LISTA 2 - CVV (Samuel)

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UFMG-ICEX-Departamento de Matema´tica
MTM123 - Ca´lculo de Va´rias Varia´veis- Prof: Samuel
Lista 2
1. Determine e fac¸a o esboc¸o do domı´nio da func¸a˜o:
(a) f(x, y) =
√
x+ y.
(b) g(x, y) = ln(9− x2 − y2).
(c) h(x, y) =
√
1− x2 −
√
1− y2.
(d) f(x, y) = xy
√
x2 + y.
(e) f(x, y) = x−3y
x+3y
.
2. Determine as derivadas parciais de primeira e segunda ordem da func¸a˜o:
(a) f(x, y) = (y − 2x)3.
(b) f(x, y) = x− ln(y).
(c) f(x, y) = yex.
(d) f(x, y) = y
x2+y2
.
(e) f(x, y) = 3x− 2y4.
(f) f(x, y) = x5 + 4x2y2 + 3xy7.
(g) z = xe2y.
(h) z = (x− y)17.
(i) f(x, y) = x−y
x
.
(j) z = sen(xy) · cos(y).
3. Determine:
(a) fx(3, 4), onde f(x, y) = ln(x+
√
x2 + y2).
(b) fy(1,−1), onde f(x, y) = tg(xy).
(c) fx(−1, 1), onde f(x, y) = x5 + 4x2y2 + 3xy7.
(d) zx(1, 1), onde z = xe
2y .
(e) zxy(1, 1), onde z = xe
2y.
4. Determine a equac¸a˜o do plano tangente a` superf´ıcie no ponto especificado:
(a) z = 4x2 + y2 − 2y no ponto (1, 1, 3).
(b) z = 4x2 + y2 − 2y no ponto (1,−1, 7).
(c) z = yln(2x2) no ponto
(√
2
2
, 1, 0
)
.
(d) z = ex
2+y2 no ponto
(√
2
2
,
√
2
2
, e
)
(e)
5. Encontre a linearizac¸a˜o da func¸a˜o no dado ponto:
(a) f(x, y) = y ln x, P = (2, 1).
(b) f(x, y) = x
y
, P = (6, 3).
(c) f(x, y) = y
√
x+ y, P = (3, 1).
(d) f(x, y) = ye−xycos(y), P = (pi, 0).
(e) f(x, y) = sen(2xpi + 3ypi), P = (−3, 2).
6. Determine dz
dt
, onde:
(a) z = x2y + xy2, x = 2 + t4, y = 1− t3.
(b) z = u2v + uv2, u = 2 + x4, v = 1− x3, x = t + t2 + t3.
(c) z = sen(x2)cos(y2), x = pit, y =
√
t.
7. Determine dz
dt
e dz
ds
, onde:
(a) z = x2y + xy2, x = 2 + t2s2, y = s− t.
(b) z = u2v + uv2, u = 2 + x4, v = 1− x3, x = t + s2.
(c) z = sen(θ)cos(φ), θ = pits, φ =
√
t + s.
8. Determine a derivada direcional da func¸a˜o no ponto P dado na direc¸a˜o do vetor −→v .
(a) f(x, y, ) = xe2y , P = (3, 2), −→u =
〈
2
3
,−
√
5
3
〉
.
(b) f(x, y) =
√
x+ y, P = (1, 3), −→u = 〈2, 3〉.
(c) f(x, y) = yln(x), P = (1,−3), −→u = 〈−4
5
, 3
5
〉
.
(d) f(x, y) =
√
x+ y, P = (1, 3), −→u = 〈2, 3〉.
(e) f(x, y) = 1 + 2x
√
y, P = (3, 4), −→v = 〈4,−3〉.
(f) f(x, y) = ln(x2 + y2), P = (2, 1), −→v = 〈−1, 2〉.
9. Determine a taxa de variac¸a˜o ma´xima de f no ponto P dado e a direc¸a˜o em que isso ocorre.
(a) f(x, y, ) = xe2y , P = (3, 2)
(b) f(x, y) = xe−y + ye−x, P = (0, 0).
(c) f(x, y) = ln(x2 + y2), P = (1, 1).
(d) f(x, y) = sen(xy), P = (1, 0).
10. Determine os valores ma´ximos e mı´nimos locais e pontos de sela da func¸a˜o:
(a) f(x, y) = 9− 2x+ 4y − x2 − 4y2;
(b) f(x, y) = xy − 2x− y;
(c) f(x, y) = (x2 + y2)ey
2−x2;
(d) f(x, y) = x3y − 12x2 − 8y;
(e) f(x, y) = e4y−x
2−y2;