Gabarito L (3)

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Ca´lculo I - Lista no¯ 3 - Gabarito
1.
(a) f
(
1
2
)
= −
5
4
(b) f
(√
2
)
= 2− 3
√
2
(c) f(a) = a(a− 3) (d) f(2a− b) = 4a2 − 6a+ 3b− 4ab+ b2
2.
(a) f
(
1
2
)
=
3
2
(b) f
(
−
√
2
)
= 3+
√
2 (c) f(3) + f(10) = 8 (d) f(−2) − f(0) = 4
3.
(a) D(f) =
{
x ∈ R | x 6= −√3 e x 6= √3
}
(b) D(f) = R
(c) D(f) = {x ∈ R | x < −3 ou x ≥ 2} (d) D(f) = {x ∈ R | x ≤ −3 ou x ≥ 0}
(e) D(f) = {x ∈ R | x > 0} (f) D(f) =
{
x ∈ R | −2 < x ≤ −√2 ou √2 ≤ x < 2
}
(g) D(f) = {x ∈ R | 1 ≤ x ≤ 3} (h) D(f) = {x ∈ R | x = 0 ou x ≥ 1}
(i) D(f) = {x ∈ R | −3 ≤ x ≤ 7} (j) D(f) =
{
x ∈ R | x > 4
3
}
(k) D(f) = {x ∈ R | x > 0} (l) D(f) =
{
x ∈ R | −√5 < x < √5
}
(m) D(f) = {x ∈ R | x < −1 ou x > 1} (n) D(f) = {x ∈ R | −2 < x < 2}
(o) D(f) = {x ∈ R | x > −1} (p) D(f) = {x ∈ R | x ≤ −4 ou x ≥ −2}
(q) D(f) = {x ∈ R | x > 1} (r) D(f) = {x ∈ R | x < −1 ou x > 1}
(s) D(f) = {x ∈ R | 2 < x < 3} (t) D(f) = R
(u) D(f) = {x ∈ R | x < ln(3)}
4.
(a) (b) (c) (d)
−6 −3
2
5
−6 −3
−3
−3
3
−6 −3
−5
−2
(e) (f) (g) (h)
−8 −5 −2
3
3 6
3
−6 −3
6
−2 1 4
3
Instituto de Matema´tica Universidade Federal do Mato Grosso do Sul
5.
D(f) = {x ∈ R | x 6= 0}
Im(f) = {y ∈ R | y 6= 0}
−1 1
6. (a) f(x) = −
√
4− x2 (b) f(x) =
√
x
−2 2
−2 1
1
7.
(a) D(f) = R (b) D(f) = R (c)D(f) = R
Im(f) = [0,∞[ Im(f) = R Im(f) = [1,∞[
2
4
−1.21 1.5
1
4
(d) D(f) = R (e) D(f) = R (f) D(f) = ] −∞, 2[ ∪ ]2,∞[
Im(f) = R Im(f) =
]
−∞, 14
3
[
Im(f) = ] −∞, 1[ ∪ ]1,∞[
0.25
−1
−3 0.5
(−0.67, 4.67)
(4,−14) 2
1
Instituto de Matema´tica Universidade Federal do Mato Grosso do Sul
(g) D(f) = R (h) D(f) = R (i) D(f) = [−4,∞[
Im(f) = ] −∞, 1] Im(f) = R Im(f) = [0,∞[
−1.41 1.41
1
−2 2 −4
2
(j) D(f) = ] −∞,−3] ∪ [3,∞[ (k) D(f) = [−√5,√5] (l) D(f) = R
Im(f) = [0,∞[ Im(f) = [−1,√5− 1] Im(f) = [√2,∞[
−3 3
−2 2
1.23
−1
1.41
(m) D(f) = [−1, 5] (n) D(f) = R (o) D(f) = R
Im(f) = [0, 3] Im(f) = [−1, 1] Im(f) = [−2, 2]
−1 52
3
−pi pi
−1.
1
−pi pi
−2
2
(p) D(f) = R (q) D(f) = R (r) D(f) = R
Im(f) = [1, 2] Im(f) = ]0,∞[ Im(f) = ]0,∞[
−pi pi
1
2
1 1
Instituto de Matema´tica Universidade Federal do Mato Grosso do Sul
(s) D(f) = R (t) D(f) = ]0,∞[ (u) D(f) = R
Im(f) = [1,∞[ Im(f) = R Im(f) = ]0,∞[
1
1
7.4
(v) D(f) = ]1,∞[ (w) D(f) = ]0,∞[ (x) D(f) = ] −∞, 0[ ∪ ]0,∞[
Im(f) = R Im(f) = R Im(f) = [0,∞[
1 2
1
−1 1
(y) D(f) = R (z) D(f) = R
Im(f) = ] −∞, 0[ ∪ {3} Im(f) = ] −∞,−1[ ∪ ]0, 1]
−1 2
−1
3 C
1 2
−2
1
−1
B
8.
(a) Im(f) = {y ∈ R | y ≥ 2} (b) Im(f) = {y ∈ R | y ≥ 0}
D(g) = {x ∈ R | x ≥ 0} D(g) = {x ∈ R | x ≥ 0}
h(x) =
√
2+ x2 h(x) =
√
x2 − x
(c) Im(f) = {y ∈ R | y 6= 1} (d) Im(f) = {y ∈ R | y 6= 2}
D(g) = {x ∈ R | x 6= 1} D(g) = {x ∈ R | x 6= 2}
h(x) = −(2x+ 1) h(x) =
2
x− 1
Instituto de Matema´tica Universidade Federal do Mato Grosso do Sul
9.
(a) A =] −∞,−2[ ∪ ] − 2,∞[ (b) A = [1,∞[
h(x) =
2
x+ 5
h(x) =
√
x2 − 1
10.
(a) f(x) =
1
x
(b) f(x) =
2− x
x− 1
(c) f(x) =
√
x+ 1+ 2
11.
(a) f−1(x) =
x+ 2
5
(b) f−1(x) =
−2x
x− 3
(c) f−1(x) = x2 + 1
−2 0.5
−2
0.5
f(x)
f−1(x)
−2 3
f−1(x)
f(x)
f(x)
f−1(x)
(d) f−1(x) =
√
x+ 4 (e) f−1(x) = −
√
x+ 4 (f) f−1(x) =
√
x
1− x
−4 2
−4
2
f(x)
f−1(x)
−4 −2
−4
−2
f(x)
f−1(x)
f(x)
f−1(x)
(g) f−1(x) = ln(x) − 4 (h) f−1(x) = 5− ex (i) f−1(x) = −
√
9− x2
f(x)
f−1(x)
4 5
4
5
f(x)
f−1(x)
−3 3
−3
3
f(x)
f−1(x)
Instituto de Matema´tica Universidade Federal do Mato Grosso do Sul
12.
(a) f−1(x) =
x− 2
2
(b) f−1(x) =
√
4− (x− 1)2 (c) f−1(x) =
√
x2 − 3
−1 21 4
−1
2
1
4
f(x)
f−1(x)
−1 1 2
−1
1
2
f(x)
f−1(x)
1.73
1.73 f(x)
f−1(x)
13.
(a)
Tempo(h) 0 1 2 3 4 5
Populac¸a˜o de bacte´rias 50 100 200 400 800 1600
(b) n = 50 · 2t
14.
(a) Q = 552.85 (b) Q = 300
15. t = 12 anos
16. h(p) = 8 Km
Instituto de Matema´tica Universidade Federal do Mato Grosso do Sul