Refresher Skills for Chapter 4
Exponents
Example
Simplify each expression by hand.
a) 82/3
b) (32)4/5
Pg. 177
36,44,48,50,52,54
Chapter 4
Exponential Functions
4.1 Introduction to The Family of
Exponential Functions
Key Points
Growth factors and growth rates
Decay factors and decay rates
The definition of an exponential function
Salary example
New Salary = 100% of Old Salary + Percent Growth Rate
Annual Growth Rate Factor
Growing at a Constant Percent
Rate
Example 2
During the 2000s, the population of Mexico increased at a
constant annual percent rate of 1.2%. Since the
population grew by the same percent each year, it can
be modeled by an exponential function. Lets calculate
the population of Mexico for the years after 2000. In
2000, the population was 100 million. The population
grew by 1.2%, so
Pop. in 2001 = Pop. in 2000 + 1.2% of Pop. in 2000
= 100 + 0.012(100)
= 100 + 1.2
= 101.2 million.
On the next slide, we extend this reasoning to estimate the
Functions Modeling
population of Mexico through 2007.
Change:
A Preparation
for Calculus,
4th
Growing at a Constant Percent
Rate
Example 2 continued Population of Mexico
The population of Mexico increased by slightly more each year than it did
the year before, because each year the increase is 1.2% of a larger
number.
Year
2000
2001
2002
2003
2004
2005
2006
2007
P, % increase
in population
1.2
1.21
1.23
1.25
1.26
1.27
1.29
P, population
(millions)
100
101.2
102.41
103.64
104.89
106.15
107.42
108.71
The projected population of
Mexico,
assuming 1.2% annual growth
P, population
(millions)
100
year
Functions Modeling
Change:
A Preparation
for Calculus,
4th
Growth Factors and Percent Growth
Rates
The Growth Factor of an Increasing Exponential
Function
In Example 2, the population grew by 1.2%, so
New Population = Old Population + 1.2% of Old
Population
= (1 + .012) Old Population
= 1.012 Old Population
We call 1.012 the growth factor.
Functions Modeling
Change:
A Preparation
for Calculus,
4th
Growth Factors and Percent Growth
Rates
The Growth Factor of a Decreasing Exponential
Function
In Example 3, the carbon-14 changes by 11.4% every
1000 yrs.
New Amount = Old Amount 11.4% of Old Amount
= (1 .114) Old Amount
= 0.886 Old Amount
Although 0.886 represents a decay factor, we use the term
growth factor to describe both increasing and decreasing
quantities.
Functions Modeling
Change:
A Preparation
for Calculus,
4th
A General Formula for the Family
of Exponential Functions
An
An exponential function Q = f(t) has the formula
f(t) = a bt, a 0, b > 0,
where a is the initial value of Q (at t = 0) and b, the
base, is the growth factor. The growth factor is given by
b=1+r
where r is the decimal representation of the percent
rate of change.
If there is exponential growth, then r > 0 and b > 1.
If there is exponential decay, then r < 0 and 0 < b < 1.
Functions Modeling
Change:
A Preparation
for Calculus,
4th
Applying the General Formula for
the Family of Exponential Functions
Example 6
Using Example 2, find a formula for P, the population
of Mexico (in millions), in year t where t = 0
represents the year 2000.
Because the growth factor may change eventually, this
formula may not give accurate results for large
values of t.
Functions Modeling
Change:
A Preparation
for Calculus,
4th
Definition of Exponential
Function
Function
A function represented by
f(t) = abt, b > 0,
0,
is an exponential function where a is the intial
value of f (at t = 0) and base b being the
growth factor.
growth
If b > 1, then f is an exponential growth function
If
then
If 0 < b < 1, then f is an exponential decay function
If
then
b = 1 + r where r is the decimal representation of the
percent change.
percent
f(t) = abt or f(t) = a(1 + r)t
The graph of f(t) = abt, b > 1 (exponential
growth function)
Range: (0, )
Domain: ( , )
Horizontal Asymptote
y=0
The graph of f(t) = abt, 0 < b < 1 (exponential
decay)
Range: (0, )
Horizontal Asymptote
y=0
x
4
Domain: ( , )
2,6,8,30,40
Chapter 4
Exponential Functions
4.2 Comparing Functions and
Linear Functions
Key Points
How to determine when a function is linear
How to determine when a function is
exponential
Finding formulas for exponential functions
Population Growth by a Constant
Number vs by a Constant
Number
Percentage
Percentage
Supposeapopulationis10,000inJanuary2004.
Supposethepopulationincreasesby
500 people per year
5% per year
What is the population in What is the population in
What
What
Jan 2005?
Jan 2005?
Jan
10,000 + 500 = 10,500
10,000 + .05(10,000) =
10,000
10,500
10,000
10,000 + 500 = 10,500
10,500
What is the population in
What
What is the population in
Jan 2006?
Jan
What
Jan 2006?
Jan
10,500 + 500 = 11,000
10,500
11,000
10,500 + .05(10,500) =
10,500
10,500 + 525 = 11,025
11,025
Suppose a population is 10,000 in Jan 2004.
Suppose the population increases
by 500 per year. What is the
by
population in .
population
Jan 2005?
10,000 + 500 = 10,500
Jan 2006?
10,000 + 2(500) = 11,000
Jan 2007?
10,000 + 3(500) = 11,500
Jan 2008?
10,000 + 4(500) = 12,000
Supposeapopulationis10,000in
Supposeapopulationis10,000in
Jan2004andincreasesby500per
year.
Let t be the number of years after 2004. Let P(t)
Let
be the population in year t. What is the symbolic
representation for P(t)? We know
Population in 2004 = P(0) = 10,000 + 0(500)
Population
10,000
Population in 2005 = P(1) = 10,000 + 1(500)
Population
10,000
Population in 2006 = P(2) = 10,000 + 2(500)
Population
10,000
Population in 2007 = P(3) = 10,000 + 3(500)
Population
10,000
Population t years after 2004 =
Population
P(t) = 10,000 + t(500)
10,000
Populationis10,000in2004;increases
Populationis10,000in2004;increases
by500peryrP(t)=10,000+t(500)
P is a linear function of t.
linear
What is the slope?
500 people/year
What is the y-intercept?
What
number of people at time 0 (the year 2004) = 10,000
When P increases
When
by a constant
number of people
per year, P is a
linear function of t.
Supposeapopulationis10,000inJan2004.
Supposeapopulationis10,000inJan2004.
Morerealistically,supposethepopulation
increasesby5%peryear.
Whatisthepopulationin.
Jan 2005?
10,000 + .05(10,000) =
10,000
10,000 + 500 = 10,500
10,000
Jan 2006?
10,500 + .05(10,500) =
10,500
10,500 + 525 = 11,025
10,500
Jan 2007?
11,025 + .05(11,025) =
11,025
11,025 + 551.25 =
11,576.25
11,576.25
Supposeapopulationis10,000inJan
Supposeapopulationis10,000inJan
2004andincreasesby5%peryear.
Let t be the number of years after 2004. Let P(t) be the
Let
be
population in year t. What is the symbolic
representation for P(t)? We know
Population in 2004 = P(0) = 10,000
Population
(0)
Population in 2005 = P(1) = 10,000 + .05 (10,000) =
Population
10,000
1.05(10,000) = 1.051(10,000) =10,500
1.05
Population in 2006 = P(2) = 10,500 + .05 (10,500) =
Population
10,500
1.05 (10,500) = 1.05 (1.05)(10,000) = 1.052(10,000) =
1.05
1.05 (1.05)(10,000) 1.05
11,025
11,025
Population t years after 2004 =
Population
P(t) = 10,000(1.05)t
P(t)
Populationis10,000in2004;increases
Populationis10,000in2004;increases
by5%peryrP(t)=10,000(1.05)t
P iis an EXPONENTIAL function of t. More specifically,
s
EXPONENTIAL
an exponential growth function.
an
What is the base of the exponential function?
1.05
What is the y-intercept?
number of people at time 0 (the year 2004) = 10,000
When P increases by a
When
constant percentage per
year, P is an exponential
function of t.
function
Linear vs. Exponential Growth
Linear
A Linear Function
Linear
adds a fixed amount
to the previous
value of y for each
unit increase in x
For example, in
For
f(x) = 10,000 + 500x
500 is added to y for
each increase of 1 in
x.
x.
An Exponential
An
Function multiplies a
fixed amount to the
previous value of y
for each unit increase
in For x.
example, in
For
f(x) = 10,000 (1.05)x y
is multiplied by 1.05
for each increase of 1
in x.
Comparison of Exponential and
Linear Functions
Linear
y = 10000(1.05) x
y = 10000 + 500 x
Linear Function
Linear
y = 10000 + 500x
x
0
1
2
3
4
5
6
y
10000
10500
11000
11500
12000
12500
13000
x
y
y
x
1
1
1
1
1
1
500
500
500
500
500
500
500/1=500
500/1=500
500/1=500
500/1=500
500/1=500
500/1=500
Linear
Function Slope is
constant.
Exponential Function
Exponential
Y = 10000 (1.05)x
x
0
1
2
3
4
5
6
y
10,000
10,500
11,025
11,576
12,155
12,763
13,401
Ratios of consecutive y-values
(corresponding to unit increases in x)
10500/10000 = 1.05
11025/10500 = 1.05
11576/11025 = 1.05
12155/11576 = 1.05
12763/12155 = 1.05
13401/12763 = 1.05
Note that this constant is the
Note
base of the exponential
function.
function.
Exponential
Exponential
Function Ratios of
consecutive
y-values
-values
(corresponding
to unit
increases in x)
are constant,
in this case
1.05.
Which function is linear and
which is exponential?
which
x
y
-3 3/8
-2 3/4
-1 3/2
0
3
1
6
2 12
3 24
xy
-3 9
-2 7
-1 5
03
11
2 -1
3 -3
For the linear function,
tell the slope and
y-intercept. For the
-intercept.
exponential function,
tell the base and the
y-intercept. Write the
-intercept.
equation of each.
equation
Identifying Linear and Exponential
Functions From a Table
For a table of data that gives y as a function
of x and in which x is constant:
If the difference of consecutive y-values is
constant, the table could represent a linear
function.
If the ratio of consecutive y-values is
constant, the table could represent an
exponential function.
Functions Modeling
Change:
A Preparation
for Calculus,
4th
2,12, 36
Chapter 4
Exponential Functions
4.3 GRAPHS OF EXPONENTIAL
FUNCTIONS
Key Points
The possible appearances of the graphs of
exponential functions
The effect of the initial value on the appearance
of the graph of an exponential function
The effect of the growth factor on the
appearance of the graph of an exponential
function
Why exponential functions have horizontal
asymptotes
Understanding limit notation and limits to infinity
Graphs of the Exponential Family:
The Effect of the Parameter a
In the formula Q = abt, the value of a tells us where the
graph crosses the Q-axis, since a is the value of Q when
t = 0.
Q
Q
Q=150 (1.2)t
Q=50 (1.4)t
Q=50 (1.2)t
Q=100 (1.2)t
150
100
50
50
Q=50 (1.2)t
0
5
10
t
Q=50 (0.8)t
0
Q=50 (0.6)t
5
t
Functions Modeling
Change:
A Preparation
for Calculus,
4th
Graphs of the Exponential Family:
The Effect of the Parameter b
The growth factor, b, is called the base of an
exponential function. Provided a is positive, if b > 1,
the graph climbs when read from left to right, and if 0
< b < 1, the graph falls when read from left to right.
Q
Q=50 (1.4)t
50
Q=50 (1.2)t
Q=50 (0.8)t
Q=50 (0.6)t
0
5
t
Functions Modeling
Change:
A Preparation
for Calculus,
4th
Horizontal Asymptotes
The horizontal line y = k is a horizontal asymptote of
The
a function, f, if the function values get arbitrarily
close to k as x gets large (either positively or
negatively or both). We describe this behavior using
the notation
f(x) k as x
or
f(x) k as x .
Alternatively, using limit notation, we write
lim f ( x) = k or lim f ( x) = k
x
x
Functions Modeling
Change:
A Preparation
for Calculus,
4th
Interpretation of a Horizontal
Asymptote
Example 1
A capacitor is the part of an electrical circuit that stores
electric charge. The quantity of charge stored decreases
exponentially with time. Stereo amplifiers provide a familiar
example: When an amplifier is turned off, the display lights
fade slowly because it takes time for the capacitors to
discharge. If t is the number of seconds after the circuit is
switched off, suppose that the quantity of stored charge (in
micro-coulombs) is given by Q = 200(0.9)t, t 0.
Q, charge (micro-coulombs)
200
The charge stored
by a capacitor over
one minute.
100
0
15
30
45
60
t (seconds)
Functions Modeling
Change:
A Preparation
for Calculus,
4th
Solving Exponential Equations
Graphically
Exercise 42
The population of a colony of rabbits grows exponentially.
The colony begins with 10 rabbits; five years later there
are 340 rabbits.
(a) Give a formula for the population of the colony of rabbits
as a function of the time.
(b) Use a graph to estimate how long it takes for the
population of the colony to reach 1000 rabbits.
R, # of rabbits
Solution
1500
(6.5+,
R = 10 (34)t/5 10 (2.0244)t
1000
Based on the graph, one
would estimate that the
population of rabbits would
reach 1000 in a little more
than 6 years.
1000)
500
(5,34)
(0,10)
0
1
2
3
4
5
6
7
t, years
Functions Modeling
Change:
A Preparation
for Calculus,
4th
Finding an Exponential Function for
Data
Example: Population data for the Houston Metro Area Since
1900
Table showing population
Graph showing population data
(in thousands) since 1900
t
0
10
20
30
40
50
N
184
236
332
528
737
1070
t
60
70
80
90
100
110
N
1583
2183
3122
3733
4672
5937
with an exponential model
P (thousands)
P = 190 (1.034)t
t (years since 1900)
Using an exponential regression feature on a calculator or
computer the exponential function was found to be P = 190
Functions Modeling
(1.034)t
Change:
A Preparation
for Calculus,
4th
18, Like 24, like 26, like 28
Chapter 4
Exponential Functions
4.5 The Number e
Key Points
Basic facts about the number e
Continuous growth rates
The Natural Number e
An irrational number, introduced by Euler in
1727, is so important that it is given a special
name, e. Its value is approximately e 2.71828 .
. .. It is often used for the base, b, of the
exponential function. Base e is called the natural
base. This may seem mysterious, as what could
possibly be natural about using an irrational base
such as e? The answer is that the formulas of
calculus are much simpler if e is used as the
base for exponentials.
Functions Modeling
Change:
A Preparation
for Calculus,
4th
Graphs of exponential functions
with various bases
Exponential Functions with Base e
For the exponential function Q = a bt, the
continuous growth rate, k, is given by
solving
ek = b. Then
Q = a ekt.
If a is positive,
If k > 0, then Q is increasing.
If k < 0, then Q is decreasing.
Functions Modeling
Change:
A Preparation
for Calculus,
4th
Exponential Functions with Base e
Example 1
Give the continuous growth rate of each of the following functions
and graph each function:
P = 5e0.2t,
Q = 5e0.3t,
and R = 5e0.2t.
20
15
R = 5e0.2t
10
P = 5e0.2t
Q = 5e0.3t
5
5
0
5
t
Functions Modeling
Change:
A Preparation
for Calculus,
4th
Exponential Functions with Base e
Example 1
Give the continuous growth rate of each of the following functions
and graph each function:
P = 5e0.2t,
Q = 5e0.3t,
and R = 5e0.2t.
Solution
The function P = 5e0.2t has a continuous growth rate of 20%, Q =
5e0.3t has a continuous 30% growth rate, and R = 5e0.2t has a
continuous growth rate of 20%. The negative sign in the
exponent tells us that R is decreasing instead of increasing.
Because a = 5 in all three
Q = 5e0.3t
functions, they each pass
P = 5e0.2t
through the point (0,5). They
are all concave up and have
R = 5e0.2t
horizontal asymptote y = 0.
20
15
10
5
5
0
5
t
Functions Modeling
Change:
A Preparation
for Calculus,
4th
Exponential Functions with Base
e
Like Example 3
Caffeine leaves the body at a continuous rate of 17%
per hour. How much caffeine is left in the body 4
hours after drinking a Monster Energy Drink
containing 160 mg of caffeine?
Functions Modeling
Change:
A Preparation
for Calculus,
4th
b = ek
Exponential Functions with Base e
Represent Continuous Growth
Any positive base b can be written as a
power of e:
b = ek
The function Q = abt = a(ek)t = aekt
S12, like10, like12, 18, 22