Chabot Mathematics
§9.1b
The Base e
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Review § 9.1
MTH 55
 Any QUESTIONS About
• §9.1 → Exponential Functions, base a
 Any QUESTIONS About HomeWork
• §9.1 → HW-42
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Compound Interest  Terms
 INTEREST ≡ A fee charged for
borrowing a lender’s money is called the
interest, denoted by I
 PRINCIPAL ≡ The original amount of
money borrowed is called the principal,
or initial amount, denoted by P
• Then Total AMOUNT, A, that accululates in
an interest bearing account if the sum of
the Interest & Principal → A = P + I
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Compound Interest  Terms
 TIME: Suppose P dollars is borrowed.
The borrower agrees to pay back the
initial P dollars, plus the interest
amount, within a specified period. This
period is called the time (or time-period)
of the loan and is denoted by t.
 SIMPLE INTEREST ≡ The amount of
interest computed only on the principal
is called simple interest.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Compound Interest  Terms
 INTEREST RATE: The interest rate is
the percent charged for the use of the
principal for the given period. The
interest rate is expressed as a decimal
and denoted by r.
 Unless stated otherwise, it is assumed
the time-base for the rate is one year;
that is, r is thus an annual interest rate.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Simple Interest Formula
 The simple interest amount, I, on a
principal P at a rate r (expressed as
a decimal) per year for t years is
I  Prt.
Chabot College Mathematics
6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Example  Calc Simple Interest

Rosarita deposited $8000 in a bank for
5 years at a simple interest rate of 6%
a) How much interest will she receive?
b) How much money will she receive at the
end of five years?

SOLUTION a) Use the simple interest
formula with:
P = 8000, r = 0.06, and t = 5
Chabot College Mathematics
7
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Example  Calc Simple Interest

SOLUTION a)
Use Formula
I  Prt
I  $8000 0.06 5 
I  $2400

SOLUTION b)
The total amount, A, A  P  I
due her in five years A  $8000  $2400
is the sum of the
A  $10, 400
original principal
and the interest earned
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Compound Interest Formula


r
A  P 1 
n

nt
 A = $-Amount after t years
 P = Principal (original $-amount)
 r = annual interest rate (expressed as a
decimal)
 n = number of times interest is compounded
each year
 t = number of years
Chabot College Mathematics
9
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Compare Compounding Periods

One hundred dollars is deposited in a
bank that pays 5% annual interest.
Find the future-value amount, A, after
one year if the interest is compounded:
a) Annually.
b) SemiAnnually.
c) Quarterly.
d) Monthly.
e) Daily.
Chabot College Mathematics
10
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Compare Compounding Periods

SOLUTION
In each of the computations that
follow, P = 100 and r = 0.05 and t = 1.
Only n, the number of times interest is
compounded each year, is changing.
Since t = 1, nt = n∙1 = n.
n
r
a) Annual

Amount: A  P  1  n 
A  100 1  0.05   $105.00
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Compare Compounding Periods
b) Semi
Annual
Amount:
r

A  P 1  

n
n
2
0.05 

A  100  1 
 $105.06


2 
c) Quarterly A  P  1  r 


Amount:
4
4
4
0.05 

A  100  1 
 $105.09


4 
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Compare Compounding Periods
d) Monthly
Amount:
r

A  P 1  
 12 
12
0.05 

A  100  1 


12
e) Daily
Amount:
r 

A  P 1 


365
13
 $105.12
365
0.05 

A  100  1 


365 
Chabot College Mathematics
12
365
 $105.13
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
The Value of the Natural Base e
 The number e, an irrational number, is
sometimes called the Euler constant.
 Mathematically speaking, e is the fixed
h
number that
1

the expression  1  

h
approaches e as h gets larger & larger
 The value of e to 15 places:
e = 2.718281828459045
Chabot College Mathematics
14
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Continuous Compound Interest
 The formula for Interest Compounded
Continuously; e.g., a trillion times a sec.
rt
A  Pe
 A = $-Amount after t years
 P = Principal (original $-amount)
 r = annual interest rate (expressed as a
decimal)
 t = number of years
Chabot College Mathematics
15
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Example  Continuous Interest
 Find the amount when a principal of
$8300 is invested at a 7.5% annual rate
of interest compounded continuously
for eight years and three months.
 SOLUTION: Convert
8-yrs & 3-months A  Pert
to 8.25 years.
0.075 8.25 
A  $8300e
P = $8300 and
r = 0.075 then
A  $15, 409.83
use Formula
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Compare Continuous Compounding
 Italy's Banca Monte dei Paschi di Siena
(MPS), the world's oldest bank founded
in 1472 and is today one the top five
banks in Italy
 If in 1797 Thomas Jefferson Placed a
Deposit of $450k the MPS bank at an
interest rate of 6%, then find the value
$-Amount for the this Account Today,
213 years Later
Chabot College Mathematics
17
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Compare Continuous Compounding
 SIMPLE
Interest
A  P  Prt  P 1  rt 
A  $450, 000 1  0.06 213
A  $6.201 million.
 YEARLY Compounding
A  P 1  r   $450, 000 1  0.06 
t
213
A  $1.105  1011
A  $110.5 million.
Chabot College Mathematics
18
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Simple
Yearly
0.11
Quarterly
0.01
145.30
Continuous
Interest Compounding
Account $Value for $450k invested at 6% Interest for 213 Years
159.80
0
10
20
30
40
50
60
70
80
90
Account Value ($B)
Chabot College Mathematics
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100
110
120
130
140
150
160
170
M55_Sec9_1_Compare_Compounding_0810.xls
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
The NATURAL Exponential Fcn
 The exponential function
f x   e
x
 with base e is so prevalent in the
sciences that it is often referred to
as THE exponential function or the
NATURAL exponential function.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Compare 2x, ex, 3x
 Several
Exponentials
Graphically
Chabot College Mathematics
21
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Example  Xlate ex, Graphs
 Use translation
to sketch the
graph of
g x   e
x 1
2
 SOLUTION 
Move ex graph:
• 1 Unit RIGHT
• 2 Units UP
Chabot College Mathematics
22
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Example  Graph Exponential
 Graph f(x) = 2 − e−3x
 SOLUTION
Make T-Table,
Connect-Dots
x
y = f(x)
−2
−401.43
−1
−18.09
0
1
1
1.95
2
2
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Exponential Growth or Decay
 Math Model for “Natural” Growth/Decay:
A t   A0 e
kt
 A(t) = amount at time t
 A0 = A(0), the initial amount
 k = relative rate of
• Growth (k > 0); i.e., k is POSITIVE
• Decay (k < 0); i.e., k is NEGATIVE
 t = time
Chabot College Mathematics
24
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Exponential Growth
 An exponential GROWTH model is a
function of the form
At   A0e
 kt
k 0
A(t)
2A0
A0
A  A0e  kt
 where A0 is the
Doubling
population at time 0, A(t)
time
is the population at time t,
and k is the exponential growth rate
t
• The doubling time is the amount of time
needed for the population to double in size
Chabot College Mathematics
25
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Exponential Decay
 An exponential DECAY model is a
function of the form
At   A0e
 kt
k 0
A(t)
A  A0e  kt
A0
 where A0 is the
½A0
population at time 0, A(t)
Half-life
is the population at time t,
and k is the exponential decay rate
t
• The half-life is the amount of time needed
for half of the quantity to decay
Chabot College Mathematics
26
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Example  Exponential Growth


In the year 2000, the human
population of the world was
approximately 6 billion and the annual
rate of growth was about 2.1 percent.
Using the model on the previous slide,
estimate the population of the world in
the years
a) 2030
b) 1990
Chabot College Mathematics
27
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Example  Exponential Growth

SOLUTION a)
Use year 2000
as t = 0 Thus
for 2030 t = 30
A0  6
k  0.021
t  30
A t   6e
0.02130 
A t   11.265663
 The model predicts there will be
11.26 billion people in the world in the
year 2030
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Example  Exponential Growth

A0  6
SOLUTION b)
Use year 2000
k  0.021
as t = 0 Thus
t  10
for 1990 t = −10
0.02110 
A t   6e
A t   4.8635055
 The model postdicted that the world
had 4.86 billion people in 1990 (actual
was 5.28 billion).
Chabot College Mathematics
29
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
WhiteBoard Work
 Problems From §9.1 Exercise Set
• 40, 58, 63
 Calculating
e
Chabot College Mathematics
30
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
All Done for Today
World
Population
Chabot College Mathematics
31
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
32
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt