f(x) = 2x− x2 (c) f(x) = x2 + x+ 1
(d) f(x) = x2 − 2x+ 1 (e) f(x) = √x (f) f(x) = lnx
(g) f(x) = x3 (h) f(x) =
1
x
(i) f(x) = e2x−2
Respostas:
(a) A = R
f :R → R
f(x) = 3− 2x
f−1:R → R
f−1(x) = 3−x
2
(b) A = [1,+∞[ f :[1,+∞[→ ]−∞, 1]
f(x) = 2x− x2
f−1:]−∞, 1]→ [1,+∞[
f−1(x) = 1 +
√
1− x
ou
A = ]−∞, 1] f :]−∞, 1]→ ]−∞, 1]
f(x) = 2x− x2
f−1:]−∞, 1]→ ]−∞, 1]
f−1(x) = 1−√1− x
(c) A = [−1/2,+∞[ f :[−1/2,+∞[→ [3/4,+∞[
f(x) = x2 + x+ 1
f−1:[3/4,+∞[→ [−1/2,+∞[
f−1(x) = − 1
2
+ 1
2
√
4x− 3
ou
A = ]−∞,−1/2] f :]−∞,−1/2]→ [3/4,+∞[
f(x) = x2 + x+ 1
f−1:[3/4,+∞[→ ]−∞,−1/2]
f−1(x) = − 1
2
− 1
2
√
4x− 3
(d) A = [1,+∞[ f :[1,+∞[→ [0,+∞[
f(x) = x2 − 2x+ 1
f−1:[0,+∞[→ [1,+∞[
f−1(x) = 1 +
√
x
ou
A = ]−∞, 1] f :]−∞, 1]→ [0,+∞[
f(x) = x2 − 2x+ 1
f−1:[0,+∞[→ ]−∞, 1]
f−1(x) = 1−√x
(e) A = [0,+∞[ f :[0,+∞[→ [0,+∞[
f(x) =
√
x
f−1:[0,+∞[→ [0,+∞[
f−1(x) = x2
(f) A = ]0,+∞[ f :]0,+∞[→ R
f(x) = lnx
f−1:R → ]0,+∞[
f−1(x) = ex
(g) A = R
f :R → R
f(x) = x3
f−1:R → R
f−1(x) = 3
√
x
(h) A = R∗
f :R∗ → R∗
f(x) = 1x
f−1:R∗ → R∗
f−1(x) = 1x
(i) A = R
f :R → ]0,+∞[
f(x) = ex+2
f−1:]0,+∞[→ R
f−1(x) = −2 + lnx
(5) Prove as identidades (onde existir):
(a)
cos2 x
1 + senx
= 1− senx (b) cos
4 x− sen4 x
1− tg4 x = cos
4 x
(c)
1− sec2 x
cosec2 x
+
1
sec2 x
= 1− tg2 x (d) tg2 x− 1
cosec2 x
+
1 + 2 sen2 x sec2 x
sec2 x
= sec2 x
(e)
cosx− senx
cosx+ senx
+
cosx+ senx
cosx− senx =
2
2 cos2 x− 1 (f) 2 cos
3 x senx− sen 2x+ 2 cosx sen3 x = 0
(6) Dada f(x) =
1− x2
x2 − 4 , calcular:
(a) lim
x→−2−
f(x) (b) lim
x→−2+
f(x) (b) lim
x→2−
f(x) (b) lim
x→2+
f(x)
Respostas: (a) −∞ (b) +∞ (c) +∞ (d) −∞
(7) Dada f(x) =
3x+ 1
2x− x2 , calcular:
(a) lim
x→0−
f(x) (b) lim
x→0+
f(x) (b) lim
x→2−
f(x) (b) lim
x→2+
f(x)
Respostas: (a) −∞ (b) +∞ (c) +∞ (d) −∞
(8) Estudar a continuidade de f no ponto a indicado:
(a) f(x) =


2x+ 1 se x ≤ 1
x2 + 2 se x > 1
(a = 1) (b) f(x) =


sen 2x
x
se x < 0
2 se x = 0
√
x+ 3 se x > 0
(a = 0)
(c) f(x) =


x3 − 3x+ 2
x− 1 se x 6= 1
0 se x = 1
(a = 1) (d) f(x) =


x sen
7
x
se x 6= 0
7 se x = 0
(a = 0)
Respostas
(a) f e´ cont´ınua no ponto a = 1 pois, lim
x→1−
f(x) = lim
x→1+
f(x) = 3 = f(1)
(b) f na˜o e´ cont´ınua no ponto a = 0 pois, lim
x→0−
f(x) = 2 6= lim
x→0+
f(x) =
√
3
(c) f e´ cont´ınua no ponto a = 1 pois, lim
x→1
f(x) = f(1) = 0
(d) f na˜o e´ cont´ınua no ponto a = 0 pois, lim
x→0
f(x) = 0 6= f(0) = 7
(9) Calcular lim
h→0
f(x+ h)− f(x)
h
para
(a) f(x) = k (constante) (b) f(x) = x (c) f(x) = x2
(d) f(x) =
√
x, x > 0 (e) f(x) =
1
x
, x 6= 0 (f) f(x) = senx
Respostas: (a) 0 (b) 1 (c) 2x (d)
1
2
√
x
(e) − 1
x2
(f) cosx
(10) Calcular:
(1) lim
x→−2
x2 − 5x− 14
x3 + x2 − 2x (2) limx→3
(
1
x− 3 −
5
x2 − x− 6
)
(3) lim
x→−2
x3 − 2x+ 4
x2 − 4
(4) lim
x→1
x4 − 4x+ 3
x3 − 3x+ 2 (5) limx→1
(
3
1− x3 −
1
1− x
)
(6) lim
x→2
√
x3 + 1 − 2x+ 1
4− x2
(7) lim
x→1
√
2x2 + x+ 1 − 3x+ 1
1− x (8) limx→2
3−√x2 + x+ 3
x− 2 (9) limx→−2
√
9x2 + 13 − 5 + x
x2 + x− 2
(10) lim
x→1
√
x+ 3 − 2
3−√x2 + 8 (11) limx→4
3 +
√
x3 − (3x− 1)
(x− 4)2 (12) limx→+∞
(√
x2 − 3 − x
)
(13) lim
x→−∞
(√
x2 − 3 − x
)
(14) lim
x→−∞
3 +
√
x2 − x+ 1√
9x2 − 7x − 2x+ 5 (15) limx→+∞
3 +
√
x2 − x+ 1√
9x2 − 7x − 2x+ 5
(16) lim
x→−∞
√
4x2 + 1 − x+ 3
2 + 7x−√x2 + 1 (17) limx→+∞
√
4x2 + 1 − x+ 3
2 + 7x−√x2 + 1 (18) limx→−∞
3 +
√
x2 − x+ 1√
9x2 − 7x − 2x+ 5
(19) lim
x→+∞
√
x2 − 5x+ 6− x (20) lim
x→−∞
√
x2 − 5x+ 6− x (21) lim
x→+∞
x(
√
x2 + 1− x)
(22) lim
x→−∞
x(
√
x2 + 1− x) (23) lim
x→+∞
(
√
x2 + 3x− x) (24) lim
x→−∞
(
√
x2 + 3x− x)
(25) lim
x→+∞
(
√
x+ 3−√x) (26) lim
x→0
1− cosx
x2
(27) lim
x→0
1− cosx√
1 + sen2x− cosx
(28) lim
x→0
√
3 + cos2x− 2
x2
(29) lim
x→0
√
4 + senx−√4− 3 senx
x
(30) lim
x→pi
senx
x− pi
(31) lim
x→pi
3
2 cosx− 1
3x− pi (32) limx→0
(√
senx+ 4 − 2
x
− senx
4x
)
(33) lim
x→1
(x2 − 4x+ 3) cos x
x− 1
(34) lim
x→0
1−√cosx
x2
(35) lim
x→+∞
[ln(2x+ 1)− ln(x+ 2)] (36) lim
x→+∞
x[ln(2x+ 1)− ln(2x)]
(37) lim
x→−∞
(
1− 2x
3
)x/7
(38) lim
x→−∞
(
1− 2
3x
)x/7
(39) lim
x→0
ln(1 + 5x)
x
(40) lim
x→+∞
(
x− 3
x+ 7
)5x−2
(41) lim
x→3
lnx− ln 3
x− 3 (42) limx→0
ex − 1
x
(43) lim
x→pi
6
2 senx− 1
6x− pi (44) limx→64
6
√
x− 2√
x− 8 (45) limx→1
3
√
x2 − 2 3√x− 1
(x− 1)2
(46) lim
x→1
√
x2 + x+ 2 − 2
x2 + 2x− 3 (47) limx→1
√
x2 + 15 − 7x+ 3
x2 + x− 2 (48) limx→−0
√
2 + 7 cos2x − 3
x senx
Respostas:
(1) -3/2 (2) 1/5 (3) -5/2 (4) 2 (5) -1 (6) 0 (7) 7/4 (8) -5/6 (9) 11/21
(10) -3/4 (11) 3/16 (12) 0 (13) +∞ (14) 1/5 (15) 1 (16) -3/8 (17) 1/6 (18) -1/2
(19) -5/2 (20) +∞ (21) 1/2 (22) −∞ (23) 3/2 (24) +∞ (25) 0 (26) 1/2 (27) 1/2
(28) -1/4 (29) 1 (30) -1 (31) 0 (32) 0 (33) 0 (34) 1/4 (35) ln 2 (36) 1/2
(37) 0 (38) e−2/21 (39) 5 (40) e−50 (41) 1/3 (42) 1 (43)
√
3/6 (44) 1/2 (45) 1/3
(46) 3/16 (47) -9/4 (48) -7/6