LISTA 2 - CVV (Samuel) - Respostas

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UFMG-ICEX-Departamento de Matema´tica
MTM123 - Ca´lculo de Va´rias Varia´veis- Prof: Samuel
Lista 2- Algumas Respostas
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
.
Respostas:
(a) Dom(f) = {(x, y)|y 6 −x}.
(b) Dom(g) = {(x, y)|x2
9
+ y2 < 1}.
(c) Dom(h) = {(x, y)| − 1 6 x 6 1, −1 6
y 6 1}.
(d) Dom(f) = {(x, y)|y ≥ −x2}.
(e) Dom(f) = {(x, y)|y 6= −x
3
}.
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).
Respostas:
(a) fx = −6(y − 2x)2, fxx = 24(y − 2x), fy = 3(y − 2x)2, fyy = 6(y − 2x), fxy = fyx =
−12(y − 2x).
(b) fx = 1, fxx = 0, fy = − 1y , fyy = 1y2 , fxy = fyx = 0.
(c) fx = ye
x, fxx = ye
x, fy = e
x, fyy = 0, fxy = fyx = e
x.
(d) fx = − 2xy(x2+y2)2 , fxx = −2y(y
2−3x2)
(x2+y2)3
= −fyy, fy = x
2−x2)
(x2+y2)2
, fxy = fyx = −2x(y
2−3x2)
(x2+y2)3
.
(e) fx = 3, fxx = 0, fy = −8y3, fyy = −24y2, fxy = fyx = 0.
(f) fx = 5x
4 + 8xy2 + 3y7, fxx = 20x
3 + 8y, fy = 8x
2y + 21xy6, fyy = 8x
2 + 126xy5,
fxy = fyx = 16xy + 21y
6.
(g) ∂z
∂x
= e2y, ∂
2z
∂x2
= 0, ∂z
∂y
= 2xe2y , ∂
2z
∂y2
= 4xe2y, ∂
2z
∂x∂y
= 2e2y.
(h) ∂z
∂x
= 17(x − y)16, ∂2z
∂x2
= 272(x − y)15, ∂z
∂y
= −17(x − y)16, ∂2z
∂y2
= 272(x − y)15,
∂2z
∂x∂y
= −272(x− y)15.
(i) fx =
y
x2
, fxx = −−2yx3 , fy = − 1x , fyy = 0, fxy = fyx = − 1x2 .
(j) ∂z
∂x
= ycos(xy)cos(y)),
∂2z
∂x2
= −y2sen(xy)cos(y),
∂z
∂y
= xcos(xy)cos(y)− sin(xy)sen(y),
∂2z
∂y2
= (−x2 − 1)sen(xy)cos(y)− 2xcos(xy)sen(y),
∂2z
∂x∂y
= cos(y)(cos(xy)− xysen(xy))− ycos(xy)sen(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.
Respostas:
(a) 1
5
. (b) sec2(−1). (c) 13. (d) e2. (e) 2e2
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
)
Respostas:
(a) z = 8x− 5.
(b) z = 8x− 4y − 5.
(c) z = 2x
√
2− 2.
(d) z = e(x+ y)
√
2− 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).
Respostas:
(a) L(x, y) = x
2
+ln(2)y−1
(b) L(x, y) = x
3
− 2y
3
+ 1
2
.
(c) L(x, y) = x
4
+ 9y
4
− 10
4
.
(d) L(x, y) = y.
(e) L(x, y) = 2pix+ 3piy.
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;
(a) Ma´ximo local :
(−1, 1
2
)
.
(b) Sela: (1, 2).
(c) Sela: (2, 4).
(d) Mı´n: (0, 0); Sela: (1, 0) e (−1, 0).
(e) Ma´x: (0, 2)