6
MATH CALCULUS 1 2024
A
B
7
} }
A
| − ≤
}
D
| − }
G
∪
≥
(
Q
=
θ | sin θ
=
1
2
}
Similarly, the
intersection of two sets
and
is the set of numbers which belong to both sets. This
notation is used:
A ∩ B
=
x
|
x belongs to both
A
and
B
.
}
2.
Exercises
1. What is the 2007th digit after the period in the expan-
sion of 1 ?
2. Which of the following fractions have finite decimal
expansions?
a
=
2 ,
b
=
3 ,
c
=
276937 .
4. Suppose A and B are intervals. Is it always true that
A ∩ B is an interval? How about A ∪ B?
5. Consider the sets
M
= x
|
x >
0
and N
=
y
|
y
>
0 .
Are these sets the same?
3 25 15625 6.
Group Problem.
3. Draw the following sets of real numbers. Each of these
sets is the union of one or more intervals. Find those
intervals. Which of thee sets are finite?
=
x x
2
3x
+
2
0
B
=
x
|
x
2
−
3x
+
2
≥ 0}
Write the numbers
x = 0.3131313131 . . . , y = 0.273273273273 . . .
and z = 0.21541541541541541 . . .
as fractions (i.e. write them as
m
, specifying m and n.)
C
=
x
|
x
−
3x
>
3
=
x x
2
5
>
2x
E
=
t
|
t
2
−
3t
+
2
≤
0
}
n
(Hint: show that
100x
= x +
31
. A similar trick
works for
y
, but
z
is a little harder.)
7.
F
=
α
|
α
2
−
3α
+
2
≥
0
}
Group Problem.
Is the number whose decimal expansion after the
=
(0,
1)
(5,
7]
H
=
{
1
}
∪
{
2, 3
}
}∩
(0,
2
√
2)
period consists only of nines, i.e.
2
R
=
ϕ
|
cos ϕ
>
0
an integer?
3.
Functions
Wherein we meet the main characters of this semester
3.1.
Definition.
To specify a
function
f
you must
(1)
give a
rule
which tells you how to compute the value
f
(x) of the function for a given real number
x, and:
(2)
say for which real numbers x the rule may be applied.
The set of numbers for which a function is defined is called its
domain
. The set of all possible numbers
f
(x)
as x runs over the domain is called the
range
of the function. The rule must be
unambiguous:
the same
xmust always lead to the same
f
(x).
For instance, one can define a function f by putting f (x) =
√
x for all x
0. Here the rule defining f is
“take the square root of whatever number you’re given”, and the function
f
will accept all nonnegative real
numbers.
The rule which specifies a function can come in many different forms. Most often it is a formula, as in
the square root example of the previous paragraph. Sometimes you need a few formulas, as in
g(x)
=
2x for x
<
0
x
2
for x
≥
0
domain of g
=
all real numbers.
x = 0.99999999999999999 . . .
}
