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| PAGE: 174 | SET: Exercises | PROBLEM: 13 |
(a)
Area of the circle is 
, Where 
is a radius.
(i)
Average rate of change if radius 
changes from 
to 
:
Average rate of change of 
with respect to 
is defined as 
on the interval 
.
Here 
, 
and 
.
Average rate of change in area is 
.
(ii)
Average rate of change if radius changes from to 
:
Average rate of change in area is
(iii)
Average rate of change if radius changes from to 
:
Average rate of change in area is
(b)
Instantaneous rate of change when 
:
Instantaneous rate of change of 
with respect to 
when 
is defined as 
.
Area of the circle is 
.
Differentiate on each side with respect to .
.
Instantaneous rate of change of area at 
is 
.
.
(c) The circumference of the circle is 
, where is the radius of the circle.
consider 
.
Therefore rate of change of area with respect to its radius is equal to the its circumference.
If the radius is increased by an amount of 
.
Geometrical view of circle with an increased radius 
.
Therefore area of circle with increased radius of is 
.
If is small, then we can approximate 
.
Rate of change of area of new circle is with respect to radius can be considered as
(a)
(i) Average rate of change if radius changes from to is 
.
(ii) Average rate of change if radius changes from to is 
.
(iii) Average rate of change if radius changes from to is 
.
(b) Instantaneous rate of change when is 
.
(c) The rate of change of area with respect to its radius is equal to the its circumference.
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