Chapter 3 Exponential, Logistic, and Logarithmic Functions

Slide 3- 2

Quick Review

3

3

4 / 3

2-3

5

Evaluate the expression without using a calculator.

1. -125

272.64

3. 27Rewrite the expression using a single positive exponent.

4.

Use a calculator to evaluate the expression.

5. 3.71293

a

Slide 3- 3

Quick Review Solutions

6

3

3

4 / 3

2-3

Evaluate the expression without using a calculator.

1. -125

272. 64

3. 27 Rewrite the expression using a single positive e

-5

3481

1xponent.

4.

Use a calculator to evaa

a

5

luate the expression.

5. 3.71293 1.3

Slide 3- 4

Exponential Functions

Let and be real number constants. An in is a function that can be written in the form ( ) , where is nonzero,

is positive, and 1. The constant is the

x

a b xf x a b a

b b a initial v

exponential function

of (the valueat 0), and is the .

alue fx b base

Determine if they are exponential functions

Answers

• Yes• No• Yes• Yes• no

Sketch an exponential function

Slide 3- 8

Example Finding an Exponential Function from its Table of Values

Determine formulas for the exponential function and whose values are given in the table below.

g h

Slide 3- 9

Example Finding an Exponential Function from its Table of Values

Determine formulas for the exponential function and whose values are given in the table below.

g h

1

Because is exponential, ( ) . Because (0) 4, 4. Because (1) 4 12, the base 3. So, ( ) 4 3 .

x

x

g g x a b g ag b b g x

1

Because is exponential, ( ) . Because (0) 8, 8.

1Because (1) 8 2, the base 1/ 4. So, ( ) 8 .4

x

x

h h x a b h a

h b b h x

Slide 3- 10

Exponential Growth and Decay

For any exponential function ( ) and any real number ,( 1) ( ).

If 0 and 1, the function is increasing and is an . The base is its .

If 0 an

xf x a b xf x b f x

a b fb

a

exponentialgrowth function growth factor

d 1, the function is decreasing and is an . The base is its .

b fb

exponentialdecay function decay factor

Sketch exponential graph and determine if they are growth or decay

Slide 3- 12

Example Transforming Exponential Functions

-2Describe how to transform the graph of ( ) 2 into the graph of ( ) 2 .x xf x g x

Slide 3- 13

Example Transforming Exponential Functions

-2Describe how to transform the graph of ( ) 2 into the graph of ( ) 2 .x xf x g x

-2The graph of ( ) 2 is obtained by translating the graph of ( ) 2 by2 units to the right.

x xg x f x

Slide 3- 14

Example Transforming Exponential Functions

-Describe how to transform the graph of ( ) 2 into the graph of ( ) 2 .x xf x g x

The graph of ( ) 2 is obtained by reflecting the graph of ( ) 2 acrossthe -axis.

x xg x f xy

Group Activity• Use this formula

• 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?

Slide 3- 16

The Natural Base e

1lim 1

x

xe

x

Slide 3- 17

Exponential Functions and the Base e

Any exponential function ( ) can be rewritten as ( ) , for any appropriately chosen real number constant .If 0 and 0, ( ) is an exponential growth function.If 0 and 0, (

x kx

kx

f x a b f x a ek

a k f x a ea k f

) is an exponential decay function.kxx a e

Slide 3- 18

Exponential Functions and the Base e

Slide 3- 19

Example Transforming Exponential Functions

3Describe how to transform the graph of ( ) into the graph of ( ) .x xf x e g x e

Slide 3- 20

Example Transforming Exponential Functions

3Describe how to transform the graph of ( ) into the graph of ( ) .x xf x e g x e

3The graph of ( ) is obtained by horizontally shrinking the graph of ( ) by a factor of 3.

x

x

g x ef x e

Slide 3- 21

Logistic Growth Functions

Let , , , and be positive constants, with 1. A

in is a function that can be written in the form ( ) or 1

( ) where the constant is the 1

x

kx

a b c k bcx f xa b

cf x ca e

logistic growth function

limit to gr

owth.

Example: Graph and Determine the horizontal asymptotes

𝑓 (𝑥 )= 71+3∗ .6𝑥

Answer

• Horizontal asymptotes at y=0 and y=7• Y-intercept at (0,7/4)

Group Work: Graph and determine the horizontal asymptotes

𝐺 (𝑥)=26

1+2𝑒−4 𝑥

Answer

• Horizontal asymptotes y=0 and y=26• Y-intercept at (0,26/3)

Word Problems:

• Year 2000 782,248 people• Year 2010 923,135 people

• Use this information to determine when the population will surpass 1 million people? (hint use exponential function)

Group Work

• Year 1990 156,530 people• Year 2000 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

• 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

If you are given a certain chemical have a half-life 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

Slide 3- 34

Quick Review

2

Convert the percent to decimal form or the decimal into a percent.1. 16%2. 0.053. Show how to increase 25 by 8% using a single multiplication.Solve the equation algebraically.4. 20 720Solve the equ

b

3

ation numerically.5. 123 7.872b

Slide 3- 35

Quick Review Solutions

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.

0.165%

25 1 4

.082

3

. 20 720 Solve the equation numerically.5. 123 7.872

6

0. 4

b

b

Slide 3- 36

Exponential Population Model

0 0

If a population is changing at a constant percentage rate each year, then( ) (1 ) , where is the initial population, is expressed as a decimal,

and is time in years.

t

P rP t P r P r

t

Example:

• You are given • Is this a growth or decay? What is the rate?

Slide 3- 38

Example Finding Growth and Decay Rates

Tell whether the population model ( ) 786,543 1.021 is an exponentialgrowth function or exponential decay function, and find the constant percentrate of growth.

tP t

Example

• You are given

• Is this a growth or decay? What is the rate?

Slide 3- 40

Example Finding an Exponential Function

Determine the exponential function with initial value=10, increasing at a rate of 5% per year.

Group Work

• Suppose 50 bacteria is put into a petri dish and it doubles every hour. When will the bacteria be 350,000?

Answer

• t=12.77 hours

Slide 3- 43

Example Modeling Bacteria Growth

Suppose a culture of 200 bacteria is put into a petri dish and the culturedoubles every hour. Predict when the number of bacteria will be 350,000.

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

Group Work

• You are given

• When will this become 150000?

Slide 3- 47

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- 48

Example Modeling a Rumor

-0.9

A high school has 1500 students. 5 students start a rumor, which spreads logistically so that ( ) 1500 /(1 29 ) models the number of studentswho have heard the rumor by the end of days, where

tS t et

0 is the day the

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?

t

Slide 3- 49

Example Modeling a Rumor: Answer

-0.9

A high school has 1500 students. 5 students start a rumor, which spreads logistically so that ( ) 1500 /(1 29 ) models the number of studentswho have heard the rumor by the end of days, where

tS t et

0 is the day the

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?

t

-0.9 ( 0 )(a) (0) 1500 /(1 29 ) 1500 /(1 29 1) 1500 / 30 50. So 50 students have heard the rumor by the end of day 0.

S e

-0.9(b) Solve 1000 1500 /(1 29 ) for .4.5. So 1000 students have heard the rumor half way

through the fifth day.

te tt

Key Word

• Maximum sustainable population

• What does this mean? What function deals with this?

Slide 3- 51

Maximum Sustainable PopulationExponential 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.

Homework Practice (Do in class also)

• P 296 #1-44 eoo

LOGARITHMIC FUNCTION, GRAPHS AND PROPERTIES

Slide 3- 54

Quick Review

-2

11

32

0

3

4

Evaluate the expression without using a calculator.1. 6

82. 2

3. 7Rewrite as a base raised to a rational number exponent.

14.

5. 10e

Slide 3- 55

Quick Review Solutions

3 / 2

1/

-2

11

3

4

2

0

3

4

Evaluate the expression without using a calculator.

1. 6

82. 2

3. 7 Rewrite as a base raised to a rational number exponent.

14.

5. 10

136

2

1

10

ee

Slide 3- 56

Changing Between Logarithmic and Exponential Form

If 0 and 0 1, then log ( ) if and only if .y

bx b y x b x

Group Work: transform logarithmic form into exponential form

• A)

• B)

• C)

• D)

Group Work: convert exponential form into logarithmic form

Slide 3- 59

Inverses of Exponential Functions

Slide 3- 60

Basic Properties of Logarithms

0

1

log

For 0 1, 0, and any real number . log 1 0 because 1. log 1 because . log because . because log log .b

b

b

y y y

b

x

b b

b x yb

b b bb y b b

b x x x

Slide 3- 61

An Exponential Function and Its Inverse

Slide 3- 62

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- 63

Basic Properties of Common Logarithms

0

1

log

Let and be real numbers with 0. log1 0 because 10 1. log10 1 because 10 10. log10 because 10 10 . 10 because log log .

y y y

x

x y x

yx x x

Slide 3- 64

Example Solving Simple Logarithmic Equations

Solve the equation by changing it to exponential form.log 4x

Slide 3- 65

Example Solving Simple Logarithmic Equations

Solve the equation by changing it to exponential form.log 4x

410 10,000x

Slide 3- 66

Basic Properties of Natural Logarithms

0

1

ln

Let and be real numbers with 0. ln1 0 because 1. ln 1 because . ln because . because ln ln .

y y y

x

x y xe

e e ee y e e

e x x x

Slide 3- 67

Graphs of the Common and Natural Logarithm

Slide 3- 68

Example Transforming Logarithmic Graphs

Describe how to transform the graph of ln into the graph of ( ) ln(2 ).

y xh x x

Slide 3- 69

Example Transforming Logarithmic Graphs

Describe how to transform the graph of ln into the graph of ( ) ln(2 ).

y xh x x

( ) ln(2 ) ln[ ( 2)]. So obtain the graph of ( ) ln(2 - ) fromln by applying, in order, a reflection across the -axis followed by

a translation 2 units to the right.

h x x x h x xy x y

Slide 3- 70

Quick Review

3

3

-2

3 3

2 2

1/ 22 4

3

Evaluate the expression without using a calculator.1. log102. ln 3. log 10Simplify the expression.

4.

5. 2

e

x yx y

x yx

Slide 3- 71

Quick Review Solutions

3

3

-2

3 3

2 2

1/ 2

5

4 22 4

3

5

Evaluate the expression without using a calculator.1. log10 2. ln 3. log

3 3

10 -Simplify the expression.

4.

2

2

5. 2

e

x yx y

xy

xx y yx

Slide 3- 72

What you’ll learn about• Properties of Logarithms• Change of Base• Graphs of Logarithmic Functions with Base b• Re-expressing Data

… and whyThe applications of logarithms are based on their many special properties, so learn them well.

Slide 3- 73

Properties of Logarithms

Let , , and be positve real numbers with 1, and any real number. : log ( ) log log

: log log log

: log ( ) log

b b b

b b b

c

b b

b R S b cRS R S

R R SS

R c R

Product rule

Quotient rule

Power rule

Slide 3- 74

Example Proving the Product Rule for Logarithms

Prove log ( ) log log .

b b bRS R S

Slide 3- 75

Example Proving the Product Rule for Logarithms

Prove log ( ) log log .

b b bRS R S

Let log and log . The corresponding exponential statementsare and . Therefore,

log ( ) change to logarithmic form log ( ) log log

b b

x y

x y

x y

b

b b b

x R y Sb R b S

RS b bRS b

RS x yRS R S

Slide 3- 76

Example Expanding the Logarithm of a Product

5

Assuming is positive, use properties of logarithms to writelog 3 as a sum of logarithms or multiple logarithms.

xx

Slide 3- 77

Example Expanding the Logarithm of a Product

5

Assuming is positive, use properties of logarithms to writelog 3 as a sum of logarithms or multiple logarithms.

xx

5 5log 3 log3 log

log3 5log

x x

x

Slide 3- 78

Example Condensing a Logarithmic Expression

Assuming is positive, write 3ln ln 2 as a single logarithm.x x

Slide 3- 79

Example Condensing a Logarithmic Expression

Assuming is positive, write 3ln ln 2 as a single logarithm.x x

3

3

3ln ln 2 ln ln 2

ln2

x xx

Group Work

𝐸𝑥𝑝𝑎𝑛𝑑 log (7 𝑥2 𝑦 𝑧 5)

Group Work

• Expand

Group Work

• Express as a single logarithm

Group Work

• Express as a single logarithm

Slide 3- 84

Change-of-Base Formula for Logarithms

For positive real numbers , , and with 1 and 1,log

log .log

a

b

a

a b x a bx

xb

Slide 3- 85

Example Evaluating Logarithms by Changing the Base

3Evaluate log 10.

Slide 3- 86

Example Evaluating Logarithms by Changing the Base

3Evaluate log 10.

3

log10 1log 10 2.096log3 log3

Solving

4𝑥=51

Solving

ln𝑒

Solving

log 1

Solving

log 5 𝑥=¿ log 4+¿ log (𝑥−3)¿¿

Solving

𝑙𝑜𝑔5 √56=𝑥

Solving

25+3𝑥=16

Homework Practice

• Pg 317 #1-50 eoe

EQUATION SOLVING AND MODELING

Slide 3- 95

Quick Review

3 1/ 3

2 / 2

Prove that each function in the given pair is the inverse of the other.1. ( ) and ( ) ln

2. ( ) log and ( ) 10Write the number in scientific notation.3. 123,400,000Write the number in

x

x

f x e g x x

f x x g x

8

-4

decimal form.4. 5.67 105. 8.91 10

Slide 3- 96

Quick Review Solutions

1 / 33ln ln

2/ 2

3 1/ 3

2 / 2

Prove that each function in the given pair is the inverse of the other.

1. ( ) and ( ) ln

2. ( ) log and ( ) 10

( ( ))

( ( )) log 1

Write the numbe

0 log1

r

0

x x

x x

x

x

f x e g x x

f x x

f g x e e x

f g x xg x

8

-

8

4

in scientific notation.3. 123,400,000 Write the number in decimal form.4. 5.67 10 5. 8.9

1.234 10

1 10567,000,000

0.0 8 91 00

Slide 3- 97

One-to-One Properties

For any exponential function ( ) , If , then .

For any logarithmic function ( ) log , If log log , then .

x

u v

b

b b

f x bb b u v

f x xu v u v

Slide 3- 98

Example Solving an Exponential Equation Algebraically

/ 2

Solve 40 1/ 2 5.x

Slide 3- 99

Example Solving an Exponential Equation Algebraically

/ 2

Solve 40 1/ 2 5.x

/ 2

/ 2

/ 2 3 3

40 1/ 2 5

11/ 2 divide by 408

1 1 1 1 2 2 8 2/ 2 3 one-to-one property

6

x

x

x

xx

Slide 3- 100

Example Solving a Logarithmic Equation

3Solve log 3.x

Slide 3- 101

Example Solving a Logarithmic Equation

3Solve log 3.x

3

3 3

3 3

log 3log log10

1010

xx

xx

Group Work

ln (3 𝑥− 2 )+ln (𝑥− 1 )=2 𝑙𝑛𝑥

Group Work: Solve for x

15( 12 )

𝑥3=5

Group Work: Solve

𝑒𝑥−𝑒−𝑥

2=5

Group Work: Solve

1.05𝑥=8

Slide 3- 106

Orders of MagnitudeThe 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.

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

Slide 3- 114

Richter Scale

The Richter scale magnitude of an earthquake is

log , where is the amplitude in micrometers ( m)

of the vertical ground motion at the receiving station, is the period of the associated seis

RaR B aT

T

mic wave in seconds, and accounts for the weakening of the seismic wave with increasingdistance from the epicenter of the earthquake.

B

Example:

• How many times more severe was the 2001 earthquake in Gujarat, India (=7.9) than the 1999 earthquake in Athens, Greece (=5.9)

Group Work: Show work

• How many times more severs was the earthquake in SF ( than the earthquake in PS ()?

Slide 3- 117

pHIn 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.

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?

Slide 3- 120

Newton’s Law of Cooling

0

An object that has been heated will cool to the temperature of the medium in which it is placed. The temperature of the object at time can be modeled by

( ) ( ) for an appropriate vakt

m m

T tT t T T T e

0

lue of , where the temperature of the surrounding medium, the temperature of the object.

This model assumes that the surrounding medium maintains a constanttemperature.

m

kTT

Slide 3- 121

Example Newton’s Law of CoolingA 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- 122

Example Newton’s Law of CoolingA 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?

0

0

5

5

5

Given 100, 15, and (5) 55.( ) ( )

55 15 8540 85

4085

40ln 5850.1507...

m

kt

m m

k

k

k

T T TT t T T T e

ee

e

k

k

0.1507

0.1507

Now find when ( ) 25.25 15 8510 85

10ln 0.15078514.2min .

t

t

t T te

e

t

t

Group Work

• A substance is at temperature is placed in . Four minutes later the temperature of the egg is Use Newton’s Law of Cooling to determine when the egg will be

Slide 3- 124

Regression Models Related by Logarithmic Re-Expression

• Linear regression: y = ax + b• Natural logarithmic regression: y = a + blnx• Exponential regression: y = a·bx

• Power regression: y = a·xb

Slide 3- 125

Three Types of Logarithmic Re-Expression

Slide 3- 126

Three Types of Logarithmic Re-Expression (cont’d)

Slide 3- 127

Three Types of Logarithmic Re-Expression(cont’d)

Homework Practice

• Pg 331 #1-51 eoe

MATHEMATICS OF FINANCE

Slide 3- 130

Interest Compounded Annually

If a principal is invested at a fixed annual interest rate , calculated at the end of each year, then the value of the investment after years is

(1 ) , where is expressed as a decimal.n

P rn

A P r r

Slide 3- 131

Interest Compounded k Times per Year

Suppose a principal is invested at an annual rate compounded times a year for years. Then / is the interest rate per compounding

period, and is the number of compounding periods. The amou

P rk t r k

kt nt

in the account after years is 1 .kt

A

rt A Pk

Slide 3- 132

Example Compounding MonthlySuppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.

Slide 3- 133

Example Compounding MonthlySuppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.

12 ( 5 )

Let 400, 0.08, 12, and 5,

1

0.08 400 112

595.9382...So the value of Paul's investment after 5 years is $595.94.

kt

P r k t

rA Pk

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.

Slide 3- 135

Compound Interest – Value of an Investment

Suppose a principal is invested at a fixed annual interest rate . The valueof the investment after years is

1 when interest compounds k times per year,

when interest co

kt

rt

P rt

rA Pk

A Pe

mpounds continuously.

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.

Slide 3- 137

Example Compounding Continuously

Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.

0.08 ( 5 )

400, 0.08, and 5,

400 596.7298...So Paul's investment is worth $596.73.

rt

P r tA Pe

e

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.

Slide 3- 139

Annual Percentage YieldA 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- 140

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- 141

Example Computing Annual Percentage Yield

Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?

4

4

4

Let the equivalent APY. The value after one year is 3000(1 ).

0.04653000(1 ) 3000 14

0.0465(1 ) 14

0.04651 1 0.047317...4

The annual percentage yield is 4.73%.

x A x

x

x

x

Slide 3- 142

Future Value of an Annuity

The future value of an annuity consisting of equal periodic paymentsof dollars at an interest rate per compounding period (payment interval) is

1 1.

n

FV nR i

iFV R

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?

Slide 3- 145

Present Value of an Annuity

The present value of an annuity consisting of equal paymentsof dollars at an interest rate per period (payment interval) is

1 1.

n

PV nR i

iPV R

i

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