Higher Engineering Mathematics, Sixth Edition

350 Higher Engineering Mathematics

2. z=2lnxy

⎡

⎢

⎣

(a)

− 2

x^2

(b)

− 2

y^2

(c) 0 (d) 0

⎤

⎥

⎦

3. 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

⎤

⎥

⎥

⎥

⎦

4. z=sinhxcosh2y

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

(a)sinhxcosh 2y

(b)4sinhxcosh 2y

(c)2coshxsinh2y

(d)2coshxsinh2y

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

5. 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)

⎤

⎥

⎥

⎦

6. 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

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

7. 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

]

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

−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

}

.