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Chapter 8
Exponential functions
In this chapter, we leave behind the power functions and polynomials, and explore a new type
of function, exponential growth. We Fnd that this function is closely related to (uncontrolled)
growth of living replicating organisms, and study one example from the popular scienceFction
literature. The inverse function, the logarithm, is also studied, and the properties of the two are
related. Understanding how to manipulate expressions involving such functions will be a skill to be
practiced
1
.
8.1
The Andromeda Strain
”The mathematics of uncontrolled growth are frightening. A single cell of the bacterium E. coli
would, under ideal circumstances, divide every twenty minutes. That is not particularly disturbing
until you think about it, but the fact is that bacteria multiply geometrically: one becomes two, two
become four, four become eight, and so on. In this way it can be shown that in a single day, one
cell of E. coli could produce a supercolony equal in size and weight to the entire planet Earth.”
Michael Crichton (1969) The Andromeda Strain, Dell, N.Y. p247
1
Lab 4 of this calculus course will exploit that skill in introducing the student to the formula for the temperature
of a cooling object. We will later have more to say about such applications, once additional tools of diFerential
equations are learned in a later chapter.
v.2005.1  September 4, 2009
1
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View Full Document Math 102 Notes
Chapter 8
8.2
Powers of 2
n
2
n
0
1
1
2
2
4
3
8
4
16
5
32
6
64
7
128
8
256
9
512
10
1024
4.0
10.0
0.0
1000.0
Figure 8.1: Powers of 2 including both negative and positive integers: Here we show 2
n
for

4
<
n <
10. Note that 2
10
≈
1000 = 10
3
.
Note that 2
10
≈
1000 = 10
3
. This is a useful approximation in converting binary numbers
(powers of 2) to decimal numbers (powers of 10).
v.2005.1  September 4, 2009
2
Math 102 Notes
Chapter 8
8.3
Growth of e. coli
•
Mass of 1 E. coli cell : 1 nanogram = 10

9
gm = 10

12
kg.
•
Mass of Planet Earth : 6
·
10
24
kg
•
Size of E. coli colony equal in mass to Planet Earth:
m
=
6
·
10
24
10

12
= 6
·
10
36
In a period of 24 hours, there are many 20minute generations. To be exact, there are 24
×
3 = 72
generations, with each one producing a doubling. This means that there would be, after 1 day, a
number of cells equal to
2
72
.
We can estimate it using the approximate decimal form as follows:
2
72
= 2
2
·
2
70
= 4
·
(2
10
)
7
≈
4
·
(10
3
)
7
= 4
·
10
21
.
The actual value is found to be 4
.
7
·
10
21
, so the approximation is relatively good.
Apparently, the estimate made by Crichton is not quite accurate. However it can be shown that
it takes less than 2 days to produce a number far in excess of the desired size. (The exact number
of generations is left as an exercise for the reader.
. but we will return to this in due time.)
8.4
The function
2
x
From previous familiarity with power functions such as
y
=
x
2
(not to be confused with 2
x
), we
know the value of
2
1
/
2
=
√
2
≈
1
.
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This note was uploaded on 05/19/2011 for the course MATH 102 Math 102 taught by Professor Allard during the Winter '09 term at The University of British Columbia.
 Winter '09
 Allard

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