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.
- Differentiate the following functions with respect
tox:
(a) 5 ln(shx) (b) 3ch^32 x(c) e^2 xsech 2x (7)- Differentiate the following functions with respect
to the variable:
(a)y=1
5cos−^1x
2
(b)y=3esin− (^1) t
(c)y=
2sec−^15 x
x
(d)y=3sinh−^1
√
( 2 x^2 − 1 ) (14)
- Evaluate the following, each correct to 3 decimal
places:
(a) sinh−^1 3(b)cosh−^1 2.5 (c) tanh−^1 0.8 (6) - Ifz=f(x,y)andz=xcos(x+y)determine
∂z
∂x,∂z
∂y,∂^2 z
∂x^2,∂^2 z
∂y^2,∂^2 z
∂x∂yand∂^2 z
∂y∂x. (12)
5. The magnetic field vectorHdue to a steady cur-
rentIflowing around a circular wire of radiusr
and at a distancexfrom 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 )- Ifxyz=c,wherecis constant, show that
dz=−z(
dx
x+dy
y)
(6)- An engineering function z=f(x,y) and
z=e
y(^2) ln( 2 x+ 3 y). Determine the rate of
increase ofz, correct to 4 significant figures,
whenx=2cm,y=3cm,xis increasing at 5cm/s
andyis increasing at 4cm/s. (8)
- The volumeVof a liquid of viscosity coefficient
ηdelivered after timetwhen passed through a
tube of lengthLand diameterdby a pressurep
is given byV=pd^4 t
128 ηL. If the errors inV,pand
Lare 1%, 2% and 3% respectively, determine the
error inη.(8)
9. Determine and distinugish between the stationary
values of the function
f(x,y)=x^3 − 6 x^2 − 8 y^2and sketch an approximate contour map to repre-
sent the surfacef(x,y).
(20)- An open, rectangular fish tank is to have a volume
of 13.5m^3. Determine the least surface area of
glass required. (12)