chapter 1 level 0 math 0 faculty of engineering - basic science department- prof h n agiza

Chapter 1

Level 0 Math 0

Faculty of Engineering - Basic Science Department- Prof H N Agiza

The slope of a line

Faculty of Engineering - Basic Science Dept- Prof H N Agiza


• We define run to be the distance we move to the right and rise to be the corresponding distance that the line rises (or falls).

The Slope of a Line :


Finding the Slope of a Line Through Two PointsExample : Find the slope of the line that passes through the points P(2, 1) and Q(8, 5).

Solution :


Point-Slope Form of the Equation of a Line• Finding the equation of the line that passes through a given point and

P(, ) has slope m.


Finding the Equation of a Line with Given Pointand SlopeExample : Find an equation of the line through (1, -3) with slope .

Solution :


Slope-Intercept Form of the Equation of a Line


Lines in Slope-Intercept Form

Example : Find the equation of the line with slope 3 and y-intercept -2.

Solution:


Vertical and Horizontal Lines

If a line is horizontal, its slope is , so its equation is , where b is the y-intercept


Vertical and Horizontal Lines

• An equation for the vertical line through (3, 5) is x = 3.• An equation for the horizontal line through (8, 2) is y =2.


Parallel and Perpendicular Lines


Finding the Equation of a Line Parallel to a Given LineExample: Find an equation of the line through the point (5,2) that is parallel to the line

Solution:


PERPENDICULAR LINES


• Example: Show that the points P(3,3), Q(8,17), and R(11,5)are the vertices of a right triangle.


Mathematical Notations• Belongs to.• Does not belong to. The complex Numbers.• For all (Universal Quantifier). The open finite interval.

{X R : a < x < b} • Empty set. The closed interval. {X R : a x b} • P P implies Q. The semi-closed interval.

{X R : a x b} • P p if and only if Q. The semi-open interval.

{X R : axb} • Natural Numbers {0,1,2,3,….}. The infinite open interval.

{X R : x a} • The Integers {…,-3,-2,-1,0,1,2,3,} The infinite closed interval.• The Rational Numbers. {X R : x a} • R The Real Numbers.


1.1 Sets and Notation• Set : a collection of well defined members or elements.

• A subset: is a sub-collection of a set.

• Example : The sets• A = {x Z : } , B={x Z : } , C={-3,-2,-1,0,1,2,3}

Solution:• The first set is the set of all integers whose square lies

between 1 and 9 inclusive, which is precisely the second set, which again is the third set.


• The union of two sets A and B, is the set This is read “A union B.”

• The intersection of two sets A and B, is This is read “A intersection B.”

• The difference of two sets A and B, is This is read “A set minus B.”


Example : If S ={1, 2, 3, 4, 5} , T={4, 5, 6, 7}, and V = {6, 7, 8}, find the sets S T, S T, and S V.

S T = {1, 2, 3, 4, 5,6,7}

S T = {4 , 5}

S v = Φ

Example :Let A={1,2,3,4,5} , B={1,3,5,7,9} .Find the sets A B

A B = {1,2,3,4,5,6,7,9}

A B = {1, 3, 5}

A B = {2, 4, 6} B A = {7, 9}


1.2 Intervals An interval is a subset of the real numbers.


Example : Graphing Intervals

Express each interval in terms of inequalities, and then graph the interval .

(a)

(b)[1.5, 4]=

(C)


• Example :Find the Intersection of this Interval, Graph each set. • The intersection of two intervals consists of the numbers that are in both intervals.

Therefore


Absolute Value• The absolute value of a number a, denoted by , is the

distance from a to 0 on the real number line.• Distance is always positive or zero, so we have

• Definition of Absolute value If a is a real number , then the absolute value of a is

Example: Evaluating Absolute Values of Numbers


Distance between points • If a and b are real numbers, then the distance between the points a and

b on the real line is

• Note that

• Example : Distance Between Points The distance between the numbers 8 and 2 is


Integer Exponents• A product of identical numbers is usually written in exponential notation.

• Exponential notation If a is any real number and n is a positive integer , then the power of a is The number is called the base , and is called the exponent

Example:


Laws of Exponents

• Example :


Adding and Subtracting Polynomials• We add and subtract polynomials using the properties of real

numbers .• The idea is to combine like terms ,using the Distributive Property.

• For instance,

• Example: Adding and Subtracting Polynomialsa)Find the sum ()+()

b) Find the difference (()


Multiplying Polynomials• To find the product of polynomials or other algebraic expressions, we need

to use the Distributive Property repeatedly.

Example: Find the product


Special Product Formulas

Example : Using the Special Product Formulas


Factorization

• We use the Distributive Property to expand algebraic expressions. We sometimes need to reverse this process by factoring an expression as a product of simpler ones.

• We say that and are factors of


• To factor a trinomial of the form we note that

• so we need to choose numbers r and s so that r + s=b and rs = c.

Example : Factor

Factor

Factoring Trinomials

chapter 1 level 0 math 0 faculty of engineering - basic science department- prof h n agiza

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