FUNDAMENTAL COUNTING
PRINCIPLE
MATH10-4
2nd quarter
SY 2015-2016
TREE DIAGRAM
This technique is a visual form of a counting
technique.
Example:
How many 3-digit number codes can be formed
using the digits 1, 2, 5 and 9 if no digits are
repeated and the fi

Mathematical
Induction
Mathematical Induction
Mathematical induction is a method of
mathematical proof which is a technique for
proving a statement: a theorem, axioms, or
a formula that is asserted about every
natural number. It is done by proving that
th

RELATIONS, FUNCTIONS AND
GRAPHS
WEEK 1 - Relations, Functions
and Graphs
(COLLEGE ALGEBRA AND TRIGONOMETRY,
Aufmann, Barker and Nation 7th ed., page
175-178, 184-189, 252-259)
1
OBJECTIVES
At the end of the lesson, the student is expected to be able to:

COUNTING TECHNIQUES
TWO WAYS OF COUNTING
LIST METHOD
CAN BE DONE BY ACTUALLY LISTING DOWN ALL THE POSSIBLE
OUTCOMES IN A GIVEN SITUATION
USE OF MATHEMATICAL FORMULAS
A SHORTER WAY OF FINDING ALL THE POSSIBLE OUTCOMES IN A
GIVEN SITUATION IS BY THE USE O

CIRCLES
MATH21-1
3rd quarter
SY 2015-2016
CIRCLE
A circle is a locus of points that moves in a plane at
a constant distance from a fixed point. The fixed point
is called the center and the distance from the center
to any point on the circle is called the

LOCUS OF A MOVING POINT
MATH21-1
3rd Quarter
SY 2015-2016
Locus of a Point
In any real-life problem which can be translated in to
a mathematical model, there are always two
important quantities involved; constants and
variables.
A constant is a quantity w

CONIC SECTION
MATH21-1
3rd quarter
SY2015-2016
Conic Section or a Conic is a locus of point that moves in a
plane so that its distance from a fixed point is in constant
positive ratio to its distance from a fixed line.
Focus is the fixed point
Directrix i

NORMAL FORM OF A LINE
MATH21-1
3rd quarter
SY 2015-2016
Since p is perpendicular to line l, then their slopes are
negative reciprocals of each other. Thus,
y
.
ml
cos
sin
Substitution ofP( p cos , p sin ) and
l
.
p
o
y0
y
p sin
x0
x
p cos
and
ml
cos

ELLIPSE
MATH21-1
3rd quarter
SY 2015-2016
Definition
The following terms are important in drawing the graph of an
ellipse:
Eccentricity measure the degree of flatness of an ellipse. The
eccentricity of an ellipse should be less than 1.
Focal chord is any

POLAR CURVES
MATH21-1
3rd quarter
SY 2015-2016
THE POLAR COORDINATE SYSTEM
THE POLAR COORDINATE SYSTEM
Examples: Plot the given points on a polar coordinate system.
a. P (3, 45)
b. P (3, 60)
c. P (3, 220)
d. P (-3, 75)
TRANSFORMATIONS BETWEEN POLAR AND
RE

MATH21-1
CALCULUS 1
3rd quarter SY 2015-2016
MATHEMATICS
A collective term for the branches of
learning that have grown out of the
ancient disciplines of geometry and
arithmetic.
CALCULUS
That branch of analysis which investigates
the infinitesimal change

MAPA INSTITUTE OF TECHNOLOGY - CIVIC WELFARE TRAINING SERVICE
MAPA INSTITUTE OF TECHNOLOGY
NATIONAL SERVICE TRAINING PROGRAM
CIVIC WELFARE TRAINING SERVICE
[MAPA-CWTS]
PROGRAM MODULE 2
MAPA INSTITUTE OF TECHNOLOGY - CIVIC WELFARE TRAINING SERVICE
MAPA INS

The Fundamental Counting Principle
YouaretravelingthroughEuropeforthesummeranddecidethebestpackingoptionistoselect
separatesthatcanbemixedandmatched.Youpackonepairofshortsandonepairofkhakipants.
YoupackapairofTevasportsandalsandapairofsneakers.Youhavethre

SEQUENCES AND SERIES
MATH10-4
2nd quarter
SY2015-2016
SEQUENCE FUNCTION
In Mathematics, sequences of numbers such as 1, 2, 3, , 998, 999, 1000 and 1,
1 1 1 1
, , , , are always encountered. These sequences are formed by special
2 3 4 5
functions called se

QUADRATIC FUNCTION
MATH10-4
2nd Quarter
SY 2015-2016
The graph of = 2 + + is a parabola. The graph
opens up if > 0, and it opens down if < 0. The vertex of a
parabola is the lowest point of the parabola that opens up or the
highest point on the parabola t

SYSTEMS OF EQUATIONS
MATH10-4
2ND QUARTER
SY 2015-2016
LINEAR SYSTEMS OF EQUATIONS
A system of equations is a set of more than one
equations.
The following are systems of equations:
CLASSIFICATIONS OF SYSTEMS OF
EQUATIONS
A. Linear system of equations. Th

3/12/2013
A
1
sequence is an ordered list of
numbers.
The sum of the terms of a sequence is
called a series.
While some sequences are simply
random values, most sequences have a
definite pattern that is used to arrive at
the sequence's terms.
Two such

WEEK 4 - Matrices and Determinants
Systems of Linear Equations in Two Variables
Systems of Linear Equations in Three Variables
WEEK 5 Nonlinear Systems of equations
Applications of Systems of Equations
Partial Fractions
WEEK 6 Inequalities in Two Variable

MAPA INSTITUTE OF TECHNOLOGY
Department of Mathematics
a.
b.
c.
VISION
Mapua shall be among the best universities in the world.
MISSION
The Institute shall provide a learning environment in order for its students to acquire the attributes
that will make t

MATHEMATICAL INDUCTION
MATH10-4
2nd quarter
SY 2015-2016
A mathematical proof used in proving that each
statement in an infinite sequence of statements is true
is called a Mathematical Induction.
Consider the following equalities:
1 = 1 = 12
2 = 1 + 3 = 2

COMBINATION
MATH10-4
2nd quarter
SY 2015-2016
So far we have been concerned with arrangements of
elements with a definite order. Consider the set , , .
Let us arrange the letters of this set without regard to the
relative order of the elements. In the pre

THE FACTORIAL OF A NUMBER
AND THE BINOMIAL THEOREM
MATH10-4
2nd quarter
SY 2015-2016
THE FACTORIAL OF A NUMBER
In different counting techniques, we often
encounter products such as 12 11 10 9 8
7 6 5 4 3 2 1 and 1 2 3 4 5 6 .
Mathematicians provided a no

ALGEBRAIC CURVES
MATH10-4
2nd Quarter
SY 2015-2016
ALGEBRAIC CURVES
An equation involving the variables x and y is
satisfied by an infinite number of values of x and
y, and each pair of values corresponds to a point.
When plotted on the Cartesian plane, t

PARTIAL FRACTIONS
MATH10-4
2nd quarter
SY 2015-2016
Given a rational expression of the form
()
()
and
assume it as a proper fraction, that is, the degree
of the numerator N(x) is less than the degree of
the denominator D(x). This rational expression
can b

MATRICES and DETERMINANTS
MATH10-4
2nd quarter
SY 2015-2016
MATRIX
A matrix is a rectangular array of numbers
written in rows and columns.
WAYS OF REPRESENTING MATRICES
The following are matrices with the
indicated orders.
TYPES OF MATRICES
SQUARE MATRIX

PERMUTATION
MATH10-4
2nd quarter
SY 2015-2016
Consider the set , , . Arranging the elements of this set
with a definite order, we get
abc acb bac bca cab cba
Here, we have six distinct arrangements. We call this kind of
arrangements of objects or elements

BINOMIAL THEOREM
OBJECTIVES:
OBJECTIVES
Recognize patterns in binomial expansions.
Evaluate a binomial coefficient.
Expand a binomial raised to a power.
Find a particular term in a binomial
expansion.
Definition:
Definition: BINOMIAL
BINOMIALTHEOREM
THEOR

Lesson 10
Sequences and Series
SPECIFIC OBJECTIVES:
At the end of the lesson, the student I expected to be able to :
Define sequences and identify the different kinds of sequences.
Find the nth term or the general term of a sequence for which
some initi

Sequences and Series
OBJECTIVES
Define sequences and identify the different kinds of
sequences.
Find the nth term or the general term of a sequence for
which some initial terms are given.
Find the common difference of an arithmetic sequence.
Find the comm

RELATIONS, FUNCTIONS AND
GRAPHS
WEEK 1 - Relations, Functions
and Graphs
(COLLEGE ALGEBRA AND TRIGONOMETRY,
Aufmann, Barker and Nation 7th ed., page
175-178, 184-189, 252-259)
1
OBJECTIVES
At the end of the lesson, the student is expected to be able to: