354 Higher Engineering Mathematics

Using equation (2), the rate of change of diagonalbis

given by:

db

dt

=

∂b

∂x

dx

dt

+

∂b

∂y

dy

dt

+

∂b

∂z

dz

dt

Sinceb=

√

(x^2 +y^2 +z^2 )

∂b

∂x

=

1

2

(x^2 +y^2 +z^2 )

− 1

(^2) ( 2 x)=
x
√
(x^2 +y^2 +z^2 )
Similarly,
∂b
∂y

y
√
(x^2 +y^2 +z^2 )
and
∂b
∂z

z
√
(x^2 +y^2 +z^2 )
dx
dt
=6mm/s= 0 .6cm/s,
dy
dt
=5mm/s= 0 .5cm/s,
and
dz
dt
=4mm/s= 0 .4cm/s
Hence
db
dt

[
x
√
(x^2 +y^2 +z^2 )
]
( 0. 6 )

• [
  y
  √
  (x^2 +y^2 +z^2 )
  ]
  ( 0. 5 )


• 
  

  [
  z
  √
  (x^2 +y^2 +z^2 )
  ]
  ( 0. 4 )
  Whenx=5cm,y=4cmandz=3cm,then:
  db
  dt

  
  

  [
  5
  √
  ( 52 + 42 + 32 )
  ]
  ( 0. 6 )

  
  


• 
  

  [
  4
  √
  ( 52 + 42 + 32 )
  ]
  ( 0. 5 )

  
  


• 
  

  [
  3
  √
  ( 52 + 42 + 32 )
  ]
  ( 0. 4 )
  = 0. 4243 + 0. 2828 + 0. 1697 = 0 .8768cm/s
  Hence the rate of increase of diagonal AC is
  0.88cm/s or 8.8mm/s, correct to 2 significant figures.
  Now try the following exercise
  Exercise 141 Further problems on rates of
  change

  
  

1. The radius of a right cylinder is increasing at
   a rate of 8mm/s and the height is decreasing
   at a rate of 15mm/s. Find the rate at which the
   volume is changing in cm^3 /s when the radius
   is 40mm and the height is 150mm.
   [+ 226 .2cm^3 /s]


2. Ifz=f(x,y)andz= 3 x^2 y^5 ,findtherateof
   change ofzwhenxis 3 units andyis 2 units
   whenx is decreasing at 5 units/s andy is
   increasing at 2.5 units/s. [2520 units/s]


3. Find the rate of change ofk, correct to 4
   significant figures, given the following data:
   k=f(a,b,c);k= 2 blna+c^2 ea;ais increas-
   ing at 2cm/s;bis decreasing at 3cm/s;cis
   decreasing at 1cm/s;a= 1 .5cm,b=6cmand
   c=8cm. [515.5cm/s]


4. A rectangular box has sides of lengthxcm,
   ycm andzcm. Sidesxandzare expanding at
   rates of 3mm/s and 5mm/s respectively and
   sideyis contractingat a rate of 2mm/s. Deter-
   mine the rate of change of volume whenxis
   3cm,yis 1.5cm andzis 6cm.
   [1.35cm^3 /s]


5. Find the rate of change of the total surface area
   of a right circular cone at the instant when the
   baseradiusis5cmandtheheight is12cmifthe
   radius is increasing at 5mm/s and the height
   is decreasing at 15mm/s.
   [17.4cm^2 /s]

35.3 Small changes

It is often useful to find an approximate value for

the change (or error) of a quantity caused by small

changes (or errors) in the variables associated with the

quantity.Ifz=f(u,v,w,...)andδu,δv,δw,...denote

small changesinu,v,w,...respectively, then the cor-

respondingapproximate changeδzinzis obtainedfrom

equation (1) by replacing the differentials by the small

changes.

Thusδz≈

∂z

∂u

δu+

∂z

∂v

δv+

∂z

∂w

δw+··· (3)