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DIFFERENTIAL CALCULUS
Study Notes 2

“THE RECTANGULAR COORDINATE
SYSTEM”

Prepared By: Tutor Win

THE RECTANGULAR COORDINATE SYSTEM

07/28/2021

In the last StudyNote 1, we have learned the importance of linear
coordinate system, absolute values, and inequality
...

Rectangular coordinate system provides an overview of how we calculate
the distance between two points, the slope, midpoints, and other geometric
properties that are essential in learning Calculus as well
...


X AND Y AXIS
In a rectangular coordinate system, we could see the two lines
intersecting at each other
...
See the figure below
...

The x-axis has values from left to right while the y-axis has values from
bottom to top
...
Negative side axes
also apply
...
It is always located on the x-axis
...
It is always located on the y-axis
...
And every point in the RCS has
always a unique x- and y- coordinates
...
Point (3, 6)
y

7
(3,6)
6
5
4
3
2
1
x
-3

-2

-1

1

2

3

4

5

6

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THE RECTANGULAR COORDINATE SYSTEM

07/28/2021

In order to plot point (3,6), we will just first locate the x-coordinate
which is 3
...
Next is to locate 6 in
the y-axis
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Make an imaginary line to
intersect the two coordinates
...


Example 2
...
That is, 2 units to the left from the origin
...
Next is to locate 4
in the y-axis
...
Make an imaginary line to
intersect the two coordinates
...


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THE RECTANGULAR COORDINATE SYSTEM

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Example 3
...
That is, 1 units to the left from the origin
...
Next is to locate
-5 in the y-axis
...
Make an imaginary line to
intersect the two coordinates
...


QUADRANTS
Quadrants are the four sections formed in a rectangular coordinate
system through the intersection of the x- and y- axis
...


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THE RECTANGULAR COORDINATE SYSTEM

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y

4
3

Quadrant 1

Quadrant 2
2
1

x
-3

-2

-1

-1

1

2

3

4

5

-2
-3

Quadrant 3

-4

Quadrant 4

-5

Points located at Quadrant 1 have coordinates (a, b)
Points located at Quadrant 2 have coordinates (-a, b)
Points located at Quadrant 3 have coordinates (-a, -b)
Points located at Quadrant 4 have coordinates (a, -b)

DISTANCE FORMULA
It is also possible for us to get the distance between two points using
this formula
...
Find the distance between the points A(3,6) and B(8,18)
...
𝐴𝐵 = (𝑥 − 𝑥 ) + (𝑦 − 𝑦 )
Step 2
...
𝐴𝐵 = (−5) + (−12)
Step 4
...
𝐴𝐵 = √25 + 144
Step 6
...
𝐴𝐵 = 𝟏𝟑 𝒖𝒏𝒊𝒕𝒔

Example 4
...

Step 1
...
𝐶𝐷 = (2 − 6) + (5 − 3)
Step 3
...
𝐶𝐷 = √16 + 4
Step 5
...
𝐶𝐷 = √4 𝑥 5
Step 7
...
47 units

MIDPOINT FORMULA
Suppose we will connect 2 points on the rectangular coordinate
system
...
We are asked to determine the coordinates of the midpoint of
this segment
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Find the midpoint of the segment connecting (3,10) and (5,4)
Step 1
...

 x=
 x=
 x=
 x=4
Step 2
...

 y=
 y=
 y=
 y=7
Step 3
...
Find the midpoint of the segment connecting (-4,2) and (2,5)
Step 1
...

 x=
 x=
 x = -1
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THE RECTANGULAR COORDINATE SYSTEM

07/28/2021

Step 2
...

 y=
 y=
 y=
 y = 3
...
The midpoint of the segment has coordinates (-1, 3
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A
...

2
...

4
...


Identify the following
...

It is the horizontal axis in the rectangular coordinates system
...
Identify the following points to what quadrant they belong
...
(9,2)
7
...
(-4,-3)
9
...

(10, 11)

C
...

11
...

C(3,4) and D(6,8)
13
...
Find the midpoint of the following
...

(3, -2) and (8, 5)
15
...
SOLUTIONS

A
...

2
...

4
...


y-axis
x-axis
ordinate
abscissa
4

B
...
Quadrant 1
7
...
Quadrant 3
9
...
Quadrant 1

C
...


Step 1
...
𝐴𝐵 = (4 − 4) + (5 − 7)
Step 3
...
𝐴𝐵 = √0 + 4
Step 5
...
𝐴𝐵 = 𝟐 𝒖𝒏𝒊𝒕𝒔

12
...
𝐶𝐷 = (𝑥 − 𝑥 ) + (𝑦 − 𝑦 )

Step 2
...
𝐶𝐷 = (−3) + (−4)
Step 4
...
𝐶𝐷 = √25
Step 6
...


07/28/2021

Step 1
...
𝐸𝐹 = (3 − 3) + (6 − (−1))
Step 3
...
𝐸𝐹 = √0 + 7
Step 5
...
𝐸𝐹 = 𝟕 𝒖𝒏𝒊𝒕𝒔
D
...


Step 1
...

 x=
 x=
 x=
 x = 5
...
Find the y coordinate
...
5
Step 3
...
5, 1
...


Step 1
...

 x=
 x=

√

Step 2
...

 y=
11 | P a g e

THE RECTANGULAR COORDINATE SYSTEM

07/28/2021

 y=
 y=
 y=3
Step 3

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