350 Higher Engineering Mathematics
- z=2lnxy
⎡
⎢
⎣
(a)
− 2
x^2
(b)
− 2
y^2
(c) 0 (d) 0
⎤
⎥
⎦
- z=
(x−y)
(x+y)
⎡
⎢
⎢
⎢
⎣
(a)
− 4 y
(x+y)^3
(b)
4 x
(x+y)^3
(c)
2 (x−y)
(x+y)^3
(d)
2 (x−y)
(x+y)^3
⎤
⎥
⎥
⎥
⎦
- z=sinhxcosh2y
⎡ ⎢ ⎢ ⎢ ⎢ ⎣
(a)sinhxcosh 2y
(b)4sinhxcosh 2y
(c)2coshxsinh2y
(d)2coshxsinh2y
⎤ ⎥ ⎥ ⎥ ⎥ ⎦
- Given z=x^2 sin(x− 2 y)find (a)
∂^2 z
∂x^2
and
(b)
∂^2 z
∂y^2
Show also that
∂^2 z
∂x∂y
=
∂^2 z
∂y∂x
= 2 x^2 sin(x− 2 y)− 4 xcos(x− 2 y).
⎡
⎢
⎢
⎣
(a)( 2 −x^2 )sin(x− 2 y)
+ 4 xcos(x− 2 y)
(b)− 4 x^2 sin(x− 2 y)
⎤
⎥
⎥
⎦
- Find
∂^2 z
∂x^2
,
∂^2 z
∂y^2
and show that
∂^2 z
∂x∂y
=
∂^2 z
∂y∂x
whenz=cos−^1
x
y
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
(a)
∂^2 z
∂x^2
=
−x
√
(y^2 −x^2 )^3
,
(b)
∂^2 z
∂y^2
=
−x
√
(y^2 −x^2 )
{
1
y^2
+
1
(y^2 −x^2 )
}
(c)
∂^2 z
∂x∂y
=
∂^2 z
∂y∂x
=
y
√
(y^2 −x^2 )^3
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
- Givenz=
√(
3 x
y
)
show that
∂^2 z
∂x∂y
=
∂^2 z
∂y∂x
and evaluate
∂^2 z
∂x^2
when
x=
1
2
andy=3.
[
−
1
√
2
]
- An equation used in thermodynamics is the
Benedict-Webb-Rubine equation of state for
the expansion of a gas. The equation is:
p=
RT
V
+
(
B 0 RT−A 0 −
C 0
T^2
)
1
V^2
+(bRT−a)
1
V^3
+
Aα
V^6
+
C
(
1 +
γ
V^2
)
T^2
(
1
V^3
)
e
−Vγ 2
Show that
∂^2 p
∂T^2
=
6
V^2 T^4
{
C
V
(
1 +
γ
V^2
)
e−
γ
V^2 −C 0
}
.