Lista de Exercícios sobre Funções Calculo I - FAENG

Prévia do material em texto

FAENG – CICLO BA´SICO – CA´LCULO I
PRIMEIRA LISTA DE EXERCI´CIOS: FUNC¸O˜ES
(1) Para cada func¸a˜o dada calcule os valores indicados:
(a) f(x) = 3x2 + 5x− 2 calcule f(1), f(0) e f(−2) Resp: 6, -2 e 0
(b) h(t) =
√
t2 + 2t+ 4 calcule h(2), h(0) e h(−4) Resp: 2√3, 2 e 2√3
(c) f(t) = (2t− 1)1/2 calcule f(1), f(2) e f(13) Resp: 1, 1√
27
e 1
125
(d) f(x) = x− |x− 1| calcule f(1), f(2) e f(−3) Resp: 0, 2 e −8
(2) Dada a func¸a˜o f(x) =
−x2 − 9
x− 2 determine:
(a) f(0) Resp: f(0) = 9
2
(b) f(1) Resp: f(1) = 10
(c) f(−1) Resp: f(−1) = 10
3
(d) f(t) Resp: f(t) =
−t2 − 9
t− 2
(e) f(−t) Resp: f(−t) = t
2 + 9
t+ 2
(f) f(x− 1) Resp: f(x− 1) = −x
2 + 2x− 10
x− 3
(3) Sendo f(x) = x2, calcule e simplifique
f(a+ b)− f(a− b)
a b
(com a b 6= 0).
Resp: 4
(4) Dada a func¸a˜o f(x) = x2, calcule e simplifique
f(x)− f(p)
x− p (com x 6= p).
Resp: x+ p
(5) Desenvolva: (a+ b)2, (a− b)2, (a+ b)3 e (a− b)3.
(6) Sendo g(t) = t3 calcule e simplifique
g(t)− g(h)
t− h , com t 6= h. Resp: t
2 + th+ h2.
(7) Calcule
f(x+ h)− f(x)
h
, com h 6= 0, nos seguintes casos:
(a) f(x) = 2x+ 1 Resp: 2
(b) f(x) = x2 Resp: 2x+ h
(c) f(x) = x3 Resp: 3x2 + 3xh+ h2
(d) f(x) =
1
x
Resp:
−1
x2 + xh
1
(8) Fac¸a o gra´fico completo, indicando o domı´nio e a imagem, das seguintes func¸o˜es:
(a) f(x) = 2x− 3 Df = R e Imf = R
(b) f(x) = −pi Df = R e Imf = {−pi}
(c) f(x) = −x+ 3 Df = R e Imf = R
(d) f(x) = (x− 1)(x+ 3) Df = R e Imf = [− 4,+∞[
(e) f(x) = −x2 + 2x+ 3 Df = R e Imf = ]−∞, 4]
(f) f(x) = x2 + 3x+ 4 Df = R e Imf = [
7
4
,+∞[
(g) f(x) = (x− 1)2 Df = R e Imf = [0,+∞[
(h) f(x) = (x− 1)2 − 2 Df = R e Imf = [− 2,+∞[
(i) f(x) =


−x2 − x, se x < 0
x2, se x ≥ 0
Df = R e Imf = R
(j) f(x) =


x− 4, se x ≤ 1
−x2 + 2x+ 3, se x > 1
Df = R e Imf = ]−∞, 5]
(k) f(x) =


x, se x ≤ 2
x+ 1, se x > 2
Df = R e Imf = ]−∞, 2] ∪ ]3,+∞[
(l) f(x) =


x+ 1, se x < −1
x2 − 1, se − 1 ≤ x ≤ 1
−x+ 1, se x > 1
Df = R e Imf = ]−∞, 0]
(m) f(x) = |2x− 3| Df = R e Imf = ]0,+∞]
(n) f(x) = |x2 − 1| Df = R e Imf = ]0,+∞]
(o) f(x) = |x2 − 1|+ 2 Df = R e Imf = ]2,+∞]
(p) f(x) =
x
|x| Df = R
∗ e Imf = {−1, 1}
(q) f(x) = x|x| Df = R e Imf = R
(r) f(x) = x|2x+ 1| Df = R e Imf = R
(s) f(x) = x2 − 3|x| − 4 Df = R e Imf = [−254 ,+∞[
2
(9) Determinar o domı´nio de f sob a forma de intervalos, sendo:
(a) f(x) =
√
3− 2x
5x+ 1
(d) f(x) =
√
x2 − 2x− 3√
1− x (g) f(x) =
1
ln(x2 + x+ 1)
(b) f(x) =
√
−3 + 5x
2− x (e) f(x) = ln(1− 4x
2) (h) f(x) =
ln(36− x2)
x+ 4
+
√
(x2 − x)(x+ 1)
(c) f(x) =
√
x2 − 2x− 3
1− x (f) f(x) =
3
√
x7 + 1 + sen3x (i) f(x) = ln(1 + x) +
√
16− x2
Respostas:
(a) ]− 1/5, 3/2] (b) [3/4, 2] (c) ]−∞,−1] ∪ ]1, 3]
(d) ]−∞,−1] (e) ]− 1/2, 1/2[ (f) R
(g) R− {−1, 0} (h) [− 1, 0] ∪ [1, 6[ (i) ]− 1, 4]
(10) Encontre o que e´ pedido em cada caso:
(a) Sendo f(x) = x2 e g(x) = x+ 1, calcule f(g(x)) e g(f(x)).
Resp: f(g(x)) = x2 + 2x+ 1 e g(f(x)) = x2 + 1
(b) Sendo f(x) = senx e g(x) =
x
1 + x2
, calcule f(g(x)) e g(f(x)).
Resp: f(g(x)) = sen
(
x
1 + x2
)
e g(f(x)) =
senx
1 + sen2x
(c) Sendo f(x) =
ax+ b
cx− a , calcule f(f(x)) e f(f(f(0))).
Resp: f(f(x)) = x e f(f(f(0))) =
−b
a
(11) Determinar g(x) nos seguintes casos:
(a) f(x) = 2x+ 3, f [g(x)] = 8− 5x (d) f(x) = lnx, g[f(x)] = 3x− 5
(b) f(x) = 5x, f [g(x)] = x (e) f(x) = e2x, f [g(x)] = x2 = 1
(c) f(x) = x+ 1, g[f(x)] = 1− 2x (f) f(x) = e3x, g[f(x)] = x2 + 1
Respostas:
(a) g(x) =
5− 5x
2
(b) g(x) = log5 x (c) g(x) = −1 + 2x
(d) g(x) = 3ex − 5 (e) g(x) = ln√x2 + 1 (f) g(x) = 1
9
ln2x+ 1
3
(12) Determine x em cada caso:
(a) log2 16 = x (b) logx 0,008 = −3 (c) logx(ac) = c (c 6= 0)
(d) log10 100 = x (e) log7 x =
2
3
(f) ln(x3) = 7
(g) log10 x = 3 (h) lnx = −2 (i) log√b( 4
√
b3 ) = x
(j) log 1
x
16 = − 4
3
(k) ln(3x) = −1 (l) aloga x = 2
(m) logb 6 =
1
2
(n) logk(x
2) = c (o) eln x = 47
Respostas: (a) 4, (b) 5, (c) a, (d) 2, (e) 3
√
49, (f) e2 3
√
e, (g) 1000, (h) 1/e2,
(i) 3/2, (j) 8, (k) 1
3e , (l) 2, (m) 36, (n)
√
kc, (o) 47.
(13) Exprimir y como func¸a˜o de x:
(a) ln y = 3 lnx+ ln 5 (b) ln y = mx+ ln c (c) 2 log y = 3 log x+ 4 log 5
(d) log2 y = 2x+ log2 7 (e) ln y = k lnx+ ln c (f) 5 log3 y = 3 log3 x− log3 2
Respostas: (a) y = 5x3, (b) y = cemx, (c) y = 25x
√
x, (d) y = 7e2x, (e) y = cxk, (f) y = 5
√
x3
2
(14) Esboce o gra´fico indicando domı´nio e imagem:
(a) y = f(x) = 3x (b) y = f(x) =
(
1
3
)x
(c) y = f(x) = log3 x (d) y = f(x) = log 1
3
x
(e) y = f(x) = 3−x (f) y = f(t) = 23t+1
(g) y = f(x) = 3x/2 (h) f(m) = e−m
(i) y = f(x) = e3x−2 (j) y = f(x) = log2(2x+ 3)
(k) f(x) = log3(5x− 1) (l) y = f(x) = ln(−x)
(m) y = f(x) = ln(1− x) (n) y = f(x) = ln(1 + x)
(o) y = f(x) = ln(3x) (p) y = ln |x|
(q) f(u) = −1 + 2u (r) y = f(x) = 3 + log2 x
Respostas:
(a) Df = R Im(f) = ]0,+∞[ (b) Df = R Im(f) = ]0,+∞[ (c) Df = ]0,+∞[ Im(f) = R
(d) Df = ]0,+∞[ Im(f) = R Para os ı´tens (e), (f), (g), (h), (i): Df = R Im(f) = ]0,+∞[
(j) Df = ]− 3/2,+∞[ Im(f) = R (k) Df = ]1/5,+∞[ Im(f) = R (l) Df = ]−∞, 0[ Im(f) = R
(m) Df = ]1,+∞[ Im(f) = R (n) Df = ]− 1,+∞[ Im(f) = R (o) Df = ]0,+∞[ Im(f) = R
(p) Df = R
∗ Im(f) = R (q) Df = R
∗ Im(f) = ]− 1,+∞[ (r) Df = ]0,+∞[ Im(f) = R
4
(15) Prove as seguintes identidades trigonome´tricas:
(a) (cosec θ − cotg θ)(sec θ − 1) = tg θ (e) sec
4θ − 1
tg2θ
= 2 + tg2θ
(b)
sen θ cotg θ + cos θ
cotg θ
= 2 sen θ (f)
cosx− senx
cosx+ senx
+
cosx+ senx
cosx− senx =
2
cos 2x
(c)
sen θ
sec θ + 1
+
sen θ
sec θ − 1 = 2 cotg θ (g)
1− sec2θ
cosec2θ
+
1
sec2θ
= 1− tg2θ
(d)
cos4θ − sen4θ
1− tg4θ = cos
4θ (h) tg2θ − 1
cosec2θ
+
1 + 2 sen2θ sec2θ
sec2θ
= sec2θ
(16) Esboce o gra´fico indicando domı´nio e imagem:
(a) y = f(x) = sen(2x) (b) y = f(x) = sen(x− pi
2
)
(c) y = f(x) = 3 + senx (d) y = f(x) = 4 cosx
(e) y = f(x) = cos(x+ pi) (f) y = f(x) = sen( 2x
3
− pi)
(g) y = f(x) = 2 cos( 3x+pi
4
) (h) y = f(x) = 4 + 3 sen(2x− pi)
Respostas: Todas as func¸o˜es tem domı´nio igual a R. As func¸o˜es dos ı´tens (a), (b),
(e) e (f) tem imagem igual a [− 1, 1]; (c) Im(f) = [2, 4], (d) Im(f) = [− 4, 4],
(g) Im(f) = [− 2, 2], (h) Im(f) = [1, 7]
(17) Determine o per´ıodo de cada func¸a˜o do exerc´ıcio anterior.
Resp. (a) pi, (b) 2pi, (c) 2pi, (d) 2pi, (e) 2pi, (f) 3pi, (g) 8pi
3
, (h) pi.
(18) Dado um triaˆngulo ABC determinar x nos seguintes casos:
(a) AB =
√
3, AC = 2, BC = 1, x = Aˆ
(b) AB = 1, AC = 25, Aˆ = 2pi
3
, x = BC
(c) AB = 7, Ac = 5, BC = 4, x = a´rea do ∆ABC
(d) Aˆ = 3pi
4
, Bˆ = pi
8
, x = AC
(e) BC = 1, Bˆ = pi
3
, Cˆ = pi
12
, x = AB
Obs. Dado um triaˆngulo ABC qualquer valem:
(a) Lei dos senos:
a
sen Aˆ
=
b
sen Bˆ
=
c
sen Cˆ
= 2R
sendo R o raio da circunfereˆncia que
circunscreve o triaˆngulo ABC
(b) Lei dos cossenos:
a2 = b2 + c2 − 2bc cos Aˆ
E´ claro que:
b2 = a2 + c2 − 2ac cos Bˆ
c2 = a2 + b2 − 2ab cos Cˆ
A
B
C
O
R
b
a
c
5