3.10 Business and Economic Applications

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...


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


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



The is the number of unitbrea s fok-even p r whichoint R C



(Marginal Revenue)dR

dx

(extra revenue from selling one additional unit)

(Marginal Cost)dC

dx

(extra cost of producing one additional unit)



(Marginal Profit)dP

dx

(extra profit from selling one additional unit)



Revenue Function

1 Unit Extra revenue

for one more

unit sold.

Marginal

Revenue

Marginal Revenue Extra Revenue


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


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


, 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


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



5000 0.56



x

C x

Marginal Profit is...

2

2.44 500020,000

xP x

2.4410,000

dP x

dx



5000 0.56



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


2500x


x


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



x


Maximum Profit occurs when:

0dP dR dC

dx dx dx

Marginal Revenue Marginal

OR

Cost


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


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


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

a) is a constant function

b)

dR

dxP R C

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

a) Demand AS MORE ITEMS ARE PRODUCED, DEMAND GOES DOWN

b) Cost AS MORE ITEMS ARE PRODUCED, COST GOES UP

c) Revenue REVENUE IS GREATER THAN PROFIT

d) Profit PROFIT IS LESS THAN REVENUE

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3.10 Business and Economic Applications