MATH1151 Student Support
Calculus Notes
•
Notation

N
=
{
0
,
1
,
2
, . . .
}
is the set of natural numbers.

Z
=
{
. . . ,

2
,

1
,
0
,
1
,
2
, . . .
}
is the set of integers.

Q
=
n
p
q
:
p, q
∈
Z
, q
6
= 0
o
is the set of rational numbers.

p
∈
Z
means
p
is an element of
Z
. i.e.
p
is an integer.

R
is the set of real numbers.

∀
means ‘for all’

∃
means ‘there exists’

x
∈
[0
,
∞
) means 0
≤
x <
∞
, whereas
x
∈
(5
,
26] means 5
< x
≤
26.

a
⇒
b
means
a
implies
b
.

a
⇔
b
means
a
iff (if and only if)
b
. i.e.
a
⇒
b
AND
b
⇒
a
The Exponential Function
•
The Least Upper Bound Axiom
 Every nonempty set of real numbers that has an upper bound has a LEAST upper bound.
•
The Exponential Function

lim
n
→∞
1 +
x
n
n
=
e
x
for any
x
∈
R
.

f
:
N
{z}
domain
→
R
{z}
codomain
If you input a value from the domain, you get back a value from the codomain.
In this example, if you input a natural number (the
x
value), you get back a real number (the
y
value).
 onetoone means
f
(
x
) =
f
(
y
) if and only if
x
=
y
.
 onto means that if
y
∈
codomain then
∃
x
∈
domain such that
f
(
x
) =
y
.
e.g. Consider
g
:
R
→
R
, defined by
g
(
x
) =
e
x
.
This function is not onto, because we can take a
y
value from the codomain (
R
), say
y
=

2 for example, and
there is no
x
value from the domain (
R
) that satisfies
e
x
=

2.
In contrast, if we have
h
:
R
→
R
+
, defined by
h
(
x
) =
e
x
, this function is onto, because for any
y
value from
the codomain (
R
+
), we can find a corresponding
x
value from the domain (
R
) that satifies
e
x
=
y
.
 If a function is onetoone and onto, it is invertible.
•
Hyperbolic and Inverse Hyperbolic Functions
 cosh
x
=
e
x
+
e

x
2
and sinh
x
=
e
x

e

x
2
 cosh is an even function, while sinh and tanh are odd functions.
 tanh
x
=
sinh
x
cosh
x
,
coth
x
=
cosh
x
sinh
x
,
sech
x
=
1
cosh
x
,
cosech
x
=
1
sinh
x
 Osborne’s Rule
Replace trig functions with hyperbolic functions, and change the sign of a product of sines or implied product
of sines (e.g. tan
2
x
). e.g. cos
2
x
+ sin
2
x
= 1 becomes cosh
2
x

sinh
2
x
= 1.

y
= sinh

1
x
means
x
= sinh
y
for any
x
∈
R

y
= cosh

1
x
means
x
= cosh
y
for
x
≥
1 and
y
≥
0.

y
= sinh

1
x
= ln(
x
+
√
x
2
+ 1)
∀
x
∈
R

y
= cosh

1
x
= ln(
x
+
√
x
2

1)
∀
x
≥
1

y
= tanh

1
x
=
1
2
ln
1 +
x
1

x
∀
x
∈
(

1
,
1)
Version 3, 5/6/08
2