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Chapter 3. Exponential, Logistic, and Logarithmic Functions. 3.1. Exponential and Logistic Functions. Quick Review. Quick Review Solutions. What you’ll learn about. Exponential Functions and Their Graphs The Natural Base e Logistic Functions and Their Graphs Population Models … and why - PowerPoint PPT Presentation
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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
Evaluate the expression without using a calculator.
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
Exponential Functions
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
Exponential Functions
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
Exponential Functions
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)
Exponential Functions
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)
Exponential Functions
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
Example Transforming Exponential Functions
-2Describe how to transform the graph of ( ) 2 into the graph of ( ) 2 .x xf x g x
Slide 3- 17
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 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
Example Transforming Exponential Functions
3Describe how to transform the graph of ( ) into the graph of ( ) .x xf x e g x e
Slide 3- 22
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 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
Exponential Functions
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
Exponential Functions
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
Exponential Functions
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
Exponential Functions
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
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.
0.16
5%
25 1
4
.082
3
. 20 720
Solve the equation numerically.
5. 123 7.872
6
0. 4
b
b
Slide 3- 32
What you’ll learn about
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
Evaluate the expression without using a calculator.
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
What you’ll learn about
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
Logarithmic Functions
•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
Logarithmic Functions
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
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 y
x y
x
y
xx y y
x
Slide 3- 64
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- 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.
Logarithmic Functions
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
Logarithmic Functions
Product Rule
nmnm bbb logloglog
49log36log 33
4log24log9log 333
Slide 3- 74
Quotient Rule nm
n
mbbb logloglog
225log5
2
50log2log50log 555
Logarithmic Functions
Slide 3- 75
Power Rule mcm b
cb loglog
ee ln4ln 4
414
Logarithmic Functions
Slide 3- 76
Expand
z
yx 23
5log
zyx 52
53
5 log)log(log
zyx 555 log)log2log3(
Logarithmic Functions
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
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 10
567,000,000
0.0 8 91 00
Slide 3- 83
What you’ll learn about
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
Solving Exponential Equations
Slide 3- 91
•Solve for x:
21log4log3 x
•Divide:
21log4log3 x
4log3
21logx 732.0x
Solving Exponential Equations
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
Solving Exponential Equations
Slide 3- 93
•Write the left side as a single logarithm:
2log1log82log xxx
2log1
82log
xx
x
Solving Exponential Equations
Slide 3- 94
•Equate the arguments:
2log1
82log
xx
x
21
82
xx
x
Solving Exponential Equations
Slide 3- 95
•Solve for x:2
1
82
xx
x
121
821
xxx
xx
Solving Exponential Equations
Slide 3- 96
1282 xxx
230 xx
2382 2 xxx
60 2 xx
Solving Exponential Equations
Slide 3- 97
230 xx 2 ,3x
2log1log82log xxx
•Check for extraneous solutions.
•x = -3, since the argument of a log cannot be negative
Solving Exponential Equations
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
Solving Exponential Equations
Slide 3- 99
•Write the left side as a single logarithm:
64loglog 2 xx
64log 2 xx
64log 3 x
Solving Exponential Equations
Slide 3- 100
64log 3 x
•Write as an exponential equation:
63 104 x
Solving Exponential Equations
Slide 3- 101
•Solve for x:
2500003 x
99.62x
Solving Exponential Equations
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
Quick Review Solutions
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
What you’ll learn about
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
Chapter Test Solutions
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
Chapter Test Solutions
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