Ms Word

1. Union and Intersection of Sets
1. Let A and B be sets. The union of sets A and B, denoted by AB, is the set that contains those
U
Relations and Functions
elements that are either in A or in B, or in both.
Venn diagram:
A B
U
A B
2. Let A and B be sets. The intersection of sets A and B, denoted by AB , is the set containing
those elements in both A and B.
Venn diagram:

2. Learning Area: Algebra
Title: Multiplication of Polynomials ( Polynomial by a Polynomial)
Rule in Multiplying Polynomial by a Polynomial:
Multiplication of two polynomials requires repeated application of the Distributive Property
Example 1. Multiply (4x  5)(x2  x  4)
Solution:
2 2 2 (4x  5)(x  x  4)  4x(4x  x  4)  5(x  x  4)
3 2 2  (4x  4x 16x)  (5x 5x  20)
3 2  4x  x 11x  20
Example 2.Multiply 2 (x  2)(x  x  3)
Solution:
2 2 2 (x  2)(x  x  3)  x(x  x  3)  2(x  x  3)
3 2 2  (x  2x  3x)  (2x  2x  6)
3 2  x 3x 5x 6
Reference: Merle S. Alferez & Ma. Cecilia A. Duro (2004) Elementary Algebra. Philippines : MSA
ACADEMIC ADVANCEMENT INSTITUTE

3. Learning Area: Trigonometry
Title of the Lesson:Degrees and Radians
Radian is the measure of a central angle of a circle whose rays subtend an arc on the circle
whose length is equal to the radius of the circle
Since the circumference of a circle is subtended by a central angle of 360 ,
it follows that
2 360
180
radian
radian




If  radian 180 is divided by  ,
180 radian 


180
1radian
 , then 1


 radian
180
To summarize the conversions:
1. To convert radians to degrees, multiply the given number of radians by
180

Example:
5
6

radians =
5 180
150
6
x



2. To convert degrees to radians multiply the given number of degrees by

180
Example: 50 = 50
180
x

=
5
18

radians
Reference:
Dilao, Soledad J. et. al. (2009) Advanced Algebra, Trigonometry and Statistics, Textbook
for Fourth Year. Philippines: SD Publications, Inc.,

4. Learning Area: Calculus
Title of the Lesson: Limit of a Constant and a Function f(x)
Definition. lim
limcf(x) = c lim
xk
lim f(x)
xk
Remember: The limit of the function x as x approaches to any constant is always equal to
the constant.
Illustrative Examples:
1.
lim
x
2
3x= 3
lim
x
2
x
= 3(-2)
= -6
2.
lim
x
1
2
6 x 
 x
6 lim
1
2
x
1
6( )
2

3
3.
lim
x
1
25
5x
 5 lim
x
1
25
x
1
5( )
25

5
=
()
25
=
1
5
Reference: Gladys Glo H. Marcelo (2009) Basic Calculus. Philippines: Rex Book Store, Inc.

5. Learning Area: Number Theory
Title of the Lesson: The Functions 휏 and 휎
Definition 6.1.Given a positive integer n , let 휏(n) denote the number of positive divisors
of n and 휎(n) denote the sum of these divisors
For example these notions , consider n=12. Since 12 has the positive divisors 1, 2, 3, 4, 6, 12
we find that
휏(12) = 6 and 휎(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28
For the first few integers;
휏(1) = 1휏(2) = 2휏(3) = 2휏(4) = 3휏(5) = 2휏(6) = 4
and
휎(1) = 1 휎(2) = 3 휎(3) = 4 휎(4) = 7 휎(5) = 6 휎(6) = 12
It is not difficult to see that 휏(푛) = 2 if and only n is a prime number, also 휎(푛) = 푛 + 1
if and only if n is a prime.
Reference:
Ymas, Sergio Jr. E. (2004). Elementary Number Theory. Philippines: Ymas Publishing House.

6. Learning Area: Linear Algebra
Title of the Lesson: Properties of Determinants
The determinant has many properties. Some basic properties of determinants are:
1. whereIn is the n × n identity matrix.
2.
3.
4. For square matrices A and B of equal size,
5. for an n × n matrix.
6. If A is a triangular matrix, i.e. ai,j = 0 whenever i>j or, alternatively, whenever i<j, then its
determinant equals the product of the diagonal entries:

7. Angle and Its Parts
An angle is the figure formed by two rays or line segments, called the sides of the angle,
sharing a common endpoint, called the vertex of the angle.
Consider the figure below:
The vertex of ∠퐴퐷퐶 is point D and its sides are ⃗퐷⃗⃗⃗퐴⃗ and ⃗퐷⃗⃗⃗퐶⃗ .
∠퐴퐷퐵 and ∠퐵퐷퐶 are called adjacent angles and their common side is ⃗퐷⃗⃗⃗⃗퐵⃗ .
Reference: http://en.wikipedia.org/wiki/Angle
A
B
C
D
D
S