EXPONENTIAL FUNCTIONS
Quick Growth
Spreading of Rumors
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A model for the number of people N in a college
community who have heard a certain rumor is
N  P 1  e
0.15 d

where P is the total population of the community and
d is the number of days that have elapsed since the
rumor began. In a community of 1000 students, how
many students will have heard the rumor after 3
days?
Exponential Functions
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What is e?
What does this equation mean?
How do I solve it?
HUH?
Exponential Functions
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Definition

An exponential function is a function of the form
f ( x)  a x
where a is a positive real number (a > 0) and a  1.
The domain of f is the set of real numbers.
Laws of Exponents
Graphing an Exponential Function
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Use an x-y chart to graph the exponential function:
f ( x)  2
x
Properties of an Exponential Function
a>1
1.
2.
3.
The domain is all real numbers; the range is the set
of positive real numbers (why?)
There are no x-intercepts, the y-intercept is 1.
(why?)
The x-axis (y=0) is a horizontal asymptote.
4. f ( x)  a x , a  1, is an increasing function and
is one-to-one.
1

5. The graph of f contains the points (0,1), (1, a), and  1,  . (why?)
a
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6. The graph of f is smooth and continuous, with no corners or gaps.
Properties of an Exponential Function
0 < a < 1(a is a Fraction)
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Differences:
The graph is a decreasing function
The points are fractions at one and whole numbers
at negative one.
The Number e
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The number e is defined as the number that the
expression  1 n
approaches as n .
1  
 n
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In calculus, this is expressed using limits notation as
 1
e  lim  1  
n 
 n
n
Graphs of Exponential Function to Base
e
Exponential Equations

If au  av , then u  v
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Examples
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More Examples
Applications
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Solve the beginning problem
Applications
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Healing of Wounds p. 299 #64
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Drug Medication p. 299 # 65
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More Examples