Transcript barnfm10e_ppt_2_2

Exponential functions
The equation
f ( x)  b x
defines the exponential function with
base b . The domain is the set of all real
numbers, while the range is the set of all
positive real numbers
( y > 0). Note y cannot equal to zero.
Riddle
 Here is a problem related to exponential functions:
 Suppose you received a penny on the first day of
December, two pennies on the second day of December,
four pennies on the third day, eight pennies on the
fourth day and so on. How many pennies would you
receive on December 31 if this pattern continues?
 2) Would you rather take this amount of money or
receive a lump sum payment of $10,000,000?
Solution (Complete the table)
Day
1
2
3
No.
pennies
1
2
4
4
5
6
7
8
16
32
64
2^1
2^2
2^3
Generalization
 Now, if this pattern continued, how many
pennies would you have on Dec. 31?
 Your answer should be 2^30 ( two raised to
the thirtieth power). The exponent on two is
one less than the day of the month. See the
preceding slide.
 What is 2^30?
 1,073,741,824 pennies!!! Move the decimal
point two places to the left to find the amount
in dollars. You should get: $10,737,418.24
Solution, continued
 The obvious answer to question two is to take
the number of pennies on December 31 and
not a lump sum payment of $10,000,000
 (although, I would not mind having either
amount!)
 This example shows how an exponential
function grows extremely rapidly. In this case,
the exponential function f ( x)  2 x

is used to model this problem.
Graph of
f ( x)  2
x
 Use a table to graph the exponential
function above. Note: x is a real
number and can be replaced with
numbers such as
as well as
2
other irrational numbers. We will use
integer values for x in the table:
Table of values
x
-4
-3
-2
-1
0
1
2
y
2 4 
2 3 
1
1

4
2
16
1
8
1
2 
4
1
1
2 
2
2
2 1
0
21  2
2
2 4
Graph of y = f ( x)  2
x
Characteristics of the graphs of
where b> 1
f ( x)  b
x
 1. all graphs will approach the x-axis as x gets large.





2.
3.
4.
5.
6.
all graphs will pass through (0,1) (y-intercept)
There are no x – intercepts.
Domain is all real numbers
Range is all positive real numbers.
The graph is always increasing on its domain.
 7. All graphs are continuous curves.
Graphs of
f ( x)  b x
if 0 < b < 1
 1. all graphs will approach the x-axis as x gets large.





2.
3.
4.
5.
6.
all graphs will pass through (0,1) (y-intercept)
There are no x – intercepts.
Domain is all real numbers
Range is all positive real numbers.
The graph is always decreasing on its domain.
 7. All graphs are continuous curves.
Graph of

1
f ( x)  2  x
2
x
Using a table of values once again, you will obtain the following graph.
x
The graphs of f ( x)  b
and f ( x )  b  x will be symmetrical with
respect to the y-axis, in general.
12
10
8
graph of y = 2^(-x)
6
approaches the positive x-axis as x gets large
4
2
passes through (0,1)
0
-4
-2
0
2
4
Graphing other exponential
functions
 Now, let’s graph
f ( x)  3
x
 Proceeding as before, we construct a
table of values and plot a few
points.Be careful not to assume that
the graph crosses the negative x-axis.
Remember, it gets close to the x-axis,
but never intersects it.
Preliminary graph of
f ( x)  3
x
Complete graph
30
y = 3^x
25
20
15
Series1
10
5
0
-4
-2
0
2
4
Other exponential graphs
 This is the graph of
x
f ( x)  4
 It is symmetric to the
graph of f ( x)  4 x
 with respect to the y-axis
 Notice that it is always
decreasing.
 It also passes through
(0,1).
Exponential function with base e
1
2
10
2.59374246
100
2.704813829
1
1000
10000
1000000
 The table to the left
illustrates what happens
to the expression
 1
1  
 x
(
+
1
2.716923932
/
x
2.718145927
)
^
x
2.718280469

x
as x gets increasingly
larger. As we can see
from the table, the
values approach a
number whose
approximation is 2.718
Leonard Euler
x
 Leonard Euler first demonstrated that  1  1 


x

 will approach a fixed constant we now call “e”.
 So much of our mathematical notation is due to Euler
that it will come as no surprise to find that the
notation e for this number is due to him. The claim
which has sometimes been made, however, that Euler
used the letter e because it was the first letter of his
name is ridiculous. It is probably not even the case
that the e comes from "exponential", but it may have
just be the next vowel after "a" and Euler was already
using the notation "a" in his work. Whatever the
reason, the notation e made its first appearance in a
letter Euler wrote to Goldbach in 1731.

(http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/e.html#s19)
Leonard Euler





He made various discoveries
regarding e in the following
years, but it was not until 1748
when Euler published
Introductio in Analysis in
infinitorum that he gave a full
treatment of the ideas
surrounding e. He showed that
e = 1 + 1/1! + 1/2! + 1/3!
+ ...
and that e is the limit of
(1 + 1/n)^n as n tends to
infinity. Euler gave an
approximation for e to 18
decimal places,
e = 2.718281828459045235
Graph of
f ( x)  e
 Graph is similar to
the graphs of
graph of y = e^x
25
f ( x)  2 x
20
 and
15
Series1
10
5
0
-4
-2
x
0
2
4
f ( x )  3x
Has same
characteristics as
these graphs
Growth and Decay applications

The atmospheric pressure p
decreases with increasing
height. The pressure is
related to the number of
kilometers h above the sea
level by the formula:
P(h)  760e0.145h
 Find the pressure at
sea level ( h =1)
 Find the pressure at
a height of 7
kilometers.
Solution:
 Find the pressure at
sea level ( h =1)
P(1)  760e
0.145(1)
 657.42
 Find the pressure at
a height of 7
kilometers
P(7)  760e0.145(7)  275.43
Depreciation of a machine
 A machine is initially
 Solution:
worth V
dollars
0
 but loses 10% of its
V (t )  V0 (0.9t )
value each year. Its value
after t years is given by
the formula
V (8)  30000(0.98 )
V (t )  V0 (0.9 )
t
 Find the value after 8
years of a machine
whose initial value is
$30,000
 $12,914
Compound interest
 The compound interest formula is
nt
 r
A  P 1  
 n
 Here, A is the future value of the
investment, P is the initial amount
(principal), r is the annual interest
rate as a decimal, n represents the
number of compounding periods per
year and t is the number of years
Problem:
 Find the amount to which $1500 will grow if deposited in
a bank at 5.75% interest compounded quarterly for 5
years.
 Solution: Use the compound interest formula:


 r
A  P 1  
 n
nt
Substitute 1500 for P, r = 0.0575,
n = 4 and t = 5 to obtain
 0.0575 
A  1500  1  

4 

(4)(5)
=$1995.55