RELATIONS, FUNCTIONS AND
GRAPHS
OBJECTIVES
At the end of the lesson, the student is expected to be able to:
distinguish functions and relations
identify domain and range of a function/relation
evaluate functions/relations
perform operation on functions/re

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

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

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

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:

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

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

FINAL EXAMINATION
MATH10 4 ADVANCED ALGEBRA
4th QUARTER SY 2014-2015
COURSE OUTCOME 1: (Total points: 40 pts.)
I. ALTERNATIVE RESPONSE: Write TRUE if the statement is correct otherwise write FALSE (2pts. @)
1. The domain of the function h ( x )= x5 is [ 5

Quadratic Functions
Even and Odd Functions
The term even is used to describe functions that are symmetric
with respect to the y axis , or vertical axis.
The term odd is used to describe functions that are symmetric
with respect to the origin.
Function
Sym

Series and Sequences
ENGR. JHA IMPERIO
1
Series and Sequences
A
sequence is a function whose
domain is a set of positive integers.
The function values, or terms of the
sequence are written as:
2
Series and Sequences
A finite sequence has a domain cfw_

Series and Sequences
ENGR. JHA IMPERIO
1
Series and Sequences
A sequence is a set of numbers with
definite order of arrangement and
formed according to a definite rule.
The numbers in a given sequence are
called terms.
2
Arithmetic Progression
An arit

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

IMPROPER INTEGRALS
OBJECTIVES
At the end of the lesson, the student is expected
to:
evaluate an improper integral that has an
infinite limit of integration.
evaluate an improper integral that has an
infinite discontinuity.
DEFINITION OF IMPROPER INTEGRA

DEFINITE INTEGRALS
OBJECTIVES
At the end of the lesson, the students are
expected to:
define and interpret definite integral.
identify and distinguish the different
properties of the definite integrals.
evaluate definite integrals.
understand and use

TOPIC
LENGTH OF AN ARC
and
AREA OF SURFACE OF
REVOLUTION
Lengths of curves
In the limit as the arc length ds approaches zero:
b
2
dy
and so: ds
ds
1 s
dx
dx
dx
dx
xa
b
xa
2
dy
1 dx
dx
Lengths of curves parametric equations
Instead of changing th

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

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

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

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

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

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

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

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

ADVANCED ALGEBRA
COUNTING METHODS
MATH 10-4: ADVANCED ALGEBRA
COUNTING METHODS
is also referred to as COMBINATORICS.
basically concerned with problems of
counting as to how many ways an event
can be done.
MATH 10-4: ADVANCED ALGEBRA
COUNTING METHODS
Mul

PLANE ANALYTIC GEOMETRY
ALGEBRAIC CURVES SKETCHING
MATH 14: PLANE ANALYTIC GEOMETRY
ALGEBRAIC CURVE SKETCHING
An algebraic equation,
simply put, is just an
equation involving algebraic expressions.
Each equation often times satisfy an infinite
number o

COLLEGE ALGEBRA
SYSTEMS of EQUATIONS AND
INEQUALITIES
MATH 10: COLLEGE ALGEBRA
SYSTEM of EQUATIONS AND INEQUALITIES
system of linear equations
2 x 2 and 3 x 3
system of nonlinear equations
system of inequalities
application problems
MATH 10: COLLEGE A

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

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