precalculus v. j. motto exponential/logarithmic part 1
TRANSCRIPT
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PrecalculusV. J. Motto
Exponential/Logarithmic Part 1
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Exponential and Logarithmic Functions4
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Chapter Overview
In this chapter, we study a new class
of functions called exponential functions.
• For example, f(x) = 2x
is an exponential function (with base 2).
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Chapter Overview
Notice how quickly the values of this function
increase:
f(3) = 23 = 8
f(10) = 210 = 1,024
f(30) = 230 = 1,073,741,824
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Chapter Overview
Compare that with the function
g(x) = x2
where g(30) = 302 = 900.
• The point is, when the variable is in the exponent, even a small change in the variable can cause a dramatic change in the value of the function.
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Chapter Overview
In spite of this incomprehensibly huge growth,
exponential functions are appropriate for
modeling population growth for all living
things—from bacteria to elephants.
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Chapter Overview
To understand how a population grows,
consider the case of a single bacterium,
which divides every hour.
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Chapter Overview
After one hour, we would have 2 bacteria;
after two hours, 22 or 4 bacteria; after three
hours, 23 or 8 bacteria; and so on.
• After x hours, we would have 2x bacteria.
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Chapter Overview
This leads us to model the bacteria
population by the function
f(x) = 2x
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Chapter Overview
The principle governing population
growth is:
• The larger the population, the greater the number of offspring.
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Chapter Overview
This same principle is present in
many other real-life situations.
• For example, the larger your bank account, the more interest you get.
• So, we also use exponential functions to find compound interest.
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Chapter Overview
We use logarithmic functions—which are
inverses of exponential functions—to help
us answer such questions as:
When will my investment grow to $100,000?
• In Focus on Modeling, we explore how to fit exponential and logarithmic models to data.
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Exponential Functions4.1
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Introduction
We now study one of the most
important functions in mathematics—
the exponential function.
• This function is used to model such natural processes as population growth and radioactive decay.
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Exponential Functions
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Exponential Functions
In Section 1-2, we defined ax for a > 0 and x
a rational number.
However, we have not yet defined irrational
powers.
• So, what is meant by or 2π?35
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Exponential Functions
To define ax when x is irrational,
we approximate x by rational numbers.
• For example, since
is an irrational number, we successively approximate by these rational powers:
3 1.73205...
3a
1.7 1.73 1.732 1.7320 1.73205, , , , , ...a a a a a
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Exponential Functions
Intuitively, we can see that these rational
powers of a are getting closer and closer
to .
• It can be shown using advanced mathematics that there is exactly one number that these powers approach.
• We define to be this number.
3a
3a
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Exponential Functions
For example, using a calculator,
we find:
• The more decimal places of we use in our calculation, the better our approximation of .
• It can be proved that the Laws of Exponents are still true when the exponents are real numbers.
3 1.7325 5 16.2411...
335
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Exponential Function—Definition
The exponential function with base a
is defined for all real numbers x by:
f(x) = ax
where a > 0 and a ≠ 1.
• We assume a ≠ 1 because the function f(x) = 1x = 1 is just a constant function.
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Exponential Functions
Here are some examples:
f(x) = 2x
g(x) = 3x
h(x) = 10x
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E.g. 1—Evaluating Exponential Functions
Let f(x) = 3x and evaluate the following:
(a) f(2)
(b) f(–⅔)
(c) f(π)
(d) f( )
• We use a calculator to obtain the values of f.
2
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E.g. 1—Evaluating Exp. Functions
Calculator keystrokes: 3, ^, 2, ENTER
Output: 9
• Thus, f(2) = 32 = 9
Example (a)
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E.g. 1—Evaluating Exp. Functions
Calculator keystrokes:
3, ^, (, (–), 2, ÷, 3, ), ENTER
Output: 0.4807498
• Thus, f(–⅔) = 3–⅔ ≈ 0.4807
Example (b)
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E.g. 1—Evaluating Exp. Functions
Calculator keystrokes: 3, ^, π, ENTER
Output: 31.5442807
• Thus, f(π) = 3π ≈ 31.544
Example (c)
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E.g. 1—Evaluating Exp. Functions
Calculator keystrokes: 3, ^, √, 2, ENTER
Output: 4.7288043
• Thus, f( ) = ≈ 4.7288
Example (d)
223
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Graphs of
Exponential Functions
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We first graph exponential functions
by plotting points.
• We will see that these graphs have an easily recognizable shape.
Graphs of Exponential Functions
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Draw the graph of each function.
(a) f(x) = 3x
(b) g(x) = (⅓)x
E.g. 2—Graphing Exp. Functions by Plotting Points
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First, we calculate values of f(x)
and g(x).
E.g. 2—Graphing Exp. Functions by Plotting Points
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Then, we the plot points to sketch
the graphs.
E.g. 2—Graphing Exp. Functions by Plotting Points
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Notice that:
• So, we could have obtained the graph of g from the graph of f by reflecting in the y-axis.
1 1( ) 3 ( )
3 3
xx
xg x f x
E.g. 2—Graphing Exp. Functions by Plotting Points
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Graphs of Exponential Functions
The figure shows the graphs of the family
of exponential functions f(x) = ax for various
values of the base a.
• All these graphs pass through the point (0, 1) because a0 = 1 for a ≠ 0.
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Graphs of Exponential Functions
You can see from the figure that there are
two kinds of exponential functions:
• If 0 < a < 1, the function decreases rapidly.
• If a > 1, the function increases rapidly.
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Graphs of Exponential Functions
The x-axis is a horizontal asymptote for
the exponential function f(x) = ax.
This is because:
• When a > 1, we have ax → 0 as x → –∞.
• When 0 < a < 1, we have ax → 0 as x → ∞.
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Graphs of Exponential Functions
Also, ax > 0 for all .
So, the function f(x) = ax has domain
and range (0, ∞).
• These observations are summarized as follows.
x
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Graphs of Exponential Functions
The exponential function
f(x) = ax (a > 0, a ≠ 1)
has domain and range (0, ∞).
• The line y = 0 (the x-axis) is a horizontal asymptote of f.
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Graphs of Exponential Functions
The graph of f has one of these
shapes.
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E.g. 3—Identifying Graphs of Exponential Functions
Find the exponential function f(x) = ax
whose graph is given.
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E.g. 3—Identifying Graphs
Since f(2) = a2 = 25, we see that
the base is a = 5.
• Thus, f(x) = 5x
Example (a)
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E.g. 3—Identifying Graphs
Since f(3) = a3 = 1/8 , we see that
the base is a = ½ .
• Thus, f(x) = (½)x
Example (b)
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Graphs of Exponential Functions
In the next example, we see how to graph
certain functions—not by plotting points—
but by:
1. Taking the basic graphs of the exponential functions in Figure 2.
2. Applying the shifting and reflecting transformations of Section 2-4.
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E.g. 4—Transformations of Exponential Functions
Use the graph of f(x) = 2x to sketch the graph
of each function.
(a) g(x) = 1 + 2x
(b) h(x) = –2x
(c) k(x) = 2x –1
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E.g. 4—Transformations
To obtain the graph of g(x) = 1 + 2x,
we start with the graph of f(x) = 2x and
shift it upward 1 unit.
• Notice that the line y = 1 is now a horizontal asymptote.
Example (a)
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E.g. 4—Transformations
Again, we start with
the graph of f(x) = 2x.
However, here,
we reflect in the x-axis
to get the graph of
h(x) = –2x.
Example (b)
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E.g. 4—Transformations
This time, we start with
the graph of f(x) = 2x
and shift it to the right
by 1 unit—to get
the graph of k(x) = 2x–1.
Example (c)
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E.g. 5—Comparing Exponential and Power Functions
Compare the rates of growth of the
exponential function f(x) = 2x and the power
function g(x) = x2 by drawing the graphs of
both functions in these viewing rectangles.
(a) [0, 3] by [0, 8]
(b) [0, 6] by [ 0, 25]
(c) [0, 20] by [0, 1000]
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E.g. 5—Exp. and Power Functions
The figure shows that the graph of
g(x) = x2 catches up with, and becomes
higher than, the graph of f(x) = 2x at x = 2.
Example (a)
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E.g. 5—Exp. and Power Functions
The larger viewing rectangle here shows
that the graph of f(x) = 2x overtakes that
of g(x) = x2 when x = 4.
Example (b)
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E.g. 5—Exp. and Power Functions
This figure gives a more global view
and shows that, when x is large, f(x) = 2x
is much larger than g(x) = x2.
Example (c)
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The Natural
Exponential Function
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Natural Exponential Function
Any positive number can be used as the base
for an exponential function.
However, some are used more frequently
than others.
• We will see in the remaining sections of the chapter that the bases 2 and 10 are convenient for certain applications.
• However, the most important is the number denoted by the letter e.
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Number e
The number e is defined as the value
that (1 + 1/n)n approaches as n becomes
large.
• In calculus, this idea is made more precise through the concept of a limit.
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Number e
The table shows
the values of the
expression (1 + 1/n)n
for increasingly large
values of n.
• It appears that, correct to five decimal places,
e ≈ 2.71828
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Number e
The approximate value to 20 decimal
places is:
e ≈ 2.71828182845904523536
• It can be shown that e is an irrational number.
• So, we cannot write its exact value in decimal form.
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Number e
Why use such a strange base for
an exponential function?
• It may seem at first that a base such as 10 is easier to work with.
• However, we will see that, in certain applications, it is the best possible base.
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Natural Exponential Function—Definition
The natural exponential function is
the exponential function
f(x) = ex
with base e.
• It is often referred to as the exponential function.
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Natural Exponential Function
Since 2 < e < 3, the graph of the natural
exponential function lies between
the graphs of y = 2x
and y = 3x.
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Natural Exponential Function
Scientific calculators have a special
key for the function f(x) = ex.
• We use this key in the next example.
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E.g. 6—Evaluating the Exponential Function
Evaluate each expression correct to five
decimal places.
(a) e3
(b) 2e–0.53
(c) e4.8
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E.g. 6—Evaluating the Exponential Function
We use the ex key on a calculator to
evaluate the exponential function.
(a) e3 ≈ 20.08554
(b) 2e–0.53 ≈ 1.17721
(c) e4.8 ≈ 121.51042
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E.g. 7—Transformations of the Exponential Function
Sketch the graph of each function.
(a) f(x) = e–x
(b) g(x) = 3e0.5x
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E.g. 7—Transformations
We start with the graph of y = ex and reflect
in the y-axis
to obtain the graph
of y = e–x.
Example (a)
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E.g. 7—Transformations
We calculate several values, plot
the resulting points, and then connect
the points with a smooth curve.
Example (b)
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E.g. 8—An Exponential Model for the Spread of a Virus
An infectious disease begins to spread
in a small city of population 10,000.
• After t days, the number of persons who have succumbed to the virus is modeled by:
0.97
10,000( )
5 1245 tv t
e
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(a) How many infected people are there
initially (at time t = 0)?
(b) Find the number of infected people after
one day, two days, and five days.
(c) Graph the function v and describe
its behavior.
E.g. 8—An Exponential Model for the Spread of a Virus
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E.g. 8—Spread of Virus
• We conclude that 8 people initially have the disease.
Example (a)
0(0) 10,000 /(5 1245 )
10,000 /1250
8
v e
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E.g. 8—Spread of Virus
Using a calculator, we evaluate v(1), v(2),
and v(5).
Then, we round off to obtain these values.
Example (b)
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E.g. 8—Spread of Virus
From the graph, we see that the number
of infected people:
• First, rises slowly.
• Then, rises quickly between day 3 and day 8.
• Then, levels off when about 2000 people are infected.
Example (c)
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Logistic Curve
This graph is called a logistic curve or
a logistic growth model.
• Curves like it occur frequently in the study of population growth.
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Compound Interest
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Compound Interest
Exponential functions occur in
calculating compound interest.
• Suppose an amount of money P, called the principal, is invested at an interest rate i per time period.
• Then, after one time period, the interest is Pi, and the amount A of money is:
A = P + Pi + P(1 + i)
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Compound Interest
If the interest is reinvested, the new principal
is P(1 + i), and the amount after another time
period is:
A = P(1 + i)(1 + i) = P(1 + i)2
• Similarly, after a third time period, the amount is:
A = P(1 + i)3
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Compound Interest
In general, after k periods,
the amount is:
A = P(1 + i)k
• Notice that this is an exponential function with base 1 + i.
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Compound Interest
Now, suppose the annual interest rate is r and
interest is compounded n times per year.
Then, in each time period, the interest rate
is i = r/n, and there are nt time periods
in t years.
• This leads to the following formula for the amount after t years.
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Compound Interest
Compound interest is calculated by
the formula
where:• A(t) = amount after t years
• P = principal
• t = number of years
• n = number of times interest is compounded per year
• r = interest rate per year
( ) 1n t
rA t P
n
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E.g. 9—Calculating Compound Interest
A sum of $1000 is invested at an interest rate
of 12% per year.
Find the amounts in the account after 3 years
if interest is compounded:• Annually• Semiannually• Quarterly• Monthly• Daily
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E.g. 9—Calculating Compound Interest
We use the compound interest formula
with: P = $1000, r = 0.12, t = 3
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Compound Interest
We see from Example 9 that the interest
paid increases as the number of
compounding periods n increases.
• Let’s see what happens as n increases indefinitely.
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Compound Interest
If we let m = n/r, then
/
( ) 1
1
11
n t
r tn r
r tm
rA t P
n
rP
n
Pm
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Compound Interest
Recall that, as m becomes large,
the quantity (1 + 1/m)m approaches
the number e.
• Thus, the amount approaches A = Pert.
• This expression gives the amount when the interest is compounded at “every instant.”
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Continuously Compounded Interest
Continuously compounded interest is
calculated by
A(t) = Pert
where:• A(t) = amount after t years
• P = principal
• r = interest rate per year
• t = number of years
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E.g. 10—Continuously Compounded Interest
Find the amount after 3 years if $1000
is invested at an interest rate of 12%
per year, compounded continuously.
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E.g. 10—Continuously Compounded Interest
We use the formula for continuously
compounded interest with:
P = $1000, r = 0.12, t = 3
• Thus, A(3) = 1000e(0.12)3 = 1000e0.36
= $1433.33
• Compare this amount with the amounts in Example 9.