Chapter 3 Exponential, Logistic, and Logarithmic Functions

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Chapter 3 Exponential, Logistic,
and Logarithmic Functions
Quick Review
Evaluate the expression without using a calculator.
1. -125
3
27
64
3. 27
Rewrite the expression using a single positive exponent.
2.
3
4/3
4.  a
-3

2
Use a calculator to evaluate the expression.
5. 3.71293
5
Slide 3- 2
Quick Review Solutions
Evaluate the expression without using a calculator.
1. -125
3
-5
27
3
64
4
3. 27
81
Rewrite the expression using a single positive exponent.
2.
3
4/3
1
a
Use a calculator to evaluate the expression.
4.  a
-3

2
5. 3.71293
5
Slide 3- 3
6
1.3
Exponential Functions
Let a and b be real number constants. An exponential function in x is a
function that can be written in the form f ( x)  a  b , where a is nonzero,
x
b is positive, and b  1. The constant a is the initial value of f (the value
at x  0), and b is the base.
Slide 3- 4
Determine if they are exponential
functions
•
•
•
•
•
 
 
 
ℎ 
 
= 4
= 6 −9
= −2 ∗ 1.5
= 7 ∗ 3−
= 5 ∗ 63
Answers
•
•
•
•
•
Yes
No
Yes
Yes
no
Sketch an exponential function
Example Finding an Exponential
Function from its Table of Values
Determine formulas for the exponential function g and h whose values are
given in the table below.
Slide 3- 8
Example Finding an Exponential
Function from its Table of Values
Determine formulas for the exponential function g and h whose values are
given in the table below.
Because g is exponential, g ( x)  a  b . Because g (0)  4, a  4.
x
Because g (1)  4  b  12, the base b  3. So, g ( x)  4  3 .
1
x
Because h is exponential, h( x)  a  b . Because h(0)  8, a  8.
x
1
Because h(1)  8  b  2, the base b  1/ 4. So, h( x)  8    .
Slide 3- 9
4
x
1
Exponential Growth and Decay
For any exponential function f ( x)  a  b and any real number x,
f ( x  1)  b  f ( x).
x
If a  0 and b  1, the function f is increasing and is an exponential
growth function. The base b is its growth factor.
If a  0 and b  1, the function f is decreasing and is an exponential
decay function. The base b is its decay factor.
Slide 3- 10
Sketch exponential graph and
determine if they are growth or decay
•   = 2
•   =
1 
3
• ℎ  = 4−
Example Transforming
Exponential Functions
Describe how to transform the graph of f ( x)  2 into the graph of g ( x)  2 .
x
x -2
Slide 3- 12
Example Transforming
Exponential Functions
Describe how to transform the graph of f ( x)  2 into the graph of g ( x)  2 .
x
x -2
The graph of g ( x)  2 is obtained by translating the graph of f ( x)  2 by
x -2
x
2 units to the right.
Slide 3- 13
Example Transforming
Exponential Functions
Describe how to transform the graph of f ( x)  2 into the graph of g ( x)  2 .
x
-x
The graph of g ( x)  2 is obtained by reflecting the graph of f ( x)  2 across
x
x
the y-axis.
Slide 3- 14
Group Activity
• Use this formula 1 +
•
•
•
•
•
•
•
•
1 

Group 1 calculate when x=1
Group 2 calculate when x=2
Group 3 calculate when x=4
Group 4 calculate when x=12
Group 5 calculate when x=365
Group 6 calculate when x=8760
Group 7 calculate when x=525600
Group 8 calculate when x=31536000
• What do you guys notice?
The Natural Base e
 1
e  lim 1  
 x
x 
Slide 3- 16
x
Exponential Functions and the Base e
Any exponential function f ( x)  a  b can be rewritten as f ( x)  a  e ,
x
for any appropriately chosen real number constant k .
If a  0 and k  0, f ( x)  a  e is an exponential growth function.
If a  0 and k  0, f ( x)  a  e is an exponential decay function.
kx
kx
Slide 3- 17
kx
Exponential Functions and the Base e
Slide 3- 18
Example Transforming
Exponential Functions
Describe how to transform the graph of f ( x)  e into the graph of g ( x)  e .
x
3x
Slide 3- 19
Example Transforming
Exponential Functions
Describe how to transform the graph of f ( x)  e into the graph of g ( x)  e .
x
3x
The graph of g ( x)  e is obtained by horizontally shrinking the graph of
3x
f ( x)  e by a factor of 3.
x
Slide 3- 20
Logistic Growth Functions
Let a, b, c, and k be positive constants, with b  1. A logistic growth function
in x is a function that can be written in the form f ( x) 
f ( x) 
Slide 3- 21
c
1 a  e
 kx
c
or
1 a b
where the constant c is the limit to growth.
x
Example: Graph and Determine the
horizontal asymptotes
•   =
7
1+3∗.6
Answer
• Horizontal asymptotes at y=0 and y=7
• Y-intercept at (0,7/4)
Group Work: Graph and determine the
horizontal asymptotes
• () =
26
1+2 −4
Answer
• Horizontal asymptotes y=0 and y=26
• Y-intercept at (0,26/3)
Word Problems:
• Year 2000
• Year 2010
782,248 people
923,135 people
• Use this information to determine when the
population will surpass 1 million people? (hint
use exponential function)
Group Work
• Year 1990
• Year 2000
156,530 people
531,365 people
• Use this information and determine when the
population will surpass 1.5 million people?
Word Problem
• The population of New York State can be modeled by
•   =
19.875
1+57.993 −0.035005
•   ℎ       ℎ     1800
• A) What’s the population in 1850?
• B) What’s the population in 2010?
• C) What’s the maximum sustainable population?
Answer
• A) 1,794,558
• B) 19,161,673
• C) 19,875,000
Group Work
In chemistry, you are given half-life formulas


  = 0 
 = ℎ ,  =
, 0 =  
If you are given a certain chemical have a halflife of 56.3 minutes. If you are given 80 g first,
when will it become 16 g?
Homework Practice
• P 286 #1-54 eoe
EXPONENTIAL AND LOGISTIC
MODELING
Review
• We learned that how to write exponential
functions when given just data.
• Now what if you are given other type of data?
That would mean some manipulation
Quick Review
Convert the percent to decimal form or the decimal into a percent.
1. 16%
2. 0.05
3. Show how to increase 25 by 8% using a single multiplication.
Solve the equation algebraically.
4. 20  b  720
Solve the equation numerically.
5. 123  b  7.872
2
3
Slide 3- 34
Quick Review Solutions
Convert the percent to decimal form or the decimal into a percent.
1. 16% 0.16
2. 0.05 5%
3. Show how to increase 25 by 8% using a single multiplication.
Solve the equation algebraically. 25 1.08
4. 20  b  720  6
Solve the equation numerically.
5. 123  b  7.872 0.4
2
3
Slide 3- 35
Exponential Population Model
If a population P is changing at a constant percentage rate r each year, then
P(t )  P (1  r ) , where P is the initial population, r is expressed as a decimal,
t
0
and t is time in years.
Slide 3- 36
0
Example:
• You are given   = 300 1.05 
• Is this a growth or decay? What is the rate?
Example Finding Growth and Decay
Rates
Tell whether the population model P(t )  786,543 1.021 is an exponential
t
growth function or exponential decay function, and find the constant percent
rate of growth.
Slide 3- 38
Example
• You are given   = 800 .15

• Is this a growth or decay? What is the rate?
Example Finding an Exponential
Function
Determine the exponential function with initial value=10,
increasing at a rate of 5% per year.
Slide 3- 40
Group Work
• Suppose 50 bacteria is put into a petri dish
and it doubles every hour. When will the
bacteria be 350,000?
Answer
•   = 50 2 
• 350000 = 50 2
• t=12.77 hours

Example Modeling Bacteria Growth
Suppose a culture of 200 bacteria is put into a petri dish and the culture
doubles every hour. Predict when the number of bacteria will be 350,000.
Slide 3- 43
Group Work: half-life
• Suppose the half-life of a certain radioactive
substance is 20 days and there are 10g
initially. Find the time when there will be 1 g
of the substance.
answer
• Just the setting up
•   = 
• 1 = 10
1
2

ℎ

20
Group Work
• You are given   = 150 1.025
• When will this become 150000?

Example Modeling U.S. Population
Using Exponential Regression
Use the 1900-2000 data
and exponential
regression to predict the
U.S. population for 2003.
Slide 3- 47
Example Modeling a Rumor
A high school has 1500 students. 5 students start a rumor, which spreads
logistically so that S (t )  1500 /(1  29  e ) models the number of students
who have heard the rumor by the end of t days, where t  0 is the day the
-0.9 t
rumor begins to spread.
(a) How many students have heard the rumor by the end of Day 0?
(b) How long does it take for 1000 students to hear the rumor?
Slide 3- 48
Example Modeling a Rumor: Answer
A high school has 1500 students. 5 students start a rumor, which spreads
logistically so that S (t )  1500 /(1  29  e ) models the number of students
who have heard the rumor by the end of t days, where t  0 is the day the
-0.9 t
rumor begins to spread.
(a) How many students have heard the rumor by the end of Day 0?
(b) How long does it take for 1000 students to hear the rumor?
(a) S (0)  1500 /(1  29  e
-0.9 ( 0 )
)
 1500 /(1  29 1)
 1500 / 30  50. So 50 students have heard the rumor by the end of day 0.
(b) Solve 1000  1500 /(1  29  e ) for t.
-0.9 t
t  4.5. So 1000 students have heard the rumor half way
through the fifth day.
Slide 3- 49
Key Word
• Maximum sustainable population
• What does this mean? What function deals
with this?
Maximum Sustainable Population
Exponential growth is unrestricted, but
population growth often is not. For many
populations, the growth begins exponentially,
but eventually slows and approaches a limit to
growth called the maximum sustainable
population.
Slide 3- 51
Homework Practice (Do in class also)
• P 296 #1-44 eoo
LOGARITHMIC FUNCTION, GRAPHS
AND PROPERTIES
Quick Review
Evaluate the expression without using a calculator.
1. 6
8
2.
2
3. 7
Rewrite as a base raised to a rational number exponent.
1
4.
e
-2
11
32
0
3
5. 10
4
Slide 3- 54
Quick Review Solutions
Evaluate the expression without using a calculator.
1
1. 6
36
8
2.
2
2
3. 7 1
-2
11
32
0
Rewrite as a base raised to a rational number exponent.
1
4.
e
3
5. 10
4
e
3 / 2
1/ 4
10
Slide 3- 55
Changing Between Logarithmic and
Exponential Form
If x  0 and 0  b  1, then y  log ( x) if and only if b  x.
y
b
Slide 3- 56
Group Work: transform logarithmic
form into exponential form
• A)5 25 = 2
• B)3 3 =
1
2
• C) 2 60 = 
• D)  = 8
Group Work: convert exponential form
into logarithmic form
• 5 = 34
• 5
−2
=
1
25
• 40 = 1
• 161 = 
Inverses of Exponential Functions
Slide 3- 59
Basic Properties of Logarithms
For 0  b  1, x  0, and any real number y.
log 1  0 because b  1.
0
b
log b  1 because b  b.
1
b
log b  y
y
b
b
logb x
Slide 3- 60
x
because b  b .
y
y
because log x  log x.
b
b
An Exponential Function and Its
Inverse
Slide 3- 61
Common Logarithm – Base 10
• Logarithms with base 10 are called common
logarithms.
• The common logarithm log10x = log x.
• The common logarithm is the inverse of the
exponential function y = 10x.
Slide 3- 62
Basic Properties of Common
Logarithms
Let x and y be real numbers with x  0.
log1  0 because 10  1.
log10  1 because 10  10.
0
1
log10  y because 10  10 .
10  x because log x  log x.
y
log x
Slide 3- 63
y
y
Example Solving Simple Logarithmic
Equations
Solve the equation by changing it to exponential form.
log x  4
Slide 3- 64
Example Solving Simple Logarithmic
Equations
Solve the equation by changing it to exponential form.
log x  4
x  10  10, 000
4
Slide 3- 65
Basic Properties of Natural
Logarithms
Let x and y be real numbers with x  0.
ln1  0 because e  1.
0
ln e  1 because e  e.
ln e  y because e  e .
e  x because ln x  ln x.
1
y
y
y
ln x
Slide 3- 66
Graphs of the Common and Natural
Logarithm
Slide 3- 67
Example Transforming
Logarithmic Graphs
Describe how to transform the graph of y  ln x into the graph of
h( x)  ln(2  x).
Slide 3- 68
Example Transforming
Logarithmic Graphs
Describe how to transform the graph of y  ln x into the graph of
h( x)  ln(2  x).
h( x)  ln(2  x)  ln[( x  2)]. So obtain the graph of h( x)  ln(2 - x) from
y  ln x by applying, in order, a reflection across the y -axis followed by
a translation 2 units to the right.
Slide 3- 69
Quick Review
Evaluate the expression without using a calculator.
1. log10
2. ln e
3
3
3. log 10
-2
Simplify the expression.
3
3
2
2
xy
4.
x y
x y
2
5.
4
2x
Slide 3- 70

3
1/ 2
Quick Review Solutions
Evaluate the expression without using a calculator.
1. log10
2. ln e
3
3
3
3
3. log 10
-2
-2
Simplify the expression.
3
3
2
2
xy
4.
x y
x y 
2
5.
x
y
4
2x
3
5
5
1/ 2
4
x y
2
2
Slide 3- 71
What you’ll learn about
•
•
•
•
Properties of Logarithms
Change of Base
Graphs of Logarithmic Functions with Base b
Re-expressing Data
… and why
The applications of logarithms are based on their many
special properties, so learn them well.
Slide 3- 72
Properties of Logarithms
Let b, R, and S be positve real numbers with b  1, and c any real number.
Product rule:
log ( RS )  log R  log S
b
Quotient rule:
Power rule:
Slide 3- 73
b
b
R
log    log R  log S
S
log ( R)  c log R
b
b
c
b
b
b
Example Proving the Product Rule for
Logarithms
Prove log ( RS )  log R  log S.
b
Slide 3- 74
b
b
Example Proving the Product Rule for
Logarithms
Prove log ( RS )  log R  log S.
b
b
b
Let x  log R and y  log S . The corresponding exponential statements
b
b
are b  R and b  S . Therefore,
x
y
RS  b  b
RS  b
x
y
x y
log ( RS )  x  y
b
change to logarithmic form
log ( RS )  log R  log S
b
Slide 3- 75
b
b
Example Expanding the Logarithm of a
Product
Assuming x is positive, use properties of logarithms to write
log  3 x
5
Slide 3- 76
 as a sum of logarithms or multiple logarithms.
Example Expanding the Logarithm of a
Product
Assuming x is positive, use properties of logarithms to write
log  3 x
5
 as a sum of logarithms or multiple logarithms.
log  3x   log 3  log  x
5
5
 log 3  5log x
Slide 3- 77

Example Condensing a Logarithmic
Expression
Assuming x is positive, write 3ln x  ln 2 as a single logarithm.
Slide 3- 78
Example Condensing a Logarithmic
Expression
Assuming x is positive, write 3ln x  ln 2 as a single logarithm.
3ln x  ln 2  ln x  ln 2
3
x 
 ln  
2
3
Slide 3- 79
Group Work
•  log(7 2  5 )
Group Work
• Expand
•
17
 2
 
Group Work
• Express as a single logarithm
•   −   + 5 
Group Work
• Express as a single logarithm
• 42  +
2
2 
5
− 32 
Change-of-Base Formula for
Logarithms
For positive real numbers a, b, and x with a  1 and b  1,
log x
log x 
.
log b
a
b
a
Slide 3- 84
Example Evaluating Logarithms by
Changing the Base
Evaluate log 10.
3
Slide 3- 85
Example Evaluating Logarithms by
Changing the Base
Evaluate log 10.
3
log 10 
3
Slide 3- 86
log10
1

 2.096
log 3 log 3
Solving
• 4 = 51
Solving
• ln 
Solving
• log 1
Solving
• log 5 = log 4 + log( − 3)
Solving
• 5 56 = 
Solving
• 25+3 = 16
Homework Practice
• Pg 317 #1-50 eoe
EQUATION SOLVING AND
MODELING
Quick Review
Prove that each function in the given pair is the inverse of the other.
1. f ( x)  e and g ( x)  ln  x
3x
1/ 3
2. f ( x)  log x and g ( x)  10
2

x/2
Write the number in scientific notation.
3. 123,400,000
Write the number in decimal form.
4. 5.67  10
5. 8.91 10
Slide 3- 95
8
-4
Quick Review Solutions
Prove that each function in the given pair is the inverse of the other.
1. f ( x)  e and g ( x)  ln  x
3x
1/ 3
2. f ( x)  log x and g ( x)  10
2

x/2
f ( g ( x))  e

3 ln x1 / 3

e
f ( g ( x))  log 10
x/2
ln  x 

2
x
 log10  x
x
Write the number in scientific notation.
3. 123,400,000 1.234  10
8
Write the number in decimal form.
4. 5.67  10
567, 000, 000
8
5. 8.91 10
-4
0.000891
Slide 3- 96
One-to-One Properties
For any exponential function f ( x)  b ,
If b  b , then u  v.
For any logarithmic function f ( x)  log x,
x
u
v
b
If log u  log v, then u  v.
b
Slide 3- 97
b
Example Solving an Exponential
Equation Algebraically
Solve 40 1/ 2   5.
x/2
Slide 3- 98
Example Solving an Exponential
Equation Algebraically
Solve 40 1/ 2   5.
x/2
40 1/ 2   5
x/2
1
1/ 2  
8
1
1

 
 
2
2
x/2  3
x6
x/2
x/2
Slide 3- 99
divide by 40
3
1 1
 
8 2
one-to-one property
3
Example Solving a Logarithmic
Equation
Solve log x  3.
3
Slide 3- 100
Example Solving a Logarithmic
Equation
Solve log x  3.
3
log x  3
3
log x  log10
x  10
3
3
3
x  10
Slide 3- 101
3
Group Work
• ln 3 − 2 + ln  − 1 = 2
Group Work: Solve for x
• 15
1
2

3
=5
Group Work: Solve
•
  − −
2
=5
Group Work: Solve
• 1.05 = 8
Orders of Magnitude
The common logarithm of a positive quantity is its order of
magnitude.
Orders of magnitude can be used to compare any like quantities:
• A kilometer is 3 orders of magnitude longer than a meter.
• A dollar is 2 orders of magnitude greater than a penny.
• New York City with 8 million people is 6 orders of magnitude
bigger than Earmuff Junction with a population of 8.
Slide 3- 106
Note:
• In regular cases, how you determine the
magnitude is by how many decimal places
they differ
• In term of Richter scale and pH level, since the
number is the power or the exponent, you just
take the difference of them.
Example:
• What’s the difference of the magnitude
between kilometer and meter?
• It is 3 orders of magnitude longer than a
meter
Example:
• The order of magnitude between an
earthquake rated 7 and Richter scale rated
5.5.
• The difference of magnitude is 1.5
Group Work
• Find the order of magnitude:
• Between A dollar and a penny
• A horse weighing 500 kg and a horse weighing
50g
• 8 million people vs population of 8
Answer
• 2 orders of magnitude
• 4 orders of magnitude
• 6 orders of magnitude
Group Work
• Find the difference of the magnitude:
• Sour vinegar a pH of 2.4 and baking soda pH
of 8.4
• Earthquake in India 7.9 and Athens 5.9
Answer
• 6 orders of magnitude
• 2 orders of magnitude
Richter Scale
The Richter scale magnitude R of an earthquake is
a
R  log  B, where a is the amplitude in micrometers ( m)
T
of the vertical ground motion at the receiving station, T is the
period of the associated seismic wave in seconds, and B
accounts for the weakening of the seismic wave with increasing
distance from the epicenter of the earthquake.
Slide 3- 114
Example:
• How many times more severe was the 2001
earthquake in Gujarat, India (1 =7.9) than the
1999 earthquake in Athens, Greece (2 =5.9)
Group Work: Show work
• How many times more severs was the
earthquake in SF (1 = 6.5) than the
earthquake in PS (2 = 3.6)?
pH
In chemistry, the acidity of a water-based solution is
measured by the concentration of hydrogen ions in the
solution (in moles per liter). The hydrogen-ion
concentration is written [H+]. The measure of acidity
used is pH, the opposite of the common log of the
hydrogen-ion concentration:
pH=-log [H+]
More acidic solutions have higher hydrogen-ion
concentrations and lower pH values.
Slide 3- 117
Example:
• Sour vinegar has pH of 2.4 and a box of Leg
and Sickle baking soda has a pH of 8.4.
• A) what are their hydrogen-ion concentration?
• B) How many more times greater is the
hydrogen-ion concentration of the vinegar
than of the baking soda?
Group Work
• A substance with pH of 3.4 and another with
pH of 8.1
• A) what are their hydrogen-ion concentration?
• B) How many more times greater is the
hydrogen-ion concentration?
Newton’s Law of Cooling
An object that has been heated will cool to the temperature of the medium in
which it is placed. The temperature T of the object at time t can be modeled by
T (t )  T  (T  T )e for an appropriate value of k , where
 kt
m
0
m
T  the temperature of the surrounding medium,
m
T  the temperature of the object.
0
This model assumes that the surrounding medium maintains a constant
temperature.
Slide 3- 120
Example Newton’s Law of Cooling
A hard-boiled egg at temperature 100ºC is placed in 15ºC water
to cool. Five minutes later the temperature of the egg is 55ºC.
When will the egg be 25ºC?
Slide 3- 121
Example Newton’s Law of Cooling
A hard-boiled egg at temperature 100ºC is placed in 15ºC water
to cool. Five minutes later the temperature of the egg is 55ºC.
When will the egg be 25ºC?
Given T  100, T  15, and T (5)  55.
0
m
T (t )  T  (T  T )e
m
0
55  15  85e
40  85e
5 k
5 k
 40 
 e
 85 
 40 
ln    5k
 85 
k  0.1507...
5 k
Slide 3- 122
 kt
m
Now find t when T (t )  25.
25  15  85e
10  85e
0.1507 t
0.1507 t
 10 
ln    0.1507t
 85 
t  14.2 min .
Group Work
• A substance is at temperature 96℃ is placed
in 16℃. Four minutes later the temperature
of the egg is 45℃. Use Newton’s Law of
Cooling to determine when the egg will be
20℃
Regression Models Related by
Logarithmic Re-Expression
•
•
•
•
Linear regression:
Natural logarithmic regression:
Exponential regression:
Power regression:
Slide 3- 124
y = ax + b
y = a + blnx
y = a·bx
y = a·xb
Three Types of Logarithmic ReExpression
Slide 3- 125
Three Types of Logarithmic ReExpression (cont’d)
Slide 3- 126
Three Types of Logarithmic ReExpression
(cont’d)
Slide 3- 127
Homework Practice
• Pg 331 #1-51 eoe
MATHEMATICS OF FINANCE
Interest Compounded Annually
If a principal P is invested at a fixed annual interest rate r , calculated at the
end of each year, then the value of the investment after n years is
A  P(1  r ) , where r is expressed as a decimal.
n
Slide 3- 130
Interest Compounded k Times per Year
Suppose a principal P is invested at an annual rate r compounded
k times a year for t years. Then r / k is the interest rate per compounding
period, and kt is the number of compounding periods. The amount A
 r
in the account after t years is A  P 1   .
 k
kt
Slide 3- 131
Example Compounding Monthly
Suppose Paul invests $400 at 8% annual interest compounded
monthly. Find the value of the investment after 5 years.
Slide 3- 132
Example Compounding Monthly
Suppose Paul invests $400 at 8% annual interest compounded
monthly. Find the value of the investment after 5 years.
Let P  400, r  0.08, k  12, and t  5,
 r
A  P 1  
 k
kt
 0.08 
 400 1 

12


 595.9382...
So the value of Paul's investment after 5 years is $595.94.
12 ( 5 )
Slide 3- 133
Group Work
• Suppose you have $10000, you invest in a
place where they give you 12% interest
compounded quarterly. Find the value of your
investment after 40 years.
Compound Interest – Value of an
Investment
Suppose a principal P is invested at a fixed annual interest rate r. The value
of the investment after t years is
 r
A  P 1   when interest compounds k times per year,
 k
A  Pe when interest compounds continuously.
kt
rt
Slide 3- 135
Example Compounding Continuously
Suppose Paul invests $400 at 8% annual interest compounded
continuously. Find the value of his investment after 5 years.
Slide 3- 136
Example Compounding Continuously
Suppose Paul invests $400 at 8% annual interest compounded
continuously. Find the value of his investment after 5 years.
P  400, r  0.08, and t  5,
A  Pe
rt
 400e
0.08 ( 5 )
 596.7298...
So Paul's investment is worth $596.73.
Slide 3- 137
Group Work
• Suppose you have $10000, you invest in a
company where they give you 12% interest
compounded continuously. Find the value of
your investment after 40 years.
Annual Percentage Yield
A common basis for comparing investments is
the annual percentage yield (APY) – the
percentage rate that, compounded annually,
would yield the same return as the given
interest rate with the given compounding
period.
Slide 3- 139
Example Computing Annual
Percentage Yield
Meredith invests $3000 with Frederick Bank at 4.65% annual
interest compounded quarterly. What is the equivalent APY?
Slide 3- 140
Example Computing Annual
Percentage Yield
Meredith invests $3000 with Frederick Bank at 4.65% annual
interest compounded quarterly. What is the equivalent APY?
Let x  the equivalent APY. The value after one year is A  3000(1  x).
 0.0465 
3000(1  x)  3000 1 

4


 0.0465 
(1  x)  1 

4


4
4
 0.0465 
x  1 
  1  0.047317...
4 

The annual percentage yield is 4.73%.
4
Slide 3- 141
Future Value of an Annuity
The future value FV of an annuity consisting of n equal periodic payments
of R dollars at an interest rate i per compounding period (payment interval) is
1  i   1
n
FV  R
Slide 3- 142
i
.
Future Value of an Annuity
• At the end of each quarter year, Emily makes a
$500 payment into the Lanaghan Mutual
Fund. If her investments earn 7.88% annual
interest compounded quarterly, what will be
the value of Emily’s annuity in 20 years?
• Remember i=r/k
Group Work
• You are currently 18 and you want to retire at
age 65. You decide to invest in your future.
You are putting in $35 month. If your
investment earn 12% annual interest
compounded monthly, what will the value of
your annuity when you retire?
Present Value of an Annuity
The present value PV of an annuity consisting of n equal payments
of R dollars at an interest rate i per period (payment interval) is
PV  R
1  1  i 
i
n
.
Slide 3- 145
Example
• Mr. Liu bought a new car for $20000. What
are the monthly payment for a 5 year loan
with 0 down payment if the annual interest
rate (APR) is 2.9%?
Homework Practice
• Pg 341 #2-56 eoe

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