Integration of Rational Functions by Partial Fractions

1.1 REAL NUMBERS AND THE REAL LINE
1. Executing long division, " 0.1, 0.2, 0.3, 0.8, 0.9
9 9 9 9 9
oe 2 oe 3 oe 8 oe 9 oe
2. Executing long division, " 0.09, 0.18, 0.27, 0.81, 0.99
11 11 11 11 11
oe 2 oe 3 oe 9 oe 11 oe
3. NT = necessarily true, NNT = Not necessarily true. Given: 2 < x < 6.
a) NNT. 5 is a counter example.
b) NT. 2 < x < 6 E 2
2 < x
2 < 6
2 E 0 < x
2 < 2.
c) NT. 2 < x < 6 E 2/2 < x/2 < 6/2 E 1 < x < 3.
d) NT. 2 < x < 6 E 1/2 > 1/x > 1/6 E 1/6 < 1/x < 1/2.
e) NT. 2 < x < 6 E 1/2 > 1/x > 1/6 E 1/6 < 1/x < 1/2 E 6(1/6) < 6(1/x) < 6(1/2) E 1 < 6/x < 3.
f) NT. 2 < x < 6 E x < 6 E (x
4) < 2 and 2 < x < 6 E x > 2 E
x <
2 E
x + 4 < 2 E
(x
4) < 2.
The pair of inequalities (x
4) < 2 and
(x
4) < 2 E | x
4 | < 2.
g) NT. 2 < x < 6 E
2 >
x >
6 E
6 <
x <
2. But
2 < 2. So
6 <
x <
2 < 2 or
6 <
x < 2.
h) NT. 2 < x < 6 E
1(2) >
1(x) <
1(6) E
6 <
x <
2
4. NT = necessarily true, NNT = Not necessarily true. Given:
1 < y
5 < 1.
a) NT.
1 < y
5 < 1 E
1 + 5 < y
5 + 5 < 1 + 5 E 4 < y < 6.
b) NNT. y = 5 is a counter example. (Actually, never true given that 4  y  6)
c) NT. From a),
1 < y
5 < 1, E 4 < y < 6 E y > 4.
d) NT. From a),
1 < y
5 < 1, E 4 < y < 6 E y < 6.
e) NT.
1 < y
5 < 1 E
1 + 1 < y
5 + 1 < 1 + 1 E 0 < y
4 < 2.
f) NT.
1 < y
5 < 1 E (1/2)(
1 + 5) < (1/2)(y
5 + 5) < (1/2)(1 + 5) E 2 < y/2 < 3.
g) NT. From a), 4 < y < 6 E 1/4 > 1/y > 1/6 E 1/6 < 1/y < 1/4.
h) NT.
1 < y
5 < 1 E y
5 >
1 E y > 4 E
y <
4 E
y + 5 < 1 E
(y
5) < 1.
Also,
1 < y
5 < 1 E y
5 < 1. The pair of inequalities
(y
5) < 1 and (y
5) < 1 E | y
5 | < 1.
5.
2x  4 E x 
2
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