Math Analysis I

Introduction to Calculus
Raymund T. de la Cruz
MAEd - Mathematics

Pure vs Applied Mathematics
Pure Mathematics - the branches of mathematics
that study and develop the principles of
mathematics for their own sake rather than for
their immediate usefulness.
Applied Mathematics - the branches of
mathematics that are involved in the study of
the physical or biological or sociological world

Pure vs Applied Mathematics
So what’s the difference?...
The difference between Pure Mathematics and
Applied Mathematics lies in application and use. Pure
Mathematics are studied with no consideration of
necessity or application; they develop the principles
of Mathematics for the sake of the principles of
Mathematics. Research in the Fibonacci Sequence is
an example of this: the Fibonacci Sequence has
almost no useful application to mankind. Applied
Mathematics, on the other hand, are studied purely
for the sake of application. Examples of this lie in
Economics, Computer Science, and Engineering.

Review of basic mathematics concept and definition
used in numerical analysis…
EXPONENTS
Simplify the following.
(a) 𝑥6 𝑥−3 (b) (𝑥−2 𝑦3)4 (c)
𝑥−1 𝑦2
𝑧−3 (d) (2𝑥2)−3 𝑦4 ÷ (𝑥−1 𝑦)2
ALGEBRAIC IDENTITIES & ALGEBRAIC EXPRESSIONS
(a) Expand 𝑥2 + 1 + 7 ( 𝑥2 + 1 − 7)
(b) Simplify
𝑥2−𝑥−6
𝑥2−6𝑥+9
SOLVING LINEAR EQUATIONS
(a) Solve 3𝑥 − 5 = 2 7 − 𝑥
(b) Solve 2 𝑥 + 4 = 7𝑥 + 2

SOLVING QUADRATIC EQUATIONS
Solve the following:
(a) 𝑥2 + 2𝑥 − 15 = 0 (b) 2𝑥2 − 9𝑥 + 10 = 0
SOLVING LINEAR INEQUALITIES
(a) 2𝑥 + 1 > 7(𝑥 + 3) (b) 3 𝑥 − 5 ≤ 7𝑥 + 1
SOLVING QUADRATIC INEQUALITIES
(a) 𝑥2 + 3𝑥 − 10 < 0 (b) 𝑥2 + 8 ≥ 6𝑥

SETS
A. Let A = {2, 3, 5}, B = {2, 5, 6, 8}, and C = {1, 2, 3}. Find the following sets.
(1) A ∩ B (2) A ∪ B (3) (A ∩ B) ∪ C
B. Let 𝑈 = 1, 2, 3, … , 12 and let
A = {𝑥 ∈ 𝑈: 𝑥 is a prime number}
B = {𝑥 ∈ 𝑈: 𝑥 is an even number}
C = {𝑥 ∈ 𝑈: 𝑥 is divisible by 3}.
(1) (A ∩ C) ∪ (B ∩ C) (2) (A ∪ B)’ (3) A’ ∩ B’

Graphical representation of different types of the
rectangular coordinate and polar coordinate system…
A RECTANGULAR COORDINATE SYSTEM consists of two perpendicular real number lines
called coordinate axes, that intersect at their origins. Generally one line is horizontal
and called the x-axis, and the other is vertical called the y-axis. The axes divide the
coordinate plane into four parts, called quadrants. Points on the axes are not in any
quadrant.
A ONE-TO-ONE CORRESPONDENCE exists between ordered pairs of numbers (𝑎, 𝑏) and
points in the coordinate plane. Thus,
1. To each point 𝑃 there corresponds an ordered pair of numbers (𝑎, 𝑏) called
coordinates of 𝑃, 𝑎 is called the abscissa; 𝑏 is called the ordinate.
2. To each ordered pair of numbers there corresponds a point, called the graph of the
ordered pair.

𝑃(𝑎, 𝑏)
THE DISTANCE BETWEEN TWO POINTS 𝑃1(𝑥1, 𝑦1) and 𝑃2(𝑥2, 𝑦2) in the plane coordinate
system is given by the distance formula:
𝑑 𝑃1, 𝑃2 = (𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2
Example: Find the distance between (3, -2) and (5, 3).
𝑑 = (3 + 2)2 + (5 − 3)2
𝑑 = 25 + 4
𝑑 = 29
𝑄𝐼V(+,-)𝑄𝐼II(-,-)
𝑄𝐼𝐼(-,+) 𝑄𝐼(+,+)
𝑥
𝑦

THE GRAPH OF AN EQUATION in two variables is the graph of its solution set, that is, of all
ordered pairs (𝑎, 𝑏) that satisfy the equation. Since there are ordinarily an infinite
number of solutions, a sketch of the graph is generally sufficient.
INTERCEPTS: The coordinate of the points where the graph of an equation crosses the
x-axis or y-axis. To find the x-intercept, set y = 0 and solve for x and to find the
y-intercept, set x = 0 and solve for y.
Example: Find the intercepts of the graph of the equation 𝑦 = 4 − 𝑥2.
Set x = 0; then 𝑦 = 4 − 02 = 4. Hence the y-intercept is 4.
Set y = 0. If 0 = 4 − 𝑥2, then 𝑥2 = 4; thus 𝑥 = ±2. Hence 2 & -2 are the x-intercepts.
THE MIDPOINT OF A LINE SEGMENT with endpoints 𝑃1(𝑥1, 𝑦1) and 𝑃2(𝑥2, 𝑦2) is given by
the midpoint formula.
Midpoint of 𝑃1 𝑃2 = (
𝑥1+𝑥2
2
,
𝑦1+𝑦2
2
)
Example: Find the midpoint between (6, -3) and (4, -5).
Midpoint =
6+4
2
,
−3+ −5
2
= (5, −4)

A point P in the Cartesian plane can be described, apart from using the known
coordinates (x, y), by polar coordinates (r, θ ), which are defined as follows. Denote
by r the distance of P from the origin O. If r > 0 we let θ be the angle, measured in
radians up to multiples of 2π, between the positive x-axis and the half-line
emanating from O and passing through P.
𝑥
𝑦
𝑂
𝑃 = (𝑥, 𝑦)
𝑟
𝜃
It is common to choose θ in (−π, π], or in [0, 2π). When r = 0, P coincides with the
origin, and θ may be any number.
The passage from polar coordinates (r, θ) to Cartesian coordinates (x, y) is given by
x = r cos θ , y = r sin θ .

Example:
1. Let 𝑃 have Cartesian coordinates (𝑥, 𝑦) = 6 2, 2 6 .
𝑟 = 72 + 24 = 96 = 4 6
As 𝑥 > 0, 𝜃 = 𝑎𝑟𝑐𝑡𝑎𝑛
2 6
6 2
= 𝑎𝑟𝑐𝑡𝑎𝑛
3
3
=
𝜋
6
.
The polar coordinates of 𝑃 are then 𝑟, 𝜃 = 4 6,
𝜋
6
.
2. Take 𝑃 of polar coordinates 𝑟, 𝜃 = 4,
2
3
𝜋 this time; in the Cartesian system
𝑥 = 4𝑐𝑜𝑠
2
3
𝜋 = 4 cos 𝜋 −
𝜋
3
= −4𝑐𝑜𝑠
𝜋
3
= −2.
y = 4𝑠𝑖𝑛
2
3
𝜋 = 4 sin 𝜋 −
𝜋
3
= 4𝑠𝑖𝑛
𝜋
3
= 2 3.

EXERCISES:
A. Find the distance between the following pair of points in the plane coordinate system.
1. (-3, 4) and (-5, -1) 2. (6, 2) and (-2, 3) 3. (1, -5) and (-2, 2)
B. Determine the intercepts of the given equations.
1. 2𝑥 + 3𝑦 = −4 2. 𝑦 − 5 = 3𝑥 3. 𝑦 = 𝑥2 − 36
C. Determine the midpoint from the given endpoints of a line.
1. (-4, 7) and (6, -3) 2. (1, 11) and (-3, -3) 3. (
3
2
, 4) and
4
3
,
1
2
D. Find the polar coordinates of the following points in the plane:
1. (5 6, 5 2) 2. (−5 6, 5 2)
E. Find the rectangular coordinates from the given polar coordinates:
1. (2,
𝜋
2
) 2. ( 20,
𝜋
4
)

Math Analysis I

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Math Analysis I