Higher Engineering Mathematics, Sixth Edition

592 Higher Engineering Mathematics

i.e. 3e−∞=( 0 )

(

3

0 + 4

)

i.e. 0 = 0 , which illustrates the theorem.

Problem 9. Verify the final value theorem for the

function( 2 +3e−^2 tsin4t)cm, which represents the

displacement of a particle. State its final steady

value.

Let f(t)= 2 +3e−^2 tsin4t

L{f(t)}=L{ 2 +3e−^2 tsin4t}

=

2

s

+ 3

(

4

(s−(− 2 ))^2 + 42

)

=

2

s

+

12

(s+ 2 )^2 + 16

from (ii) of Table 61.1, page 584 and (ii) of

Table 62.1 on page 587.

By the final value theorem,

limit

t→∞

[f(t)]=limit

s→ 0

[sL{f(t)}]

i.e. limit

t→∞

[2+3e−^2 tsin4t]

=limit

s→ 0

[

s

(

2

s

+

12

(s+ 2 )^2 + 16

)]

=limit

s→ 0

[

2 +

12 s

(s+ 2 )^2 + 16

]

i.e. 2+ 0 = 2 + 0

i.e. 2 = 2 , which verifies the theorem in this case.

The final value of the displacement is thus 2 cm.

The initial and final value theorems are used in pulse

circuit applications where the response of the circuit

for small periods of time, or the behaviour immediately

after the switch is closed, are of interest. The final value

theorem is particularly useful in investigating the sta-

bility of systems (such as in automatic aircraft-landing

systems) and isconcerned withthesteady state response

for large values of timet, i.e. after all transient effects

have died away.

Now try the following exercise

Exercise 222 Further problems on initial

and final value theorems

1. Statetheinitialvaluetheorem.Verify thetheo-
   rem for the functions (a) 3−4sint(b)(t− 4 )^2
   and state their initial values.
   [(a) 3 (b) 16]


2. Verify the initial value theorem for the voltage
   functions: (a)4+2cost(b)t−cos3tandstate
   their initial values. [(a) 6 (b)−1]


3. State the final value theorem and state a prac-
   tical application where it is of use. Verify the
   theoremforthefunction4+e−^2 t(sint+cost)
   representing a displacement and state its final
   value. [4]


4. Verify the final value theorem for the function
   3 t^2 e−^4 tand determine its steady state value.
   [0]