Higher Engineering Mathematics

354 DIFFERENTIAL CALCULUS

and

∂t

∂g

=−π

√

l

g^3

(from Problem 6, Chapter 34)

δl=

0. 2

100

l= 0. 002 land δg=− 0. 001 g

henceδt≈

π

√

lg

(0. 002 l)+−π

√

l

g^3

(− 0. 001 g)

≈ 0. 002 π

√

l

g

+ 0. 001 π

√

l

g

≈(0.001)

[

2 π

√

l

g

]

+ 0. 0005

[

2 π

√

l

g

]

≈ 0. 0015 t≈

0. 15

100

t

Hence the approximate error intis a 0.15%
increase.

Now try the following exercise.

Exercise 143 Further problems on small

changes

1. The powerPconsumed in a resistor is given
   byP=V^2 /Rwatts. Determine the approxi-
   mate change in power whenVincreases by
   5% andRdecreases by 0.5% if the original
   values ofVandRare 50 volts and 12.5 ohms
   respectively. [+21 watts]


2. An equation for heat generatedHisH=i^2 Rt.
   Determine the error in the calculated value of
   Hif the error in measuring currentiis+2%,
   the error in measuring resistanceRis−3%
   and the error in measuring timetis+1%.
   [+2%]

3.fr=

1

2 π

√

LC

represents the resonant

frequency of a series connected circuit

containing inductance L and capacitance

C. Determine the approximate percentage

change infrwhenLis decreased by 3% and

Cis increased by 5%. [−1%]

4. The second moment of area of a rectangle
   about its centroid parallel to sidebis given by
   I=bd^3 /12. Ifbanddare measured as 15 cm
   and 6 cm respectively and the measurement
   errors are+12 mm inband−1.5 mm ind,
   find the error in the calculated value ofI.
   [+1.35 cm^4 ]


5. The sidebof a triangle is calculated using
   b^2 =a^2 +c^2 − 2 accosB.Ifa,candBare
   measured as 3 cm, 4 cm andπ/4 radians
   respectively and the measurement errors
   which occur are +0.8 cm, −0.5 cm and
   +π/90 radians respectively, determine the
   error in the calculated value ofb.
   [−0.179 cm]

6.Qfactor in a resonant electrical circuit is

given by:Q=

1

R

√

L

C

. Find the percentage

change inQwhenLincreases by 4%, R

decreases by 3% andCdecreases by 2%.

[+6%]

7. The rate of flow of gas in a pipe is given

by:v=

C

√

d

√ 6

T^5

, whereCis a constant,dis

the diameter of the pipe andTis the ther-

modynamic temperature of the gas. When

determining the rate of flow experimentally,

d is measured and subsequently found to

be in error by+1.4%, andThas an error of

−1.8%. Determine the percentage error in the

rate of flow based on the measured values of

dandT.[+2.2%]