Slide 3- 1. Chapter 3 Exponential, Logistic, and Logarithmic Functions

Slide 3- 1

Chapter 3

Exponential, Logistic, and Logarithmic Functions

3.1

Exponential and Logistic Functions

Slide 3- 4

Quick Review

3

3

4 / 3

2-3

5

Evaluate the expression without using a calculator.

1. -125

272.

643. 27

Rewrite the expression using a single positive exponent.

4.

Use a calculator to evaluate the expression.

5. 3.71293

a

Slide 3- 5

Quick Review Solutions

6

3

3

4 / 3

2-3


1. -125

272.

643. 27

Rewrite the expression using a single positive e

-5

3

481

1

xponent.

4.

Use a calculator to evaa

a

5

luate the expression.

5. 3.71293 1.3

Slide 3- 6

What you’ll learn about

Exponential Functions and Their Graphs The Natural Base e Logistic Functions and Their Graphs Population Models

… and whyExponential and logistic functions model many growth patterns, including the growth of human and animal populations.

Slide 3- 7

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 x

f x a b a

b b a initial v

exponential function

of (the value

at 0), and is the .

alue f

x b base

Slide 3- 8


Rules For ExponentsIf a > 0 and b > 0, the following hold true for all real numbers x and y.

yxyx aa. a 1

yxy

x

aa

a. 2

xyyx aa. 3

xxx (ab)b. a 4

x

xx

b

a

b

a.

5

16 0 . a

x-x

a. a

17

q pq

p

a. a 8

Slide 3- 9

Use the rules for exponents tosolve for x

•4x = 128•(2)2x = 27

•2x = 7•x = 7/2

•2x = 1/32•2x = 2-5

•x = -5


Slide 3- 10

•(x3y2/3)1/2

•x3/2y1/3

•27x = 9-x+1

•(33)x = (32)-x+1

•33x = 3-2x+2

•3x = -2x+ 2•5x = 2•x = 2/5


Slide 3- 11

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- 12

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 a

g 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- 13

5

4

3

2

1

-2

-3

-4

-5

y

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

x

y = 2 xIf b > 1, then the graph of b x will:

•Rise from left to right.

•Not intersect the x-axis.

•Approach the x-axis.

•Have a y-intercept of (0, 1)


Slide 3- 14

5

4

3

2

1

-2

-3

-4

-5

y

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

x

y = (1/2) xIf 0 < b < 1, then the graph of

b x will:

•Fall from left to right.

•Not intersect the x-axis.

•Approach the x-axis.

•Have a y-intercept of (0, 1)


Slide 3- 15

Example Transforming Exponential Functions

Describe how to transform the graph of f(x) = 2x into the graph g(x) = 2x-2

The graph of g(x) = 2x-2 is obtained by translating the graph of f(x) = 2x

by 2 units to the right.

Slide 3- 16


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

Slide 3- 17


-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 across

the -axis.

x xg x f x

y

Slide 3- 18

The Natural Base e

1lim 1

x

xe

x

Slide 3- 19

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 e

k

a k f x a e

a k f

) is an exponential decay function.kxx a e

Slide 3- 20

Exponential Functions and the Base e

Slide 3- 21


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

Slide 3- 22


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 e

f x e

Slide 3- 23

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 b

cx f x

a bc

f x ca e

logistic growth function

limit to gr

owth.

Slide 3- 24

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 x

f x b f x

a b f

b

a

exponential

growth function growth factor

d 1, the function is decreasing and is an

. The base is its .

b f

b

exponential

decay function decay factor

Slide 3- 25


Definitions Exponential Growth and Decay

The function y = k ax, k > 0 is a model for exponential growth if a > 1, and a model for exponential decay if 0 < a < 1.

h

t

Obyy y new amountyO original amountb baset timeh half life

Slide 3- 26


An isotope of sodium, Na, has a half-life of 15 hours. A sample of this isotope has mass 2 g.

(a) Find the amount remaining after t hours.

(b) Find the amount remaining after 60 hours.

• a. y = yobt/h

• y = 2 (1/2)(t/15)

• b. y = yobt/h

• y = 2 (1/2)(60/15)

• y = 2(1/2)4

• y = .125 g

Slide 3- 27


A bacteria double every three days. There are 50 bacteria initially present

(a) Find the amount after 2 weeks.

(b) When will there be 3000 bacteria?

• a. y = yobt/h

• y = 50 (2)(14/3)

• y = 1269 bacteria

Slide 3- 28


A bacteria double every three days. There are 50 bacteria initially present

When will there be 3000 bacteria?

• b. y = yobt/h

• 3000 = 50 (2)(t/3)

• 60 = 2t/3

•

3.2

Exponential and Logistic Modeling

Slide 3- 30

Quick Review

2

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. 20 720

Solve the equ

b

3

ation numerically.

5. 123 7.872b

Slide 3- 31


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.

0.16

5%

25 1

4

.082

3

. 20 720

Solve the equation numerically.

5. 123 7.872

6

0. 4

b

b

Slide 3- 32


Constant Percentage Rate and Exponential Functions Exponential Growth and Decay Models Using Regression to Model Population Other Logistic Models

… and whyExponential functions model many types of unrestricted growth; logistic functions model restricted growth, including the spread of disease and the spread of rumors.

Slide 3- 33

Constant Percentage Rate

Suppose that a population is changing at a constant percentage

rate r, where r is the percent rate of change expressed in decimal

form. Then the population follows the pattern shown.

0

0 0 0

Time in years Population

0 (0) initial population

1 (1) (1 )

2 (2) (1)

P P

P P Pr P r

P P

2

0

3

0

0

(1 ) (1 )

3 (3) (2) (1 ) (1 )

( ) (1 ) t

r P r

P P r P r

t P t P r

Slide 3- 34

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 r

P t P r P r

t

Slide 3- 35

Example Finding Growth and Decay Rates

Tell whether the population model ( ) 786,543 1.021 is an exponential

growth function or exponential decay function, and find the constant percent

rate of growth.

tP t

Because 1 1.021, .021 0. So, is an exponential growth function

with a growth rate of 2.1%.

r r P

Slide 3- 36

Example Finding an Exponential Function

Determine the exponential function with initial value = 10,

increasing at a rate of 5% per year.

0Because 10 and 5% 0.05, the function is ( ) 10(1 0.05) or

( ) 10(1.05) .

t

t

P r P t

P t

Slide 3- 37

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- 38

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.

2

400 200 2

800 200 2

( ) 200 2 represents the bacteria population hr after it is placed

in the petri dish. To find out when the population will reach 350,000, solve

350,000 200 2 for using

t

t

P t t

t

a calculator.

10.77 or about 10 hours and 46 minutes.t

Slide 3- 39

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- 40

Example Modeling U.S. Population Using Exponential Regression

Use the 1900-2000 data and exponential regression to predict the

U.S. population for 2003.

103

Let ( ) be the population (in millions) of the U.S. years after 1900.

Using exponential regression, find a model ( ) 80.5514 1.01289 .

To find the population in 2003 find (103) 80.5514 1.01289 3

t

P t t

P t

P

01.3.

Slide 3- 41

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- 42

Example Modeling a Rumor

A high school has 1500 students. 5 students start a rumor which spreadslogistically so that s(t) = 1500/(1 + 29 e.-.09t) models the number of students who have heard the rumor at the end of t days, where t = 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?

50291

1500

291

1500

291

1500)0(

291

1500)((a)

009.

09.

eeS

etS

t

t

Slide 3- 43

Example Modeling a Rumor

A high school has 1500 students. 5 students start a rumor which spreadslogistically so that s(t) = 1500/(1 + 29 e.-.09t) models the number of students who have heard the rumor at the end of t days, where t = 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?

-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 t

t

3.3

Logarithmic Functions and Their Graphs

Slide 3- 45

Quick Review

3 / 2

1/

-2

11

3

4

2

0

3

4


1. 6

82.

23. 7

Rewrite as a base raised to a rational number exponent.

14.

5. 10

1

36

2

1

10

ee

Slide 3- 46


Inverses of Exponential Functions Common Logarithms – Base 10 Natural Logarithms – Base e Graphs of Logarithmic Functions Measuring Sound Using Decibels

… and whyLogarithmic functions are used in many applications, including the measurement of the relative intensity of sounds.

Slide 3- 47

Logarithmic Functions

The inverse of an exponential function is called a logarithmic function.

Definition: x = a y if and only if y = log a x

Slide 3- 48

Changing Between Logarithmic and Exponential Form

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

bx b y x b x

Slide 3- 49


•log 4 16 = 2 ↔ 42 = 16

•log 3 81 = 4 ↔ 34 = 81

•log10 100 = 2 ↔ 102 = 100

Slide 3- 50

Inverses of Exponential Functions

Slide 3- 51


The function f (x) = log a x is called a logarithmic function.

•Domain: (0, ∞)•Range: (-∞, ∞)

• Asymptote: x = 0• Increasing for a > 1

• Decreasing for 0 < a < 1• Common Point: (1, 0)

Slide 3- 52

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 y

b

b b b

b y b b

b x x x

Slide 3- 53

An Exponential Function and Its Inverse

Slide 3- 54

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- 55

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

y

x x x

Slide 3- 56

Example Solving Simple Logarithmic Equations

Solve the equation by changing it to exponential form.

log 4x

410 10,000x

Slide 3- 57

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 x

e

e e e

e y e e

e x x x

Slide 3- 58

Graphs of the Common and Natural Logarithm

Slide 3- 59

Example Transforming Logarithmic Graphs

Describe how to transform the graph of ln into the graph of

( ) ln(2 ).

y x

h x x

Slide 3- 60

Example Transforming Logarithmic Graphs

Describe how to transform the graph of ln into the graph of

( ) ln(2 ).

y x

h x x

( ) ln(2 ) ln[ ( 2)]. So obtain the graph of ( ) ln(2 - ) from

ln by applying, in order, a reflection across the -axis followed by

a translation 2 units to the right.

h x x x h x x

y x y

Slide 3- 61

Decibels

0

2 12 2

0

The level of sound intensity in (dB) is

10log , where (beta) is the number of decibels,

is the sound intensity in W/m , and 10 W/m is the

threshold of human hearing (the qu

I

I

I I

decibels

ietest audible sound

intensity).

3.4

Properties of Logarithmic Functions

Slide 3- 63

Quick Review

3

3

-2

3 3

2 2

1/ 2

5

4 22 4

3

5


1. log10

2. ln

3. log

3

3

10 -

Simplify the expression.

4.

2

2

5. 2

e

x y

x y

x

y

xx y y

x

Slide 3- 64


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- 65

1. loga(ax) = x for all x 2. alog ax = x for all x > 03. loga(xy) = logax + logay4. loga(x/y) = logax – logay5. logaxn = n logax

Common Logarithm: log 10 x = log xNatural Logarithm: log e x = ln x

All the above properties hold.


Slide 3- 66

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 c

RS R S

RR S

S

R c R

Product rule

Quotient rule

Power rule

Slide 3- 67

Example Proving the Product Rule for Logarithms

Prove log ( ) log log .b b b

RS R S

Slide 3- 68

Example Proving the Product Rule for Logarithms

Prove log ( ) log log .

b b bRS R S

Let log and log . The corresponding exponential statements

are 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 S

b R b S

RS b b

RS b

RS x y

RS R S

Slide 3- 69

Example Expanding the Logarithm of a Product

5

Assuming is positive, use properties of logarithms to write

log 3 as a sum of logarithms or multiple logarithms.

x

x

Slide 3- 70

Example Expanding the Logarithm of a Product

5

Assuming is positive, use properties of logarithms to write

log 3 as a sum of logarithms or multiple logarithms.

x

x

5 5log 3 log3 log

log3 5log

x x

x

Slide 3- 71

Example Condensing a Logarithmic Expression

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

Slide 3- 72

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 x

x

Slide 3- 73


Product Rule

nmnm bbb logloglog

49log36log 33

4log24log9log 333

Slide 3- 74

Quotient Rule nm

n

mbbb logloglog

225log5

2

50log2log50log 555


Slide 3- 75

Power Rule mcm b

cb loglog

ee ln4ln 4

414


Slide 3- 76

Expand

z

yx 23

5log

zyx 52

53

5 log)log(log

zyx 555 log)log2log3(


Slide 3- 77

Change-of-Base Formula for Logarithms

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

loglog .

loga

b

a

a b x a b

xx

b

Slide 3- 78

Example Evaluating Logarithms by Changing the Base

3Evaluate log 10.

Slide 3- 79

Example Evaluating Logarithms by Changing the Base

3Evaluate log 10.

3

log10 1log 10 2.096

log3 log3

3.5

Equation Solving and Modeling

Slide 3- 81

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 ( ) 10

Write the number in scientific notation.

3. 123,400,000

Write the number in

x

x

f x e g x x

f x x g x

8

-4

decimal form.

4. 5.67 10

5. 8.91 10

Slide 3- 82


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 10

567,000,000

0.0 8 91 00

Slide 3- 83


Solving Exponential Equations Solving Logarithmic Equations Orders of Magnitude and Logarithmic Models Newton’s Law of Cooling Logarithmic Re-expression

… and whyThe Richter scale, pH, and Newton’s Law of Cooling, are among the most important uses of logarithmic and exponential functions.

Slide 3- 84

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 b

b b u v

f x x

u v u v

Slide 3- 85

Example Solving an Exponential Equation Algebraically

/ 2

Solve 40 1/ 2 5.x

Slide 3- 86

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 40

8

1 1 1 1

2 2 8 2

/ 2 3 one-to-one property

6

x

x

x

x

x

Slide 3- 87

Example Solving a Logarithmic Equation

3Solve log 3.x

Slide 3- 88

Example Solving a Logarithmic Equation

3Solve log 3.x

3

3 3

3 3

log 3

log log10

10

10

x

x

x

x

Slide 3- 89

Solving Exponential Equations

To solve exponential equations, pick a convenient base (often base 10 or base e) and take the log of both sides.

Solve:

2143 x

Slide 3- 90

2143 x

•Take the log of both sides: 21log4log 3 x

•Power rule: 21log4log3 x


Slide 3- 91

•Solve for x:

21log4log3 x

•Divide:

21log4log3 x

4log3

21logx 732.0x


Slide 3- 92

To solve logarithmic equations, write both sides of the equation as a single log with the same base, then equate the arguments of the log expressions.

Solve:

2log1log82log xxx


Slide 3- 93

•Write the left side as a single logarithm:

2log1log82log xxx

2log1

82log

xx

x


Slide 3- 94

•Equate the arguments:

2log1

82log

xx

x

21

82

xx

x


Slide 3- 95

•Solve for x:2

1

82

xx

x

121

821

xxx

xx


Slide 3- 96

1282 xxx

230 xx

2382 2 xxx

60 2 xx


Slide 3- 97

230 xx 2 ,3x

2log1log82log xxx

•Check for extraneous solutions.

•x = -3, since the argument of a log cannot be negative


Slide 3- 98

To solve logarithmic equations with one side of the equation equal to a constant, change the equation to an exponential equation

Solve:

64loglog 2 xx


Slide 3- 99

•Write the left side as a single logarithm:

64loglog 2 xx

64log 2 xx

64log 3 x


Slide 3- 100

64log 3 x

•Write as an exponential equation:

63 104 x


Slide 3- 101

•Solve for x:

2500003 x

99.62x


63 104 x

Slide 3- 102

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- 103

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

R

aR B a

TT

mic wave in seconds, and

accounts for the weakening of the seismic wave with increasing

distance from the epicenter of the earthquake.

B

Slide 3- 104

5.5 Graphs of Logarithmic Functions

BT

aR

log

What is the magnitude on the Richter scale of anearthquake if a = 300, T = 30 and B = 1.2?

2.130

300log

R

2.110log R

2.11R

2.2R

Slide 3- 105

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- 106

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 t

T 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 constant

temperature.

m

k

T

T

Slide 3- 107

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- 108

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? 0

0

5

5

5

Given 100, 15, and (5) 55.

( ) ( )

55 15 85

40 85

40

85

40ln 5

85

0.1507...

m

kt

m m

k

k

k

T T T

T t T T T e

e

e

e

k

k

0.1507

0.1507

Now find when ( ) 25.

25 15 85

10 85

10ln 0.1507

85

14.2min.

t

t

t T t

e

e

t

t

Slide 3- 109

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- 110

Three Types of Logarithmic Re-Expression

Slide 3- 111

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

Slide 3- 112

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

3.6

Mathematics of Finance

Slide 3- 114

Quick Review

1. Find 3.4% of 70.

2. What is one-third of 6.25%?

3. 30 is what percent of 150?

4. 28 is 35% of what number?

5. How much does Allyson have at the end of 1 year

if she invests $400 at 3% simple interest?

Slide 3- 115


1. Find 3.4% of 70.

2. What is one-third of 6.25%?

3. 30 is what percent of 150?

4. 28 is 35% of what number?

5. How much does Allyson have at the end of 1

2.38

0.02083

20%

8

year

if

0

she invests $400 at 3% simple interest? $412

Slide 3- 116


Interest Compounded Annually Interest Compounded k Times per Year Interest Compounded Continuously Annual Percentage Yield Annuities – Future Value Loans and Mortgages – Present Value

… and whyThe mathematics of finance is the science of letting your money work for you – valuable information indeed!

Slide 3- 117

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 r

n

A P r r

Slide 3- 118

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 r

k t r k

kt nt

in the account after years is 1 .kt

A

rt A P

k

Slide 3- 119

Example Compounding Monthly

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

Slide 3- 120

Example Compounding Monthly

Suppose 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 1

12

595.9382...

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

kt

P r k t

rA P

k

Slide 3- 121

Compound Interest – Value of an Investment

Suppose a principal is invested at a fixed annual interest rate . The value

of the investment after years is

1 when interest compounds k times per year,

when interest co

kt

rt

P r

t

rA P

k

A Pe

mpounds continuously.

Slide 3- 122

Example Compounding Continuously

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

Slide 3- 123

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 t

A Pe

e

Slide 3- 124

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- 125

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- 126

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 1

4

0.0465(1 ) 1

4

0.04651 1 0.047317...

4

The annual percentage yield is 4.73%.

x A x

x

x

x

Slide 3- 127

Future Value of an Annuity

The future value of an annuity consisting of equal periodic payments

of dollars at an interest rate per compounding period (payment interval) is

1 1.

n

FV n

R i

iFV R

i

Slide 3- 128

Present Value of an Annuity

The present value of an annuity consisting of equal payments

of dollars at an interest rate per period (payment interval) is

1 1.

n

PV n

R i

iPV R

i

Slide 3- 129

Chapter Test

4-1. State whether ( ) 2 is an exponential growth function or an

exponential decay function, and describe its end behavior using limits.

2. Find the exponential function that satisfies the conditio

xf x e

ns:

Initial height = 18 cm, doubling every 3 weeks.

3. Find the logistic function that satisfies the conditions:

Initial value = 12, limit to growth = 30, passing through (2,20).

4. Describe how to transf2

2

orm the graph of log into the graph of

( ) log ( 1) 2.

5. Solve for : 1.05 3.x

y x

h x x

x

Slide 3- 130

Chapter Test

6. Solve for : ln(3 4) - ln(2 1) 5

7. Find the amount accumulated after investing a principal for

years at an interest rate compounded continuously.

8. The population of Preston is 89,000 and i

x x x

A P

t r

s decreasing by 1.8% each year.

(a) Write a function that models the population as a function of time .

(b) Predict when the population will be 50,000?

9. The half-life of a certain substance is 1.5 sec

t

0

. The initial amount of

substance is grams.

(a) Express the amount of substance remaining as

a function of time .

(b) How much of the substance is left after 1.5 sec?

(c) How much of the substance

S

t

0

is left after 3 sec?

(d) Determine if there was 1 g left after 1 min.

S

Slide 3- 131

Chapter Test

10. If Joenita invests $1500 into a retirement account with an 8% interest rate

compounded quarterly, how long will it take this single payment to grow to

$3750?

Slide 3- 132

Chapter Test Solutions

-

4-1. State whether ( ) 2 is an exponential growth function or an

exponential decay function, and

exponential decay

describe its end behavior u

; li

sing limits.

2. Find

m ( ) , lim

the e

(

x

) 2x x

x

f

f x e

x f x

/ 21

ponential function that satisfies the conditions:

Initial height = 18 cm, doubling every 3 weeks.

3. Find the logistic function that satisfies the conditions:

Initial value = 12, limit t

( 2

o

18

) xf x

2

2

0.55

growth = 30, passing through (2,20).

4. Describe how to transform the g

( ) 30 /(1 1.5 )

translate right 1 unit, relect across the -a

raph of log into the graph of

( ) log ( 1) 2. xis,

transl

x

y x

h

f x e

xx x

5. Solve for : 1.

ate up 2 un

05 3.

its.

22.5171xx x

Slide 3- 133


6. Solve for : ln(3 4) - ln(2 1) 5

7. Find the amount accumulated after investing a principal for

years at an interest rate compounded continuously.

8. The population of

-0.49

Pres

15

t

rt

x x x

A P

t r

x

Pe

on is 89,000 and is decreasing by 1.8% each year.

(a) Write a function that models the population as a function of time .

(b) Predict when the population will be

( ) 89,000(0.9

50,000?

82)

31.74 year

tP

t

t

0

/1.5

0

9. The half-life of a certain substance is 1.5 sec. The initial amount of

substance is grams.

(a) Express the amount of substance remaining as

a functi

s

1( )

2on of time .

(b) How

t

t t S

S

S

0

0

0

much of the substance is left after 1.5 sec?

(c) How much of the substance is left after 3 sec?

(d) Determine if there was 1 g left after 1 mi

/2

/ 4

1,009,500 metric n. t ns

o

S

S

S

Slide 3- 134


10. If Joenita invests $1500 into a retirement account with an 8% interest rate

compounded quarterly, how long will it take this single payment to grow t

11.5

o

$375 7 y0? ears

Documents

Slide 3- 1. Chapter 3 Exponential, Logistic, and Logarithmic Functions