Higher Engineering Mathematics

Assign-09-H8152.tex 23/6/2006 15: 11 Page 365

G

Differential calculus

Assignment 9

This assignment 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
   tox:
   (a) 5 ln (shx) (b) 3 ch^32 x
   (c) e^2 xsech 2x (7)


2. Differentiate the following functions with respect
   to the variable:

(a)y=

1

5

cos−^1

x

2

(b)y=3esin

− (^1) t
(c)y=
2 sec−^15 x
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. Ifz=f(x,y) andz=xcos(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 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)

6. Ifxyz=c, wherecis constant, show that

dz=−z

(

dx

x

+

dy

y

)

(6)

7. An engineering function z=f(x,y) and
   z=e

y

(^2) ln (2x+ 3 y). Determine the rate of increase
of z, correct to 4 significant figures, when
x=2cm,y=3cm,xis increasing at 5 cm/s and
yis increasing at 4 cm/s. (8)

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^2
   and sketch an approximate contour map to repre-
   sent the surfacef(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)