ClassMpeting
#24
POPIILAR
INTEGRALS
IN
BUSIhI-ESS
-Since
integration
takes
a derivative
of
a
function
to
the
originat
functioq
many
applications
to economics
are
used for
this
purpose.
-Derivatives
are
"rates
of changd',
so, integration
will
give
the accumulated (total
change)
over
a
period
oftime.
-Most
involve
the
use of the
definite integral.
-Applications
to
ilIarginal
Cost
& Marginal
profit
-traample
#l:
.
A company's
rnarginal
cost
function
is
C
'(x)
:
2x-3,
where x represents
the
rumber
of units
produced.
Fixed
costs
aro
$1,000.
rilhat
is
the total
cost,
C(x),
ofproducing
100
units?
.-loo
'-r"o
c(x)=fixedcosts+variablecosts:1,000+-\{zx-r)dx=I,000+1x2-3x)f:1,000+(10,000-300):$10,700.
uo
)o
-Example
#2:
A
manufacturer's
marginal
profil
function
is
f['
=
-3x2
+
60x
+150.
Find
the
profit,
fI,
achieved
by increasing
production
from 3
to
6
units.
Il(6)
-
fl(r;
=
.l
Ur.'+6ox+150)61
=
(-x3*:0,.'+tsox;-l
=
t,071
u3
j3
-Note:
The
TI-83
can
be used for numerical
integration
by
inserting
the
function (Y=),
setting
the
proper
viewing
window,
then 2d
calg menu
7,
enter,
lower
limit,
enter,
upp"r
limit,
enter.
*Applications
to average
value
of
a
function.
rb
-The
average
value
(y-value)
of a
funcrioq
y
=
(x),
over
[qb]
is
given
by l(b-a)
I
(x)a*
u,t
-Example
#3:
According
to
estinrates,
the
cost
of
a
new
product
is C(x)
=
7rji'+
g,
where x represents
the
number
of units-
Find
thq
average cost
of
producing
the
first
50
units.
fro
Average
Cost
=
1(50-0)
|
qZ{il+Sydx:
$41
(use
calculator).
uo
Example
#4:
*
T*':
,I$toO
aftert months
is
given
by I(t)
:
20
+
40t
-
3tz for
0
<
t
<
l!..Find
the average
inventory
during
the
2*
quarter
of the
year
(i.e.,
from
3
to
6
months).
Pe'
Average
Inventory:
l(6-3)
|
(zo
*
4at
4f
)dt:
l3Z.
J,
