350 Higher Engineering Mathematics
2. z = 2 ln x y
⎡
⎢
⎣
(a)
−2
x 2 (b)
−2
y 2
(c) 0
(d) 0
⎤
⎥
⎦
3. z =
(x − y)
(x + y)
⎡
⎢
⎢
⎢
⎣
(a)
−4y
(x + y) 3 (b)
4x
(x + y) 3
(c)
2(x − y)
(x + y) 3 (d)
2(x − y)
(x + y) 3
⎤
⎥
⎥
⎥
⎦
4. z = sinh x cosh 2y
⎡
⎢
⎢
⎢
⎢
⎣
(a) sinh x cosh 2y
(b) 4 sinh x cosh 2y
(c) 2 cosh x sinh 2y
(d) 2 cosh x sinh 2y
⎤
⎥
⎥
⎥
⎥
⎦
5. Given z = x 2 sin(x − 2y) find (a)
∂ 2 z
∂x 2 and
(b)
∂ 2 z
∂ y 2
Show also that
∂ 2 z
∂x∂ y
=
∂ 2 z
∂ y∂x
= 2x
2 sin(x − 2y) − 4x cos(x − 2y).
⎡
⎢
⎢
⎣
(a) (2 − x 2 ) sin(x − 2y)
+ 4x cos(x − 2y)
(b) − 4x
2 sin(x − 2y)
⎤
⎥
⎥
⎦
6. Find
∂ 2 z
∂x 2 ,
∂ 2 z
∂ y 2 and show that
∂ 2 z
∂x∂ y
=
∂ 2 z
∂ y∂x
when z = 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
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
7. Given z =
3x
y
show that
∂ 2 z
∂x∂ y
=
∂ 2 z
∂ y∂x
and evaluate
∂ 2 z
∂x 2 when
x =
1
2
and y = 3.
−
1
√
2
8. 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
−
γ
V 2
Show that
∂ 2 p
∂T 2
=
6
V 2 T 4
C
V
1 +
γ
V 2
e
−
γ
V 2 − C 0
.