Applied Calculus: An introduction about Real Numbers and Real Line

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Applied Calculus
1st
Semester
Lecture -01
Muhammad Rafiq
Assistant Professor
University of Central Punjab
Lahore Pakistan

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Preliminaries
REAL NUMBERS AND REAL LINE:
The numbers which can be represented on real line are called
real numbers.
Set of real numbers is denoted by “ ”
NOTE:
Set of Real numbers is union of Rational and Irrational numbers.

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ORDERED PROPERTIES OF REAL NUMBERS OR
PROPERTIES OF INEQUILITIES.
NOTE: If a , b , c
1. a
2. a
3. a
4. a
5. a
6. If a and b are both positive or both negative then
a
NOTE: Ordered properties hold only for real numbers and do
not hold for complex numbers.

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Intervals
Finite Interval:
Notation Set Graph Name
(a ,b) {x/a<x<b} Open interval
[a ,b] {x/a≤x≤b} Closed interval
(a , b] {x/a<x≤b} Open Closed
interval
[a , b) { Closed Open
interval

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Infinite Intervals:
Notation Set Graph
(a ,∞) {x / > }
a ∞
[a ,∞)
a ∞
(-∞, b) {x/x<b}
-∞ b
(-∞ , b] {x/x≤b}
-∞ b
(-∞ , ∞) -∞ +∞

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Example 1:
Solve the following inequalities and graph the solution on
number line.
(a) 2x-1
Solution:
2x-x
x
Solution = (-∞, 4) -∞ 4

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(b)
6
6+5
x
Solution: (1, ]
0
Absolute value of real number:The absolute value of a real
number ‘x’ is denoted by and is denoted as

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=
If
Then x =
Note: =
Absolute value properties.
1.
2.

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3.
4.
Inequalities involving absolute values.
1.
2. -a
e.g
1.

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2.
(-∞, U
Solution:
(-∞,-2] [2, ∞)
Example 2:
Solve the inequality and graph the solution on number lines.
a)

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-1
-1+3 2x+3-3
2
1
Solution: [1, 2]
b)
2x - 3 2x - 3
2x 2x
x x

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(-∞,1] [2,∞)
Solution: (-∞, 1] [2,∞)
c)
-
Solution: (- )

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CARTESSION CO-ORDINATE SYSTEM:
For example:
1- P(4,5) x- coordinate (abscissa) y- coordinate(ordinate)
This point lies in I quadrant.
IV quad
III quad
I quadII quad

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2- Q(-4,5)
This point lies in II quadrant.
3- S(-4,-5)
This point lies in III quadrant.
4- R(4,-5)
This point lies in IV quadrant.
DISTANCE FORMULA: The distance between two points in
a plane is given by. IABI= ′ ′

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INCLINATION OF A LINE:
An angle made by the line
with positive direction of x- axis is called inclination of that
line.
SLOPE:
If is the inclination of a line “L” then its slope is given by
m =

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If a line of inclination 450
then, the slope is
m = 0
m = 1 0
=1)
When 0
or on x-axis then slope is zero.
When 0
or y-axis then slope is undefined.

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Physically slope=
m =
′
′
PARALLEL AND PERPENDECULAR LINES:
If two non-vertical lines L1 and L2 have slopes m1 and m2 resp.
then,
1- L1 ll L2 iff m1=m2
2- L1 L2 iff m1 m2=-1

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EQUATIONS OF LINES:
Lines (x and y axis)
Y-AXIS:

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X-AXIS:
POINT SLOPE FORM:
When one point and slope is given
then equation of line is

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SLOPE – INTERCEPT FORM:
When slop and y-intercept is given
EXERCISE
Q1-Solve the following inequalities and represent the
solution on number line
i.

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viii.
Q2-Find equations of following lines which
i. passes through (2,-3) with slope m =
ii. passes (3,4) and (-2,5)
iii. has slope m = and y-intercept = 6
iv. (-12,-9) and II to x- axis
v. (-12,-9) and II to y –axis

Applied Calculus: An introduction about Real Numbers and Real Line

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