微乙小考一 (2016/3/10)
1. (6%) 給定 f (x, y, z) = ln(xy + z). 求
∂f
(1, 2, 1)
∂y
及
∂f
(1, 2, 1).
∂z
sol:
∂f
1
(1, 2, 1) =
· x(x,y,z)=(1,2,1) =
∂y
xy + z
∂f
1
(1, 2, 1) =
· 1(x,y,z)=(1,2,1) =
∂z
xy + z
1
3
1
3
2. (7%) 求與 z = x2 + y 2 切於點 (1, 4, 17) 之切平面方程式。
sol:
∂z
(1, 4) = 2x(x,y)=(1,4) = 2
∂x
∂z
(1, 4) = 2y (x,y)=(1,4) = 8
∂y
∴ Tangent plane: E : z − 17 = 2(x − 1) + 8(y − 4)
∂z ∂z −1 y
及 ∂v .
3. (7%) 令 z = tan ( x ), x = u + v, y = uv, 求 ∂u u=1,v=1
sol:
∂z
∂u
∂z
∂v
=
=
∂z ∂x
∂x ∂u
∂z ∂x
∂x ∂v
+
+
∂z ∂y
∂y ∂u
∂z ∂y
∂y ∂v
=
=
1
y 2
1+( x )
1
y 2
1+( x
)
⇒
·
·
−y
x2
−y
x2
·1+
·1+
1
y 2
1+( x )
1
y 2
1+( x
)
·
·
u=1,v=1
1
x
·v =
−uv
(u+v)2 +(uv)2
+
(u+v)v
(u+v)2 +(uv)2
1
x
·u=
−uv
(u+v)2 +(uv)2
+
(u+v)u
(u+v)2 +(uv)2
1
1
∂z ∂z = and
=
u=1,v=1
u=1,v=1
∂u
5
∂v
5
1