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3.10 Business and Economic Applications. 3.10 Business and Economic Applications. Marginal Cost. Marginal Profit. Marginal Revenue. 3.10 Business and Economic Applications. 3.10 Business and Economic Applications. 3.10 Business and Economic Applications. - PowerPoint PPT Presentation
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Knowing the rates of change of , , and
with respect to th number of units produced ore is one
of the keys
s
to
profit rev
business
o
enue c
succe
ost
ld
ss.
It's the difference between big $$$$$$$ BANKRUP & TCY.
Do Not Write; Definitions to Follow
3.10 Business and Economic Applications
Economists refer to these concepts as...
3.10 Business and Economic Applications
Do Not Write; Definitions to Follow
Summary of Business Terms and Formulas
is the number of units produced (or sold)x
is the price per unitp
is the total revenue for selling unitsxR
R xp
3.10 Business and Economic Applications
is the total profit.P
P R C xp C
is the total cost of producing unitsxC
is the average cost per unitC
CC
x
3.10 Business and Economic Applications
Summary of Business Terms and Formulas
The is the number of unitbrea s fok-even p r whichoint R C
3.10 Business and Economic Applications
Summary of Business Terms and Formulas
(Marginal Revenue)dR
dx
(extra revenue from selling one additional unit)
(Marginal Cost)dC
dx
(extra cost of producing one additional unit)
3.10 Business and Economic Applications
Summary of Business Terms and Formulas
(Marginal Profit)dP
dx
(extra profit from selling one additional unit)
3.10 Business and Economic Applications
Summary of Business Terms and Formulas
Revenue Function
1 Unit Extra revenue
for one more
unit sold.
Marginal
Revenue
Marginal Revenue Extra Revenue
3.10 Business and Economic Applications
A manufacturer determines that the derived from
selling of a certain item
profit
un is givets yi n bx30.0002 10P x x
marginal prA. Find the for 50 units sofit old.
dP
dx 20.0006 10x
$11.50
We’ll profit approximately $11.50 more if
we sell 51 units.
50x
dP
dx
3.10 Business and Economic Applications
A manufacturer determines that the profit derived from
selling units of a certain item is given byx30.0002 10P x x
B. Compare the $11.50 with the actual gain in profit obtained by
increasing 50 tosales 51 u f nrom its.
50P $525.00
51P $536.53
the additional profit obtained is actually
$11.53
$11.5050x
dP
dx
The price written in terms of the # of units sold
is called DEMAND FUNCthe N.TIO
p
x
3.10 Business and Economic Applications
, 2000,10 & 2250,9.75
.25 25 1
250 25,000 1000
110 2000
1000
10 21000
121000
x
x p
m
y x
xy y
Finding the demand function
Find the corresponding to thd iemand funct s prediction ion.
It is predicted that monthly sales will increase
250 items $0.25 reducti
by
for each on in price.
2000 items per A business sells at a price omonth $10f
e
.00
ach.
}LINEAR
We're not leaving in
point-slope, so we
can easily apply this
demand function.
&
are constant
x y
,x p
Two points determine a line :)
Finding the demand function
121000
xp
So what would you suggest we set the price at if we want to
sell appr 30ox 00imat itely ems?
300012
1000p $9.00
# of items
Price per item
3.10 Business and Economic Applications
Demand for burgers is60,000
2 000.
0,
xp
(marginal revenFind the increase in revenue per burger
for monthly sales of
ue)
20,000 burg ers.
R xp 60,000
20,000
xx
Profit
2
320,000
xx
310,000
dR x
dx
$1/ unitIf we sell 20,001 burgers we'll bring in $1.00 more.
Suppose the cost of producing burgers is
5000 0.56
Find the total profit and the marginal profit for
20,000, for 24,400, and for 30,000 units (burgers).
x
C x
P R C 2
3 5000 0.5620,000
xP x x
Revenue
2
2.44 500020,000
xx
3.10 Business and Economic Applications
Suppose the cost of producing burgers is
5000 0.56
Find the total profit and the marginal profit for
20,000, for 24,400, and for 30,000 units (burgers).
x
C x
Marginal Profit is...
2
2.44 500020,000
xP x
2.4410,000
dP x
dx
3.10 Business and Economic Applications
Suppose the cost of producing burgers is
5000 0.56
Find the total profit and the marginal profit for
20,000, for 24,400, and for 30,000 units (burgers).
x
C x
Demand 20,000 24,400 30,000
Profit $23,800 $24,768 23,200
Marginal Profit
$0.44 $0.00 -$0.56
2
2.44 500020,000
xP x 2.44
10,000
dP x
dx
The maximum profit corresponds to the point where
the marginal profit is 0. When more than 24,400 burgers
are sold, the marginal profit is negative--increasing
production beyond this point will rareduce ther than
increase profit.
x
P
# of Units
Pro
fit (
in d
olla
rs)
5k
20k 40k
24,400, 24,768
2
2.44 500020,000
xP x
Demand Function: 50p
x
Cost of producing items:x 0.5 500C x Find the price per unit that yields the maximum profit.
(0.5 500)P R C xp x 50
0.5 500x xx
50 0.5 500x x
3.10 Business and Economic Applications
2500x
Demand Function: 50p
x
Cost of producing items:x 0.5 500C x Find the price per unit that yields the maximum profit.
Setting the marginal profit equal to will
give us the that maximizes pr .
0
ofitx
50 0.5 500P x x 25
0.5dP
dx x 0
Whiteboard Sub for x in the
demand function
$1.00
3.10 Business and Economic Applications
Demand Function: 50p
x
Cost of producing items:x 0.5 500C x Find the price per unit that yields the maximum profit.
Maximum Profit occurs when:
0dP dR dC
dx dx dx
Marginal Revenue Marginal
OR
Cost
3.10 Business and Economic Applications
Find the production level that minimizes average cost per unit.
CC
x
8000.04 0.0002x
x
Set 0dC
dx 2
8000.0002 0
dC
dx x
2800 0.04 0.0002C x x
2800 0.04 0.0002x x
x
Whiteboard
3.10 Business and Economic Applications
Find the production level that minimizes average cost per unit.
2800 0.04 0.002C x x
2000 unitsx
HW 3.10/1,2,5,9,13,15,19,21,23,39
3.10 Business and Economic Applications
HW 3.10/1,2,5,9,13,15,19,21,23,39
a) Fixed Cost
b) is strictly increasing and possibly cubic
is quadratic and positive.
c) has a relative min at the location where
costs are increasing at their slowest rate.
C
dC
dxdC
dx
HW 3.10/1,2,5,9,13,15,19,21,23,39
2
2 2
2 2
2
1,000,000
0.02 1800
0.02 1800 0.041,000,000 0
(0.02 1800)
18
By the first derivative test, 300 is the locati
00 0.02 0
on of a max
0
.
30
xR
x
dR x x
dx x
x x
x
HW 3.10/1,2,5,9,13,15,19,21,23,39
1/ 2 1/ 2
1/ 2
1/ 2 1/ 2
3000 300
1300 300 1 300
2 2 300
2 300 3 6000
2 300 2 300
20
By the First Derivative Test, 200 yields the min average cost
0
.
C x x
dC xx x x x
dx x
x x x
x x
x
x
HW 3.10/1,2,5,9,13,15,19,21,23,39
2
22 2
4000 40 0.02 , 50100
50 4000 40 0.02 0.03 90 4000100
0.06 90 0 150 350
xC x x p
xP x x x x x
dPx x
dxp
HW 3.10/1,2,5,9,13,15,19,21,23,39
2
1
2
2 2
2 5 18
2 5 18
18 2 182 3
3 17
5 7
0
4 1
C x x
C x x
dC x
dx x xx
C
dCx
dx
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
2
2
2
2
90 100 0.15 60
90 0.15 15 60 45 0.15
150
150 is the loc
45 0.
ation o
3
f a .0 a
0
m x
P x x x x
x x x x
dPx
dx
x x
x
xd P
dx
HW 3.10/1,2,5,9,13,15,19,21,23,39
21
22
2
2 3
110 115 550
600 60
11 5500 3
54
3,000 1160
11000 54.8 yields a
.8 mp
mi
h
n
v vC v
v
dCv
dv v
d Cv
dv v
v
2
0.52
0.5 22
2
2
2 2
2 2
2 2
12 5280 6 16 5280 0.25
12 5280 16 5280 0.5 0.25 2
16 5280 1612 5280 5280 12 0
1/ 40.25
1612 3 1/ 4 4
1/ 4
9 1/ 4 16
9 9 / 4 16
37 9 / 4 9 / 2 0.57 m8
2 7i
C x x
dCx x
dx
x x
xx
xx x
x
x x
x x
x x x
0.5 mi
6 mi
6 x
x
2 2 2
2 2
0.5
0.5
y x
y x
We are back in miles
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