Revision Test 10
This Revision Test covers the material contained in Chapters 32 to 36. The marks for each question are shown in
brackets at the end of each question.
1. Differentiate the following functions with respect
to x:
(a) 5 ln (shx) (b) 3 ch
3 2x
(c) e
2x sech 2x
(7)
2. Differentiate the following functions with respect
to the variable:
(a) y =
1
5
cos −1 x
2
(b) y = 3e sin −1 t
(c) y =
2 sec −1 5x
x
(d) y = 3 sinh −1
(2x 2 − 1)
(14)
3. Evaluate the following, each correct to 3 decimal
places:
(a) sinh −1 3 (b) cosh −1 2.5 (c) tanh −1 0.8
(6)
4. If z = f (x, y) and z = x cos(x + y) determine
∂z
∂x
,
∂z
∂ y
,
∂ 2 z
∂x 2 ,
∂ 2 z
∂ y 2 ,
∂ 2 z
∂x∂ y
and
∂ 2 z
∂ y∂x
.
(12)
5. The magnetic field vector H due to a steady current I flowing around a circular wire of radius r
and at a distance x from its centre is given by
H = ±
I
2
∂
∂x
x
√
r 2 + x 2
Show that H = ±
r 2 I
2
(r 2 + x 2 ) 3
(7)
6. If x yz = c, where c is constant, show that
dz = −z
dx
x
+
d y
y
(6)
7. An engineering function z = f (x, y) and
z = e
y
2
ln(2x + 3y). Determine the rate of
increase of z, correct to 4 significant figures,
when x = 2 cm, y = 3 cm, x is increasing at 5 cm/s
and y is increasing at 4 cm/s.
(8)
8. The volume V of a liquid of viscosity coefficient
η delivered after time t when passed through a
tube of length L and diameter d by a pressure p
is given by V =
pd 4 t
128ηL
. If the errors in V , p and
L are 1%, 2% and 3% respectively, determine the
error in η.
( 8 )
9. Determine and distinugish between the stationary
values of the function
f (x, y) = x
3
− 6x
2
− 8y
2
and sketch an approximate contour map to represent the surface f (x, y).
(20)
10. An open, rectangular fish tank is to have a volume
of 13.5 m 3 . Determine the least surface area of
glass required.
(12)
This Revision Test covers the material contained in Chapters 32 to 36. The marks for each question are shown in
brackets at the end of each question.
1. Differentiate the following functions with respect
to x:
(a) 5 ln (shx) (b) 3 ch
3 2x
(c) e
2x sech 2x
(7)
2. Differentiate the following functions with respect
to the variable:
(a) y =
1
5
cos −1 x
2
(b) y = 3e sin −1 t
(c) y =
2 sec −1 5x
x
(d) y = 3 sinh −1
(2x 2 − 1)
(14)
3. Evaluate the following, each correct to 3 decimal
places:
(a) sinh −1 3 (b) cosh −1 2.5 (c) tanh −1 0.8
(6)
4. If z = f (x, y) and z = x cos(x + y) determine
∂z
∂x
,
∂z
∂ y
,
∂ 2 z
∂x 2 ,
∂ 2 z
∂ y 2 ,
∂ 2 z
∂x∂ y
and
∂ 2 z
∂ y∂x
.
(12)
5. The magnetic field vector H due to a steady current I flowing around a circular wire of radius r
and at a distance x from its centre is given by
H = ±
I
2
∂
∂x
x
√
r 2 + x 2
Show that H = ±
r 2 I
2
(r 2 + x 2 ) 3
(7)
6. If x yz = c, where c is constant, show that
dz = −z
dx
x
+
d y
y
(6)
7. An engineering function z = f (x, y) and
z = e
y
2
ln(2x + 3y). Determine the rate of
increase of z, correct to 4 significant figures,
when x = 2 cm, y = 3 cm, x is increasing at 5 cm/s
and y is increasing at 4 cm/s.
(8)
8. The volume V of a liquid of viscosity coefficient
η delivered after time t when passed through a
tube of length L and diameter d by a pressure p
is given by V =
pd 4 t
128ηL
. If the errors in V , p and
L are 1%, 2% and 3% respectively, determine the
error in η.
( 8 )
9. Determine and distinugish between the stationary
values of the function
f (x, y) = x
3
− 6x
2
− 8y
2
and sketch an approximate contour map to represent the surface f (x, y).
(20)
10. An open, rectangular fish tank is to have a volume
of 13.5 m 3 . Determine the least surface area of
glass required.
(12)