Chabot Mathematics
§5.5 Int Apps
Biz & Econ
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Review §
5.4
 Any QUESTIONS About
• §5.4 → Applying the Definite Integral
 Any QUESTIONS About HomeWork
• §5.4 → HW-25
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
§5.5 Learning Goals
 Use integration to compute the future
and present value of an income flow
 Define consumer willingness to spend
as a definite
integral, and use
it to explore
consumers’
surplus and
producers’ surplus
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Time Value of Money
 We say that money has a time value
because that money can be invested
with the expectation of earning a
positive rate of return
• In other words, “a dollar received today is
worth more than a dollar to be received
tomorrow”
– That is because today’s dollar can be
invested so that we have more than one dollar
tomorrow
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Terminology of Time Value
 Present Value → An amount of money today,
or the current value of a future cash flow
 Future Value → An amount of money at some
future time period
 Period → A length of time (often a year, but can
be a month, week, day, hour, etc.)
 Interest Rate → The compensation paid to a
lender (or saver) for the use of funds expressed
as a percentage for a period (normally
expressed as an annual rate)
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Time Value Abreviations




PV → Present value
FV → Future value
Pmt → Per period payment $-amount
N → Either the
• total number of cash flows or
• the number of a Payment periods
 i → The interest rate per period
• Usually %/Yr expressed as a fraction
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
TimeLines
 A timeline is a graphical device used to
clarify the timing of the cash flows for an
investment
 Each tick represents one time period
PV
0
Today
Chabot College Mathematics
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FV
1
2
3
4
5
5 TimePeriods Later
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Finding Future Value by Arith
 Consider $100 ($1 cNote) invested
today at an interest rate of 10% per year
or 0.10/yr as decimal
 Then the $-Value expected in 1 year
$100
?
0
1
2
3
4
5
FV1  1001  010
.   110
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Finding Future Value by Arith
 Now Extend the Investment for another
Year (“Let it Ride”)
 Then the $-Value expected after the 2nd
year with no additional investment
$110
?
1
2
0
3
4
5
FV2  1001  010
. 1  0.10  121
or
2
FV2  1001  010
.   121
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Recognize Future Value Pattern
 Engaging in the EXTREMELY
VALUABLE Practice of PATTERN
RECOGNITION surmise The Pattern
that is developing for FV in year-N
FVN  PV1  i
N
 If the $1c investment were extended for
a 3rd Year then the FV
FV3  1001  010
.   13310
.
3
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Present Value by Arithmetic
 What is the $Amount TODAY (the
Present Value, PV) needed to realize a
FV $Goal after N-years invested at
interest rate i per year?
 Solve the FV equation for PV
FVN  PV 1  i 
N
Divide Both Sides by 1  i 
FVN
PV 
N
1  i 
N
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Example  Present Value
 Tadesuz has a 5-year old daughter for which
he now plans for college $-expenses.
 Tadesuz lives in San Leandro, and he
develops this college plan for his daughter
• She can live at home until she is 22
• Attends Chabot and takes the Lower-Division
Courses needed for University Transfer
• Transfers to UCBerkeley (Go Bears!) where
she earns her Bachelors Degree
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Example  Present Value
 Tadesuz estimates that he will need
about $30k on her 18th birthday to pay
for her bacaluarate Education.
 If he can earn 8% per year on his ONETime Initial investment, then how much
must he invest today to achieve the
$30k Future-Value Goal?
 SOLUTION: After 18-5 periods the PV:
Chabot College Mathematics
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$30k
PV 
 $11.031k
13
1  .0.08 $30k
PV 
 $11.031k
Bruce Mayer, PE
13
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Annuities
 An annuity is a series of nominally equal
$-payments equally spaced in time
• Annuities are very common: Bldg Leases,
Mortgage payments, Car payments,
Pension income
 The timeline shows an example of a 5year, $100 ($1 cNote) annuity
0
Chabot College Mathematics
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100
100
100
100
100
1
2
3
4
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Principle of Value Additivity
 To find the value (PV or FV) of an
annuity first consider principle of value
additivity:
• The value of any stream of cash flows is
equal to the sum of the values of the
components
 Thus can move the cash flows to the
same time period, and then simply add
them all together to get the total value
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
PRESENT Value of an Annuity
 Use the principle of value additivity to
find the present value of an annuity, by
simply summing the present values of
each of the components:
N
PVA 
 1  i
Pmt t
t 1
Chabot College Mathematics
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t

Pmt 1
1  i
1

Pmt 2
1  i
2
  
Pmt N
1  i
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
N
Present Value of an Annuity
 Using the example, and assuming a
discount rate (a.k.a., interest rate) of
10% per year, find that the PVA as:
100
PVA 
. 
110
1

100
. 
110
2

100
. 
110
3

100
. 
110
4

100
. 
110
5
 379.08
62.09
68.30
75.13
82.64
90.91
379.08
100
100
100
100
100
1
2
3
4
5
0
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Present Value of an Annuity
 Actually, there is no need to take the
present value of each cash flow
separately
 Using Convergent Series Analysis find a
closed-form of the PVA equation
instead:
N
PVA 
 1  i
t 1
Chabot College Mathematics
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Pmt t
t
1  1

N

1  i 

 Pmt 

i




N
Pmtt
PVA  
t
1  i 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Present Value of an Annuity
 Using the PVA equation in the $1c
1  1

example
5

PVA  Pmt 


. 
110
010
.

  379.08


• Thus a 5yr constant yearly income Annuity
of $100/yr, discounted by 10% has PV of
$379
 The PVA equation works for all regular
annuities, regardless of the number
of payments
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
FUTURE Value of an Annuity
 Use the principle of value additivity to
find the Future Value of an annuity, by
simply summing the future values of
each of the components
N
FVA 
 Pmt 1  i
t
Nt
 Pmt 1 1  i
N 1
 Pmt 2 1  i
N 2
    Pmt N
t 1
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Future Value of an Annuity
 Again consider a $1c annuity, and
assume a discount rate of 10% per
year, find that the future value:
FVA  100110
.   100110
.   100110
.   100110
.   100  610.51
4
3
2
1
146.41
133.10
121.00
110.00
0
100
100
100
100
100
1
2
3
4
5
Chabot College Mathematics
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}
= 610.51
at year 5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Future Value of an Annuity
 As was done for the PVA equation, use
series convergence to find a closedform of the FVA equation:
N
FVA 
 Pmt 1  i
t
t 1
Nt
 1  i N  1 

 Pmt 
i


 As with The PVA, the FVA eqn works for
all regular annuities, regardless of
the number of payments
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
PV of an Income Stream
 Now assume the Pmt is divided into k
payments per year (say 12) and then
the discount is also applied k times a yr
• Call this an Income Stream as the
payments are NO Longer made annually
 N  t k
 Then the
Nk
i

FVA → FVIS FVIS   Pmt kt  1  
 k
kt 1
eqn
• Note that Pmtkt may VARY in time;
e.g., Pmt73≠ Pmt74
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
PV of an Income Stream
 The discounts Occur infinitely often, and
the Pmtkt becomes continuously
variable in time, then the PVIS equation
becomes
N
FVIS  e
i N

0
f t   e dt
it
 Or textbook notation
FV  e
Chabot College Mathematics
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rT
T

0
f t  e dt
 rt
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Example  PV of Income Stream
 Yasiel’s grandfather promises to
contribute continuously at a rate of
$10,000 per year to a trust fund earning
3% interest as long as the boy
maintains a 3.0 GPA in school.
 If Yasiel maintains the required grades
for 8 years during high school and
college, what is the value of the trust at
the end of that period?
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Example  PV of Income Stream
 The trust’s value is the future value of
the annuity into which Yasiel’s
grandfather is paying. Since the money
accrues at a rate of $10,000 per year,
and is simultaneously invested, its
T
future value is
rT
-rt
FV = e ò 10000e dt
0
e
Chabot College Mathematics
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0.03(8)
8
 10000e
0.03t
0
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
dt
Example  PV of Income Stream
 Running the Numbers
8
FV  e

0.24
e
 1
0.03t 
10000
e

  0.03
0

10000 0.03(8) 0.03( 0)
e
e
 0.03
0.24

 90,416.38
 Thus the fund is worth about $90,416 at
the end of eight years.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Present Value of an Income Stream
 From the PV discussion, taking
payment infinitely often, and the
payments to becomes continuously
variable in time, then the FVIS equation
becomes
PV   f t  e dt
T
 rt
0
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Example  Present Value
 Instead of investing at a continuous rate
of $10,000 per year over the eight
years, Yasiel’s grandfather in could
have invested for eight years a lump
sum of
8
0.03t
PV   10000e
dt
0
10000 é -0.03(8) -0.03(0) ù
=
-e
ëe
û
-0.03
= $71,124.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
CAL Football Tickets
Value of Ticket to Potential Demanders
Peter
$200
Paul
Mary
Jack
Jill
$150
$100
$50
$50
Price
Peter
Value of Ticket to Potential Suppliers:
Professor V
$50
Professor W
Professor X
Professor Y
Professor Z
$50
$100
$150
$200
Z
200
Paul
Y
150
Mary
100
X
Jack and Jill
50
V and W
0
1
2
3
4
5
Tickets
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
CAL Football Tickets
Equilibrium Price = $100
Peter, Paul and Mary buy tickets from
Professors V, W and X. If they all Buy & Sell
at the equilibrium price, does it matter who
buys from whom? → No
Price
Consumer Surplus
Peter
Gains:
Peter
Paul
Mary
V
W
X
Z
200
= $200 - $100 = $100
= $150 - $100 = $50
= $100 - $100 = $0
= $100 - $50 = $50
= $100 - $50 = $50
= $100 - $100 = $0
Paul
Y
150
Mary
100
X
Jack and Jill
50
V and W
Total Gain: $250
0
1
2
3
4
Tickets
Producer Surplus
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5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Consumers’ Surplus
Price
($/unit)
Consumer Surplus
Maximum Willingness to Pay
(or Spend) for Qo
WS  
Q Q0
Q 0
Po
Q Q  dQ
What is Actually Paid
AP  
Q Q0
Q 0
P0 dQ  P0  Q0
D(Q)
Qo
Quantity (Units)
• By Supply & Demand the Price settles at
P0, but SOME Consumers are willing to
pay MUCH MORE, thus these consumers
save $-Money
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Consumers’ Surplus
 Thus the Surplus Total $-Funds kept by
the consumers:
CS = [Amount Willing to Spend] − [Amount Actually Paid]
 With Reference to the Areas shown on
the Supply-n-Demand Graph
CS  WS  AP  
Q Q0
Q 0
Chabot College Mathematics
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QQ  dQ  P0  Q0
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Producers’ Surplus
Price
($/Unit)
Producer Surplus
S(Q)
$-Amount
Actually paid
Po
AP  
Q Q0
Q 0
P0 dQ  P0  Q0
Minimum $-Amount Needed
to Supply Qo
AN  
Q Q0
Q 0
Qo
S Q  dQ
Quantity (units)
• By Supply & Demand the Price settles at
P0, but SOME Producrs are willing to
accept a LOWER Price, thus these
Producers make extra $-Money
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Consumer & Producer Surplus
 The Two Surpluses usually exist
simultaneously
Price
($/Unit)
Consumer
Surplus
S(Q)
Equilibrium
Price-Point
Po
Producer Surplus
D(Q)
Qo
Chabot College Mathematics
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Quantity (Units)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
WhiteBoard Work
 Problems From §5.5
• P18 → Supply &
Demand
• P26 → Construction
Decision
• P42 → Invention
Profit
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
All Done for Today
Tree
NPV
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
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BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
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BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
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BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
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BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
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BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
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BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
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BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
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BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
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BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
P55_42_AirPurifiers_1307.mn
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53
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx
Chabot College Mathematics
54
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-26_sec_5-5_Integral_Apps_Biz-n-Econ.pptx