Transcript e – x

CHAPTER 5:
Exponential and
Logarithmic Functions
5.1
5.2
5.3
5.4
5.5
Inverse Functions
Exponential Functions and Graphs
Logarithmic Functions and Graphs
Properties of Logarithmic Functions
Solving Exponential and Logarithmic
Equations
5.6 Applications and Models: Growth and Decay;
and Compound Interest
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5.2
Exponential Functions
and Graphs


Graph exponential equations and exponential
functions.
Solve applied problems involving
exponential functions and their graphs.
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Exponential Function
The function f(x) = ax, where x is a real number, a > 0
and a  1, is called the exponential function, base a.
The base needs to be positive in order to avoid the
complex numbers that would occur by taking even
roots of negative numbers.
The following are examples of exponential functions:
x
 1
x
f (x)  2
f (x)   
f (x)  (3.57)x
 2
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Slide 5.2 - 4
Graphing Exponential Functions
To graph an exponential function, follow the steps
listed:
1. Compute some function values and list the results in
a table.
2. Plot the points and connect them with a smooth
curve. Be sure to plot enough points to determine
how steeply the curve rises.
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Example
Graph the exponential function y = f (x) = 2x.
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Example (continued)
As x increases, y increases
without bound. As x decreases,
y decreases getting close to 0;
as x g ∞, y g 0.
The x-axis, or the line
y = 0, is a horizontal
asymptote. As the x-inputs
decrease, the curve gets
closer and closer to this line,
but does not cross it.
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Example
x
 1
Graph the exponential function y  f (x)    .
 2
x
 1
1 x
Note y  f (x)     2
 2  x.
 2
 
This tells us the
graph is the
reflection of the
graph of y = 2x
across the yaxis. Selected
points are listed
in the table.
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Example (continued)
As x increases, the
function values
decrease, getting
closer and closer
to 0. The x-axis,
y = 0, is the
horizontal
asymptote. As x
decreases, the
function values
increase without
bound.
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Graphs of Exponential Functions
Observe the following graphs of exponential functions
and look for patterns in them.
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Example
Graph y = 2x – 2.
The graph is the graph of y = 2x shifted to right 2 units.
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Example
x
 1
Graph y = 5 –
y  5  0.5  5     5  2  x.
 2
The graph is a reflection of the graph of y = 2x across
the y-axis, followed by a reflection across the x-axis
and then a shift up 5 units.
0.5x .
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x
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Application
The amount of money A that a principal P will
grow to after t years at interest rate r (in decimal
form), compounded n times per year, is given by
the formula
nt
r

A  P 1   .

n
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Slide 5.2 - 13
Example
Suppose that $100,000 is invested at 6.5% interest,
compounded semiannually.
a) Find a function for the amount to which the
investment grows after t years.
b) Graph the function.
c) Find the amount of money in the account at t = 0,
4, 8, and 10 yr.
d) When will the amount of money in the account
reach $400,000?
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Example (continued)
Solution:
a) Since P = $100,000, r = 6.5%=0.65, and n = 2, we
can substitute these values and write the following
function
0.065 

A t   100,000  1 


2
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2t
 $100,000 1.0325 
2t
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Example (continued)
Solution continued:
b) Use the graphing calculator with viewing window
[0, 30, 0, 500,000].
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Example (continued)
Solution continued:
c) We can compute function values using function
notation on the home screen of a graphing calculator.
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Example (continued)
Solution continued:
c) We can also calculate the values directly on a
graphing calculator by substituting in the expression
for A(t):
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Example (continued)
Solution continued:
d) Set 100,000(1.0325)2t = 400,000 and solve for t,
which we can do on
the graphing
calculator. Graph the
equations
y1 = 100,000(1.0325)2t
y2 = 400,000
Then use the intersect
method to estimate the
first coordinate of the
point of intersection.
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Slide 5.2 - 19
Example (continued)
Solution continued:
d) Or graph y1 = 100,000(1.0325)2t – 400,000 and use
the Zero method to
estimate the zero of
the function
coordinate of the
point of intersection.
Regardless of the
method, it takes
about 21.67 years, or
about 21 yr, 8 mo,
and 2 days.
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The Number e
e is a very special number in mathematics. Leonard
Euler named this number e. The decimal representation
of the number e does not terminate or repeat; it is an
irrational number that is a constant;
e  2.7182818284…
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Example
Find each value of ex, to four decimal places, using the
ex key on a calculator.
a) e3
b) e0.23
c) e2
d) e1
Solution:
a) e3 ≈ 20.0855
b) e0.23 ≈ 0.7945
c) e0 = 1
d) e1 ≈ 2.7183
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Slide 5.2 - 22
Graphs of Exponential Functions, Base e
Example
Graph f(x) = ex and g(x) = e–x.
Use the calculator and enter y1 = ex and y2 = e–x. Enter
numbers for x.
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Graphs of Exponential Functions,
Base e - Example (continued)
The graph of g is a reflection of the graph of f across
they-axis.
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Example
Graph f(x) = ex + 3.
Solution: The graph f(x) = ex + 3 is a translation of
the graph of y = ex left 3 units.
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Example
Graph f(x) = e–0.5x.
Solution: The graph f(x) = e–0.5x is a horizontal
stretching of the graph of y = ex followed by a
reflection across the y-axis.
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Example
Graph f(x) = 1  e2x.
Solution: The graph f(x) = 1  e2x is a horizontal
shrinking of the graph of y = ex
followed by a reflection across
the y-axis and then across the
x-axis,
followed
by a
translation
up 1 unit.
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