FIRST YEAR CALCULUS
W W L CHEN
c
W W L Chen, 1982, 2005.
This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990.
It is available free to all individuals, on the understanding that it is not to be used for financial gain,
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Chapter 6
SOME SPECIAL FUNCTIONS
6.1. Exponential Functions
In this section, we construct a class of functions of the form
f
a
:
R
→
R
, where for every
x
∈
R
,
f
a
(
x
) =
a
x
.
Here
a >
0 denotes a positive real constant.
Let us state very carefully what we mean by
a
x
. We would like to define
a
x
appropriately so that
a
x
+
y
=
a
x
a
y
for every
x, y
∈
R
. To do so, we must have
a
x
+0
=
a
x
a
0
. This forces us to write
(1)
a
0
= 1
.
Also, it seems reasonable to write
(2)
a
n
=
a . . . a
n
times
for every
n
∈
N
.
Next, it is clear that it is necessary to define, for every
p, q
∈
N
,
(3)
y
=
a
1
/q
>
0
if and only if
y
q
=
a,
and
(4)
a
p/q
= (
a
1
/q
)
p
.
Chapter 6 : Some Special Functions
page 1 of 4
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First Year Calculus
c
W W L Chen, 1982, 2005
Note that (2)–(4) give
a
x
for every
x
∈
Q
+
, the set of all positive rational numbers. Our definition is
now extended to cover the set all negative rational numbers
Q
−
by
(5)
a
x
=
1
a
−
x
for every
x
∈
Q
−
.
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 Spring '09
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 Math, Calculus, Derivative, W W L Chen, R+

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