Lista 4 Funções de Várias Variáveis Reais

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Universidade Federal de Uberlaˆndia - Lista 4 - Ca´lculo 2
Profa. Dra. Taciana Oliveira Souza
Observac¸a˜o: Use as Propriedades dos Limites para calcular os limites desta lista de exerc´ıcios.
Um arquivo pdf com as Propriedades dos Limites esta´ disponivel na pa´gina:
sites.google.com/site/olitaciana/disciplinas/calculo-2—engenharia-quimica
(1) Determine e fac¸a um esboc¸o do domı´nio da func¸a˜o.
(a) f(x, y, z) =
1√
9− x2 − y2 − z2 (b) f(x, y) =
4
x+ y
(c) f(x, y) =
√
1− x2 −
√
1− y2 (d) f(x, y) = ln(4−√x2 + y2)
(e) f(x, y) =
1√
x2 − y2 (f) f(x, y) =
√
y − x2
1− x2
(g) f(x, y) =
√
y +
√
25− x2 − y2 (h) f(x, y) = ln(16− 4x2 − 4y2 − z2)
(2) Descreva as curvas de n´ıvel e fac¸a um esboc¸o do gra´fico da func¸a˜o.
(a) z = −2x2 − 2y2 (b) z = x2 + y2 − 1
(c) z = 1− x2 − y2 (d) z = (x− 1)2 + (y − 2)2
(e) z = 1− (x− 1)2 − (y − 2)2 (f) f(x, y) = 1− x2
(g) f(x, y) = cos(x) (h) f(x, y) = y2 + 1
(3) Calcule o limite, se existir, ou mostre que o limite na˜o existe.
(a) lim
(x,y)→(0,0)
x− y
2x+ y
(b) lim
(x,y)→(0,0)
3xy
4x2 + 5y2
(c) lim
(x,y)→(0,0)
y4 + 3x2y2 + 2yx3
(y2 + x2)2
(d) lim
(x,y)→(1,0)
(x− 1)2y
(x− 1)4 + y2
(e) lim
(x,y)→(0,0)
2x√
x2 + y2
(f) lim
(x,y)→(0,0)
−x2y
2x2 + 2y2
(g) lim
(x,y)→(0,1)
x2(y − 1)2
x4 + (y − 1)4 (h) lim(x,y)→(0,0)
3xy√
x2 + y2
(4) Calcule os seguintes limites:
(a) lim
(x,y)→(1,2)
(
2xy + x2 − x
y
)
(b) lim
(x,y)→(0,0)
√
x− 1
x2y2 + xy − 1
(c) lim
(x,y)→(∞,∞)
(
1
x+ y
− 10
)
(d) lim
(x,y)→(∞,∞)
e
1
x+y
(e) lim
(x,y)→(pi,pi
2
)
sen(x+ y)
x
(f) lim
(x,y)→(4,2)
ln
(
x2 + y2
x− y + 1
)
(g) lim
(x,y)→(0+,0+)
√
xy2 + y3 − xy3
x2 + y2
(h) lim
(x,y)→(0,0)
xy
√
x+ y
x2 + y2
(i) lim
(x,y)→(2,3)
x2y − 3x2 − 4xy + 12x+ 4y − 12
xy − 3x− 2y + 6 (j) lim(x,y)→(0,0)
√
x+ 3−√3
xy + x
(k) lim
(x,y)→(0,1)
ysen(x)
xy + 2x
(l) lim
(x,y)→(1,2)
(exy − ey + 1)
(m) lim
(x,y)→(−1,2)
(x3y3 + 2xy2 + y) (n) lim
(x,y)→(−2,1)
xy2 − 5x+ 8
x2 + y2 + 4xy
(o) lim
(x,y)→(1,1)
x2 − yx
x2 − y2 (p) lim(x,y)→(1,2) ln
(
xy − 1
2xy + 4
)
(q) lim
(x,y)→(0,0)
xsen
(
1
y
)
(r) lim
(x,y)→(0,0)
cos
(
x3
x2 + y2
)
(5) Verifique se as func¸o˜es dadas sa˜o cont´ınuas no ponto P indicado:
(a) f(x, y) =

xsen
(
1
y
)
se y 6= 0
, P = (0, 0)
0 se y = 0
(b) f(x, y) =

y4 + 3x2y2 + 2yx3
(x2 + y2)2
se (x, y) 6= (0, 0)
, P = (0, 0)
0 se (x, y) = (0, 0)
(c) f(x, y) =

x2 − yx
x2 − y2 se x 6= ±y
, P = (1, 1)
1
4
(x+ y) se x = ±y
(d) f(x, y) =

x2 − y2
x2 + y2
se (x, y) 6= (0, 0)
, P = (0, 0)
0 se (x, y) = (0, 0)
(e) f(x, y) =
x3 − 3xy2 + 2
2xy2 − 1 , P = (1, 2)
(6) Escreva o conjunto de todos os pontos onde a func¸a˜o dada e´ cont´ınua:
(a) f(x, y) =
x− 2
(xy − 2x− y + 2)(y + 1).
(b) f(x, y) = ln
(
x+ y
x2 − y2
)
.
(c) f(x, y) = exsen(y).
(d) f(x, y, z) = arcsen(x2 + y2 + z2).
(e) f(x, y, z) =
√
y − x2 ln(z).
(7) Determine o valor de a para que a func¸a˜o dada seja cont´ınua em (0, 0):
(a) f(x, y) =
 (x2 + y2)sen
(
1
x2 + y2
)
se (x, y) 6= (0, 0)
a se (x, y) = (0, 0)
(b) f(x, y) =

x2y2√
y2 + 1 − 1 se (x, y) 6= (0, 0)
a− 4 se (x, y) = (0, 0)
Respostas de alguns exerc´ıcios
(1)
(a)D(f) = {(x, y, z) ∈ R3/9− x2 − y2 − z2 > 0} (b)D(f) = {(x, y) ∈ R2/x+ y 6= 0}
(c)D(f) = {(x, y) ∈ R2/1− x2 ≥ 0 e 1− y2 ≥ 0} (d)D(f) = {(x, y) ∈ R2/4−√x2 + y2 > 0}
(e)D(f) = {(x, y) ∈ R2/x2 + y2 > 0} (f)D(f) = {(x, y) ∈ R2/y − x2 ≥ 0 e 1− x2 6= 0}
(g)D(f) = {(x, y) ∈ R2/x2 + y2 ≤ 25 e y ≥ 0} (h)D(f) = {(x, y, z) ∈ R3/4x2 + 4y2 + z2 < 16}
(3)
(a) na˜o existe (b) na˜o existe (c) na˜o existe (d) na˜o existe
(e) na˜o existe (f) existe (g) na˜o existe (d) existe
(4)
(a) 9/2 (b) 1 (c) − 10 (d) 1 (e) − 1/pi (f) ln(20/3)
(g) 0 (h) 0 (i) 0 (j) 1/2
√
3 (k) 1/3 (l) 1
(m) − 14 (n) 7/13 (o) 1/2 (p) ln(1/8) (q) 0 (r) 1
(5) (a) Cont´ınua (b) Descont´ınua (c) Cont´ınua (d) Descont´ınua (e) Cont´ınua
(6) (a) {(x, y) ∈ R2/(x− 1)(y − 2)(y + 1) 6= 0}
(b) {(x, y) ∈ R2/y > x e y 6= −x}
(c) R2
(d) {(x, y, z) ∈ R3/x2 + y2 + z2 ≤ 1}
(e) {(x, y, z) ∈ R3/y ≥ x2 e z > 0}
(7) (a) a = 0; (b) a = 4.