Lecture 3: The Exponential Function
3.1 The exponential function
Definition The exponential function, with value at a real number x denoted by exp(x),
is the inverse of the exponential function.
Some immediate consequences of the definition are:
1. y = exp(x) if and only if log(y) = x
2. exp(log(x)) = x and log(exp(x)) = x
3. exp(0) = 1
4. exp(1) = e
5. The function f (x) = exp(x) is differentiable and increasing on the interval (−∞, ∞).
The range of f is (0, ∞). Moreover,
lim exp(x) = ∞
x→∞
and
lim exp(x) = 0.
x→−∞
The graph of f is as shown below.
20
15
10
5
-3
-2
1
-1
2
3
Graph of f (x) = exp(x)
6. Since
log(exp(x) exp(y)) = log(exp(x)) + log(exp(y)) = x + y,
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Lecture 3: The Exponential Function
it follows that
exp(x + y) = exp(x) exp(y).
7. Since
log
exp(x)
exp(y)
= log(exp(x)) − log(exp(y)) = x − y,
it follows that
exp(x − y) =
exp(x)
.
exp(y)
As a particular case,
exp(−x) = exp(0 − x) =
1
exp(0)
=
.
exp(x)
exp(x)
8. If x is a real number and r is a rational number, then
log(exp(x)r ) = r log(exp(x)) = rx.
Hence
exp(x)r = exp(rx).
9. Since log(exp(x)) = x for all x, it follows that
d
d
log(exp(x)) =
x.
dx
dx
Thus
1
d
exp(x) = 1,
exp(x) dx
that is,
d
exp(x) = exp(x).
dx
Example
If f (x) = x2 exp(x2 ), then
f 0 (x) = x2 exp(x2 )(2x) + 2x exp(x2 ) = 2x(x2 + 1) exp(x2 ).
3.2 The exponential function and e
If r is a rational number, then
log(er ) = r log(e) = r.
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Lecture 3: The Exponential Function
3-3
Hence
exp(r) = er .
We now make this the basis for defining irrational exponents when the base is e.
Definition
If x is an irrational number, we define
ex = exp(x).
With this definition, we have the following proposition.
Proposition
For any real number x, exp(x) = ex .
We may now restate our properties of the exponential function:
1. y = ex if and only if log(y) = x
2. elog(x) = x and log(ex ) = x
3. e0 = 1
4. e1 = e
5. The function f (x) = ex is differentiable and increasing on the interval (−∞, ∞) with
range (0, ∞). Moreover,
lim ex = ∞
x→∞
and
lim ex = 0.
x→−∞
The graph of f is as shown above.
6. ex+y = ex ey
7. ex−y =
ex
ey
8. For any real number x and rational number r,
r
(ex ) = erx .
9.
d x
e = ex
dx
Example
If f (x) = esin(3x) , then
f 0 (x) = 3 cos(3x)esin(3x) .
Lecture 3: The Exponential Function
Example
We have
ex − e−x
1 − e−2x
1+0
=
lim
=
= 1.
x→∞ ex + e−x
x→∞ 1 + e−2x
1+0
lim
3.3 Integrals
The differentiation formula
d x
e = ex
dx
yields the integration formula
Z
Example
We have
Z
Example
ex dx = ex + c.
e4x dx =
1 4x
e + c.
4
Using the substitution u = −x2 , we have
Z
Z
2
1
1
1
−x2
xe
dx = −
eu du = − eu + c = − e−x + c.
2
2
2
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