Chapter 3: Exponential, Logistic, and Logarithmic Functions 3.1a

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Chapter 3 Exponential, Logistic, and Logarithmic
Functions3.1a bHomework p. 286-287 1-39 odd
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Overview of Chapter 3
So far in this course, we have mostly studied
algebraic functions, such as polys, rationals,
and power functions w/ ratl exponents
The three types of functions in this chapter
(exponential, logistic, and logarithmic) are
called transcendental functions, because they go
beyond the basic algebra operations involved in
the aforementioned functions
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Consider these problems
Evaluate the expression without using a
calculator.
1.
2.
3.
4.
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We begin with an introduction to exponential
functions
First, consider
Now, what happens when we switch the base and
the exponent ???
This is a familiar monomial, and a power
function one of the twelve basics?
This is an example of an exponential function
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Definition Exponential Functions
Let a and b be real number constants. An
exponential function in x is a function that can
be written in the form
where a is nonzero, b is positive, and b 1.
The constant a is the initial value of f (the
value at x 0), and b is the base.
Note Exponential functions are defined and
continuous for all real numbers!!!
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Identifying Exponential Functions
Which of the following are exponential functions?
For those that are exponential functions, state
the initial value and the base. For those that
are not, explain why not.
1.
4.
Initial Value 1, Base 3
Initial Value 7, Base 1/2
2.
Nope! ? g is a power func.!
5.
Nope! ? q is a const. func.!
3.
Initial Value 2, Base 1.5
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More Practice with Exponents
, find an exact value for
Given
2.
1.
3.
4.
5.
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Finding an Exponential Function from its Table of
Values
Determine the formula for the exp. func. g
General Form
x
g(x)
2
4/9
x 3
Initial Value
1
4/3
x 3
0
4
x 3
Solve for b
1
12
x 3
2
36
The Pattern?
Final Answer
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Finding an Exponential Function from its Table of
Values
Determine the formula for the exp. func. h
General Form
x
h(x)
Initial Value
2
128
x 1/4
Solve for b
1
32
x 1/4
0
8
x 1/4
Final Answer
1
2
x 1/4
2
1/2
The Pattern?
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How an Exponential Function Changes (a recursive
formula)
For any exponential function
and any real number x,
If a gt 0 and b gt 1, the function f is increasing
and is an exponential growth function. The base
b is its growth factor.
If a gt 0 and b lt 1, f is decreasing and is an
exponential decay function. The base b is its
decay factor.
Does this formula make sense with our previous
examples?
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Graphs of Exponential Functions
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We start with an Exploration
Graph the four given functions in the same
viewing window 2, 2 by 1, 6. What point
is common to all four graphs?
Graph the four given functions in the same
viewing window 2, 2 by 1, 6. What point
is common to all four graphs?
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We start with an Exploration
Now, can we analyze these graphs???
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x
Exponential Functions f(x) b
Domain
Continuity
Continuous
Symmetry
None
Range
Boundedness
Below by y 0
Extrema
None
None
H.A.
y 0
V.A.
If b gt 1, then also
If 0 lt b lt 1, then also

• f is an increasing func.,

• f is a decreasing func.,

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In Sec. 1.3, we first saw the The Exponential
Function
Natural
(we now know that it is an exponential growth
function ? why?)
But what exactly is this number e???
Definition The Natural Base e
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Analysis of the Natural Exponential Function
The graph
Domain All reals
Range
Continuous
Increasing for all x
No symmetry
Bounded below by y 0
No local extrema
H.A. y 0
V.A. None
End behavior
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Guided Practice
Describe how to transform the graph of f into the
graph of g.
1.
Trans. right 1
2.
Reflect across y-axis
3.
Horizon. shrink by 1/2
4.
Reflect across both axes, Trans. right 2
5.
Reflect across y-axis, Vert. stretch by 5, Trans.
up 2
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Guided Practice
Determine a formula for the exponential function
whose graph is shown.
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Whiteboard
State whether the given function is exp. growth
or exp. decay, and describe its end behavior
using limits.
Exponential Decay
Exponential Growth
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Whiteboard
Solve the given inequality graphically.
The graph?
The graph?
x gt 0
x gt 0