3.10 Business and Economic Applications
K now ing the rates of change of profit, rev enue , and c ost
w ith respect to th e num ber of units produced or s o ld is one
of the keys to business succe ss.
It's the difference betw een big $$$$$$$
&
BANKRUPTCY.
D o N ot W rite; D efinitions to Follow
3.10 Business and Economic Applications
E conom ists refer to these concepts as...
D o N ot W rite; D efinitions to Follow
3.10 Business and Economic Applications
S um m ary of B usiness T erm s and Form ulas
x is the num ber of units produced (or sold)
p is the price per unit
P is the total profit.
P  R  C  xp  C
R is the total revenue for selling x units
R  xp
3.10 Business and Economic Applications
S um m ary of B usiness T erm s and Form ulas
C is the total cost of producing x units
C is the average cost per unit
C 
C
x
3.10 Business and Economic Applications
S um m ary of B usiness T erm s and Form ulas
T he brea k-even p oint is the num ber of unit s fo r w hich R  C

3.10 Business and Economic Applications
S um m ary of B usiness T erm s and Form ulas
dR
 (M arginal R evenue)
dx
 (extra revenue from selling one addition al unit)
dC
 (M arginal C ost)
dx
 (extra cost of producing one additional unit)
3.10 Business and Economic Applications
S um m ary of B usiness T erm s and Form ulas
dP
 (M arginal P rofit)
dx
 (extra profit from selling one additiona l unit)
3.10 Business and Economic Applications
M arg in al
R even u e
1 U n it
E xtra revenue
for one m ore
R even u e F u n ctio n
unit sold.
M arginal R evenue  E xtra R evenue
3.10 Business and Economic Applications
A m anufacturer determ ines that the profit derived from
selling x un i ts of a certain item is give n b y
P  0.0002 x  10 x
3
A . Find the m arginal pr ofit for 50 units s old.
dP
 0.0006 x 2  10
dx
dP
dx

x  50
$ 1 1 .5 0
We’ll profit
approximately
$11.50 more if
we sell 51 units.
A m anufacturer determ ines that the profit derived from
selling x units of a certain item is given by
P  0.0002 x  10 x
3
B . C om pare the $11.50 w ith the actual g ain in profit obtained by
increasing sales f rom 50 to 51 u n its.
P  5 0   $ 5 2 5 .0 0
P  5 1   $ 5 3 6 .5 3
dP
dx

$ 1 1 .5 0
x  50
 the additional profit obtained is actua lly
$ 1 1 .5 3
3.10 Business and Economic Applications
T he price
x
 p
w ritten in term s of the # of units sold
is called the D E M A N D FU N C T IO N .
 x, p 
Finding the dem and function
A business sells 2000 item s per m onth at a price o f $10 .00
T w o points determ ine a line :)
e ach.
LINEAR
It is predicted that m onthly sales w ill increase by
x & y
250 item s for each $0.25 reducti on in price.
are co n stan t
}
Find the d em and funct ion corresponding to th i s predict ion.
 x , p    2000,10 
m 
 .25
25

250
&
25, 000
1
y  10  
1000
y  10  2 
 2250,9.75 
1

W e're not leaving in
1000
point-slope, so w e
 x  2000 
x
1000
can easily apply this
dem and function.
 y  12 
x
1000
3.10 Business and Economic Applications
Finding the dem and function
p  12 
# of
items
x
Price per
item
1000
S o w hat w ould you suggest w e set the price at if w e w ant to
sell appr ox im at ely 30 00 it em s?
p  12 
3000
1000

$ 9 .0 0
D em and for burgers is
p
60, 000  x
.
2 0, 000
Find the increase in revenue per burger (m arginal reven ue)
for m onthly sales of 20,000 burg ers.
2
R  xp 
dR
dx
 3
x
 60, 000  x 
x
  3 x  20, 000
 20, 000 
x
10, 000
 $ 1 / u n it
If w e sell 20,001 burgers w e'll bring in  $1.00 m ore.
Profit
3.10 Business and Economic Applications
S uppose the cost of producing x burgers is
C  5000  0.56 x
Find the total profit and the m arginal p rofit for
20,000, for 24,400, and for 30,000 units (burgers).
P  RC
P  3x 
x
2
 5000  0.56 x
20, 000
 2.44 x 
x
2
 5000
20, 000
Revenue
3.10 Business and Economic Applications
S uppose the cost of producing x burgers is
C  5000  0.56 x
Find the total profit and the m arginal p rofit for
20,000, for 24,400, and for 30,000 units (burgers).
P  2.44 x 
x
2
 5000
20, 000
 M arginal P rofit is...
dP
dx
 2.44 
x
10, 000
S uppose the cost of producing x burgers is
C  5000  0.56 x
Find the total profit and the m arginal p rofit for
20,000, for 24,400, and for 30,000 units (burgers).
Demand
20,000
24,400
30,000
Profit
$23,800
$24,768
23,200
Marginal
Profit
$0.44
$0.00
-$0.56
P  2.44 x 
x
2
20, 000
 5000
dP
dx
 2.44 
x
10, 000
P
Profit (in dollars)
 2 4, 4 0 0,
2 4, 7 6 8 
P  2.44 x 
x
2
 5000
20, 000
5k
20k
40k
x
# o f U n its
T he m axim um profit corresponds to the po int w here
the m arginal profit is 0. W hen m ore than 24,400 burgers
are sold, the m arginal profit is negativ e--increasing
production beyond this point w ill reduce ra ther than
increase profit.
3.10 Business and Economic Applications
D em an d F u n ctio n :
p
50
x
C ost of producing x item s: C  0 .5 x  5 0 0
Find the price per unit that yields the m axim um profit.
P  R  C  xp  (0.5 x  500)
 50 
 x
  0.5 x  500
 x 
 50 x  0.5 x  500
3.10 Business and Economic Applications
D em an d F u n ctio n :
p
50
x
C ost of producing x item s: C  0 .5 x  5 0 0
Find the price per unit that yields the m axim um profit.
S etting the m arginal profit equal to 0 w ill
give us the x that m axim izes pr ofit.
P  50 x  0.5 x  500
dP
dx

25
x
 0.5  0
W h iteb o a rd
x  2500
Sub for x
in the
demand
function
$ 1 .0 0
3.10 Business and Economic Applications
D em an d F u n ctio n :
p
50
x
C ost of producing x item s: C  0 .5 x  5 0 0
Find the price per unit that yields the m axim um profit.
M ax im u m P ro fit o ccu rs w h en :
dP

dx
dR
dx

dC
0
dx
OR
M arginal R evenue  M arginal C ost
3.10 Business and Economic Applications
C  800  0.04 x  0.0002 x
2
Find the production level that m inim izes average cost per unit.
C 
C
x

800  0.04 x  0.0002 x
x

800
 0.04  0.0002 x
x
S et
dC
dx
2
0
W h iteb o a rd
dC
dx

 800
x
2
 0.0002  0
3.10 Business and Economic Applications
C  800  0.04 x  0.002 x
2
Find the production level that m inim izes average cost per unit.
 x  2 0 0 0 u n its
HW 3.10/1,2,5,9,13,15,19,21,23,39
HW 3.10/1,2,5,9,13,15,19,21,23,39
a) F ix ed C o st
b ) C is strictly in creasin g an d p o ssib ly cu b ic 
dC
is q u ad ratic an d p o sitive.
dx
c)
dC
h as a relative m in at th e lo catio n w h ere
dx
co sts are in creasin g at th eir slo w es t rate.
HW 3.10/1,2,5,9,13,15,19,21,23,39
a)
dR
is a constant function
dx
b) P  R  C
HW 3.10/1,2,5,9,13,15,19,21,23,39
R 
1, 000, 000 x
0.02 x  1800
2
 0.02 x 2  1800  0.04 x 2 
 1, 000, 000 
 0
2
2
dx
(0.02 x  1800)


dR
 18 00  0.02 x  0  x  30 0
2
B y the first derivative test, x  300 is the locati on of a m ax .
HW 3.10/1,2,5,9,13,15,19,21,23,39
C  3000  x 300  x

 
dx

dC

x
1 / 2
1/ 2
1

300  x  x   300  x 
  300  x 
  1   
1/ 2
2
  2  300  x 
x  2  300  x 
2  300  x 
1/ 2

3 x  600
2  300  x 
1/ 2
 0
 x  20 0
B y the First D erivative T est, x  200 yields the m in average cost .
HW 3.10/1,2,5,9,13,15,19,21,23,39
C  4000  40 x  0.02 x , p  50 
2
x
100
P  50 x 
x
2
 4000  40 x  0.02 x   0.03 x  90 x  4000
2
100
dP
dx
  0.06 x  90  0  x  150 0  p  35
2
HW 3.10/1,2,5,9,13,15,19,21,23,39
C  2 x  5 x  18
2
C  2 x  5  18 x
dC
 2
dx
18
x
2
1
2 x  18
2

C  3   17
dC
dx
 4 x  5  17
x
2
 0 x 3
HW 3.10/1,2,5,9,13,15,19,21,23,39
x
Price
Profit
102 90-2(0.15)
102[90-2(0.15)]-102(60)=3029.40
104 90-4(0.15)
104[90-4(0.15)]-104(60)=3057.60
106 90-6(0.15)
106[90-6(0.15)]-106(60)=3084.60
108 90-8(0.15)
108[90-8(0.15)]-108(60)=3110.40
110 90-10(0.15)
110[90-10(0.15)]-110(60)=3135.00
112 90-12(0.15)
112[90-12(0.15)]-112(60)=3158.40
P  x   x  90   x  100   0.15    60 x
 90 x  0.15 x  15 x  60 x  45 x  0.15 x
2
dP
2
 45  0. 3 x  0  x  150
dx
2
d P
dx
2
 0  x  150 is the loc ation o f a m a x .
HW 3.10/1,2,5,9,13,15,19,21,23,39
 v
  110  11v
1
C 
 5
 550 v

60
 600
 v 
2
dC

dv
11

550
60
v
2
 0  3 3, 000  11v
2
 v  5 4 .8 m p h
2
d C
dv
2

1100
v
3
 0  v  54.8 yields a m i n
C  12  5280   6  x   16  5280 
dC
We are
back in
miles
dx
2
2
y 
2
2
0.5  x
2

2
2
  12  5280  16  5280  0.5  x  0.25 
  12  5280 
y  0.5  x
x  0.25
16 x
x  1/ 4
16 x  5280
x
2
 0.25 
0.5

 5280   12 

 0.5
2x

  0
2
x  1/ 4 
16 x
 12  3 x  1 / 4  4 x
2
2
 9  x  1 / 4   16 x
2
 9 x  9 / 4  16 x
2
2
2
 7 x  9 / 4  x  9 / 28  x 
2
2
3
2 7
0 .5 m i
6 x
x
6 mi
 0.57 m i
a) D em and  A S M O R E IT E M S A R E P R O D U C E D , D E M A N D G O E S D O W N
b) C ost  A S M O R E IT E M S A R E P R O D U C E D , C O S T G O E S U P
c) R evenue  R E V E N U E IS G R E A T E R T H A N P R O FIT
d) P rofit  P R O FIT IS LE S S T H A N R E V E N U E