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FIRST YEAR CALCULUS
W W L CHEN
c


W W L Chen, 1982, 2008
...

It is available free to all individuals, on the understanding that it is not to be used for financial gain,
and may be downloaded and/or photocopied, with or without permission from the author
...


Chapter 4
SOME SPECIAL FUNCTIONS

4
...
Exponential Functions
In this section, we construct a class of functions of the form fa : R → R, where for every x ∈ R,
fa (x) = ax
...

Let us state very carefully what we mean by ax
...
To do so, we must have ax+0 = ax a0
...


(1)

Also, it seems reasonable to write
an = a

...
{z

for every n ∈ N
...

Chapter 4 : Some Special Functions

(4)
page 1 of 8

c


First Year Calculus

W W L Chen, 1982, 2008

Note that (2)–(4) give ax for every x ∈ Q+ , the set of all positive rational numbers
...


(5)

Hence we have, by (1)–(5), defined ax for every x ∈ Q
...
Without giving all the details,
we claim that it is possible to define ax for all irrational numbers x so that the function fa (x) = ax is
continuous and differentiable everywhere in R
...

Now let us consider the derivative fa0 (x)
...

y→x y − x
h→0
h→0
h
h

fa0 (x) = lim
Let us write

ah − 1

...
Indeed, it can be shown that there exists a
unique e ∈ (2, 3) such that c(e) = 1
...

The results below are easy consequences of our discussion
...
The function f : R → R, defined for every x ∈ R by f (x) = ex , has the following
properties:
(a) f (x) > 0 for every x ∈ R, and f (0) = 1
...

(c) f (x) is differentiable, and f 0 (x) = f (x) for every x ∈ R
...

(e) f (x) → 0 as x → −∞
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4
...
The Exponential and Logarithmic Functions
It is easy to see that the function considered in Proposition 4A is one-to-one, in view of part (d)
...
However, this “mishap” can be corrected
easily by changing the codomain to R+ = f (R), the set of all positive real numbers
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PROPOSITION 4B
...

Definition
...

It now follows from Proposition 2C that the exponential function exp : R → R+ has an inverse
function
...
Hence
y = exp(x)
Chapter 4 : Some Special Functions

if and only if

x = log(y)
...

discussion
...

4C
...

(a) log(y) > 0 for every y > 1, log(y) < 0 for every positive y < 1, and log(1) = 0
...

!0
+
(c)
log(y)
is
differentiable,
and
log
(y)
=
1/y
for
every
∈R
R+
(c) log(y) is differentiable, and log (y) = 1/y for every yy ∈

...

(d)
log(y)
is
strictly
increasing
in
R
(d) log(y) is strictly increasing in R ; in other words, log(y1 ) < log(y2 ) whenever 0 < y1 < y2
...

0+
...

(f ) log(y) → +∞ as y → +∞
...

(c)
...

1/(dy/dx)
...

=
dy
dy
yy

2
2

−2x
4
...
1
...
2
...

Consider the
the function
function ff (x)
(x) =
= eexx2 −2x
Example

...
Also, f (x) → +∞
as x → +∞ and as x → −∞
...


Hence there is a stationary point at x = 1
...
Now f 00!! (x) = ((2x − 2)22 + 2)exx −2x
> 0 always
...
Furthermore, the slope of the tangent is always increasing
...
2
...
Consider the (even) function f (x) = log(x22 + 1)
...
Also, f (x) → +∞ as x → +∞ and as
x → −∞
...


Hence there is a stationary point at x = 0
...
Now

< 0 if x < −1,



2  = 0 if x = −1,
2 − 2x
f 00 (x) = 2
> 0 if −1 < x < 1,
(x + 1)2 


 = 0 if x = 1,
< 0 if x > 1
...
Also it has points of inflection at x = −1
and at x = 1
...


4
...
Derivatives of the Inverse Trigonometric Functions
The purpose of this section is to determine the derivatives of the inverse trigonometric functions by using
implicit differentiation and our knowledge on the derivatives of the trigonometric functions
...
The six inverse trigonometric functions are well
defined, provided that we restrict the values for x to suitable intervals of real numbers
...

Example 4
...
1
...
Differentiating with respect to x, we obtain
1 = cos y

dy
,
dx

so that

dy
1
1
1
=
=p
=√

...
Differentiating with respect to x, we obtain
1 = − sin y

dy
,
dx

so that

dy
1
1
1
=−
= −p
= −√

...
3
...
If y = tan−1 x, then x = tan y
...

2
2
dx
sec y
1 + x2
1 + tan y

If y = cot−1 x, then x = cot y
...

2
dx
csc y
1
+
x2
1 + cot y
page 4 of 8

c

!

First Year Calculus

2008
W W L Chen, 1982, 2006

Example 4
...
3
...
Differentiating with respect to x, we obtain
1 = tan y sec y

dy
,
dx

dy
1
1
1
= √ 2
=
=

...
Differentiating with respect to x, we obtain
1 = − cot y csc y

dy
,
dx

dy
1
1
1
=−
=−
=− √

...
4
...
These functions are all exponential in nature
...
4
...
Consider the function f (x) = ekx , where k ∈ R is fixed
...
Note also that
!
n > 0 if
f "0 (x)
< 0 if

k > 0,
k < 0
...

−1
...
What
happens in the case k = 0?
Example 4
...
2
...
Then f (x) = eα log x , and so
f "0 (x) =

α α log x
α
e
= f (x)
...

page 5 of 8

cc W
!
WW
W LL Chen,
Chen, 1982,
1982, 2008
2006


First Year
Year Calculus
Calculus
First

Below we
Below
we show
show the
the graphs
graphs in
in the
the cases
cases αα =
= 1
...
1, αα =
= 1
...
9, αα =
= 0
...
5 and
and αα =
= 0
...
1 where
where the
the functions
functions are
are
increasing
...


y = x1
...
9

y = x0
...
1

Below
Below we
we show
show the
the graphs
graphs in
in the
the cases
cases αα =
= −1
...
1, αα =
= −1
...
9, αα =
= −0
...
5 and
and αα =
= −0
...
1 where
where the
the functions
functions
are
are decreasing
...


y = x−1
...
9

y = x−0
...
1

What happens
happens in
in the
the case
case αα =
= 0?
0?
What
2

−(x−a)2 /b
Example 4
...
3
...
4
...
Consider
Consider the
the function
function ff(x)
(x) =
= ee−(x−a)
/b , where a, b ∈ R are fixed and b > 0
...
Then

2(x − a) −(x−a)2 /b
2(x − a)
f " (x) = −2(x − a) e−(x−a)
2
/b = −2(x − a) f (x)
...

b
b
b
b
It is easy to see that
It is easy to see that

Chapter 44 :: Some
Chapter
Some Special
Special Functions
Functions

n! > 0 if x < a,
"
(x) > 0 if x < a,
ff0 (x)
< 00 ifif xx >
> a
...

<
page 66 of
page
of 88

cc W
!

WW
WL
L Chen,
Chen, 1982,
1982, 2006
2008

First
First Year
Year Calculus
Calculus

Below
Below we
we show
show the
the graphs
graphs in
in the
the cases
cases aa =
= 0,
0, aa =
= 11 and
and aa =
= −1,
−1, with
with the
the same
same value
value of
of bb =
= 2
...

2
2 /2

y = e−x

y = e−(x−1)

/2

2
2
/2

y = e−(x+1)

/2

2
2
/2

/2

Note that the shape of the graph is independent of the choice of the paramater a which in fact determines
the horizontal positioning of the graph
...
Below we show the graphs in the cases b = 1, b = 0
...
1 and b = 0
...

2
2

2
2
/0
...
1

y = e−x

/0
...
01

y = e−x

/0
...
01

What happens if the parameter b is a very large positive number?
x
x

x
Example 4
...
4
...
Then
Then
x
bx
log a

x log
log b
b
ex
log a

b x log a
e x log b log
log a
a,
ff (x)
(x) =
= eeb log a =
= eee
,

so
so that
that
x log
log b
b
x

x
x

"
x log b e x log b log
log a
a = (log a)(log b)bx
xabbx = (log a)(log b)bx
xf (x)
...

0
...

1
...
1
and
b
=
0
...
1
and
a
=
0
...

show the graphs in the cases b = 1
...
9, with a = 1
...
9
...
11
...
91
...
10
...
90
...
Find a largest domain, the corresponding range and derivative of each of the following functions:
a) f (x) = 4 sin−1 (5x)
b) f (x) = 12 cos−1 (x2 )
c) f (x) = tan−1 (x2 − 1)
2
...
Find the inverse function of f (x) if it exists
...
Does the function y = x3 − 2 have an inverse function on the real line? Give your reasons
...

4

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