MATH 17 First Long Examination
PART 1. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. C B B 1 D 2 3 A C D B 11. C 12. x x 4 x 2 13. D 14. 15. 16. 17. 18. 19. 20.
x 12 x11 B A D A B A
3 2
1st Semester 2010-2011 02 August 2010
21. 22. 23. 24. 25. 26.
C C D B A 20 2 x 12 2
8/2/2013
Inequalities
A statement that one mathematical
expression is greater than or less than
another is called an inequality.
Goal:
Find solutions and solution sets
for inequalities.
Interval Notation
x a x b a,b
x a x b a,b
x x a , a
x x a a,
x a
CHEM 1412 FORMULA SHEET
mass of proton = 1.007276470 amu
c = 2.998 X 108 m/s
K = oC + 273.15
1 g = 6.022 X 1023 amu
Sg = kHPg
PA = XAPAo
= MRT
PV = nRT
mass of neutron = 1.008664904 amu
F = 96485 C/(mol of e) = 96485 J/(V mol of e)
R = 0.08206 (L atm)/(m
For Na+ and K+
Clean the nichrome wire by using concd HCl and heating
1st Trial
2nd Trial
(Preparation of Solution)
(Preparation of Solution)
Evaporate CC1 to about 2 drops
and cool
Get 4 drops of the sample
Add 6M HCl, to the centrifugate add
6M NH3, to
New York City College of Technology
Department of Chemistry
CHEM 2223 Organic Chemistry I
(5 credits, 4 hours lecture, 3 hours lab)
Course Description:
Fundamental concepts of nomenclature, formulae, preparation and properties of organic compounds.
Modern
MATH 17 Y. Algebra and Trigonometry
Exercise 8. Prefinal Exercise
21 November 2015
General Instructions: Do as indicated. Write your answers, neatly, orderly and completely in your bluebook(s). Deadline is on Wednesday, 25 November 2015, recitation hour.
2/26/2015
Addition of Rational
Expressions
Example 1.5.3
Perform the indicated operation.
Definition
Let P x ,Q x , and R x be polynomials
in x then
P x
Q x
R x
Q x
P x R x
Q x
Q x 0
1
x
x 1 x 13
LCD : x 1
3
1 x 1 x
2
1
x
LCD : x 2 x 2
x 2 x2 4
1
Double Measure
Identities
Double Measure
Identities
If
, then
Double Measure
Identities
If ,then
sin sin cos cos sin
sin 2 sin cos sin cos
2sin cos
tan tan
tan
1 tan tan
2 tan
tan 2
2
1 tan
Example
Given:
Determine
Solve for
first.
Since
Which q
9/4/15
Solvable Equations in R
CHAPTER 1.3
THE COMPLEX
NUMBER SYSTEM
Mathematics Division, IMSP, UPLB
ax b 0 is always solvable in R.
b
x
a
1
Solvable Equations in R
If p 0, the solutions to the
equation x 2 p are
x 2 3 is solvable in R.
Mathematics Divis
Circular
Functions
Recall
The wrapping function P is a
function from
to U such that
where (x, y) is the terminal
point of the arc with length t.
Is P one-to-one?
Special Real Numbers
Examples
4 1 3
P
,
3
2
2
7 3 1
P
,
6 2 2
7
P
4
2 , 2
2
2
08/10/2015
OBJECTIVES
Upon completion, you should be able to
RADICALS
Simplify radicals; and
Perform addition, subtraction, multiplication
and division of radicals.
2
RATIONAL EXPONENTS & RADICALS
1
RADICALS
A radical (or irrational expression) is an
alge
9/4/15
1
Types of Quantities
Chapter 1.4
POLYNOMIAL
Expressions
Variables
represented only by letters
and whose values may be
arbitrarily chosen depending
on the situation.
3
Mathematics Division, IMSP, UPLB
2
4
1
9/4/15
Types of Quantities
Algebraic Expr
Chapter 4.3
Circular Functions
1
Domain, Range, and Graphs
1. ft
sin t
Dom f
Rng f 1,1
P t x,y
sin t y
2
Domain, Range, and Graphs
2. ft
cos t
Dom f
Rng f 1,1
P t x,y
cos t x
3
Periodic Functions
Periodic functions are functions whose values
repea
Chapter 5.3
System of Inequalities
Solutions of Inequalities
A solution of an inequality with two
variables is an ordered pair which
satisfies the inequality.
We will represent solutions of an
inequality using graphs.
Example
In the Cartesian plane, th
10/8/2015
Objectives
Chapter 2
Relations and
Functions
1. define relations, functions and inverse
functions;
2. state the domain, range, intercepts and
symmetry of the functions and relations;
3. differentiate relations from functions;
4. perform operatio
For Na+ and K+
Clean the nichrome wire by using concd HCl and heating
1st Trial
2nd Trial
(Preparation of Solution)
(Preparation of Solution)
Evaporate CC1 to about 2 drops
and cool
Get 4 drops of the sample
Discard any precipitate and add 1
drop of concd
9/2/2013
Inverse Functions
If f is a function consisting of ordered pairs x , y ,
Inverse Functions
then there is a relation called the inverse of f ,
whose elements are the oredered pairs y , x .
Inverse Functions
If f is one-to-one, then the inverse of
7/22/2013
OBJECTIVES
RATIONAL EXPONENTS &
RADICALS
Upon completion, you should be able to
Simplify radicals; and
Perform addition, subtraction, multiplication
and division of radicals.
2
RATIONAL EXPONENTS & RADICALS
RADICALS
Examples
A radical (or irr
8/16/2013
Objectives
Chapter 2
Relations and
Functions
1. define relations, functions and inverse
functions;
2. state the domain, range, intercepts and
symmetry of the functions and relations;
3. differentiate relations from functions;
4. perform operatio
6/26/2013
Chapter 1
Section 1.2
The Real Number System
as a Number Field
Uses of Number
Definition
The Set of Real Numbers
1.
Naming
2.
Ordering
3.
Counting
4.
Measuring
Numbers that will be used for counting and
measuring will be collectively called real
8/18/2013
OPERATIONS ON FUNCTIONS
Definition. Sum, Difference, Product,
Quotient and Composite Functions
OPERATIONS ON FUNCTIONS
Let f and g be functions of the variable x.
1. The sum function f+g is the function
defined by
f g x f x gx
OPERATIONS ON FU
7/8/2012
Solvable Equations in R
CHAPTER 1.3
THE COMPLEX
NUMBER SYSTEM
ax + b = 0 is always solvable in R.
b
x=
a
Solvable Equations in R
Solvable Equations in R
x 2 = 9 is solvable in R.
The solutions are 3 and 3
If p 0, the solutions to the
equation x 2
9/2/2013
One-to-One Functions
Chapter 2.3
Properties of
Functions
if whenever a and b are two numbers in the
One-to-One Functions
Example 2.3.1
A function f is one-to-one 1 1 if and only
domain of f and a b then f a f b .
Determine if the following functi
7/22/2013
Types of Quantities
Chapter 1.4
POLYNOMIALS
Variables
represented only by letters
and whose values may be
arbitrarily chosen depending
on the situation.
Types of Quantities
Constant
a quantity whose value is fixed
and may not be changed during
a
9/2/2013
Chapter 3
Exponential and
Logarithmic Functions
Chapter 3.1
Exponential Functions
Exponential Functions
Example 3.1.1
If f x 2 x , complete the given table.
If b 0 and b 1, then the exponential
function with base b is defined by
f x bx
x
-4
-3
-2
6/20/2013
MATH 17. COLLEGE ALGEBRA AND TRIGONOMETRY
Chapter 1
Algebra as the
Study of Structures
Chapter Outline
1.
2.
3.
4.
6.
7.
Sets, Set Operations and Number Sets
The Real Number System
The Complex Number System
Field of Algebraic Expressions
Equatio
7/22/2013
Rational Expressions
Rational Expressions are algebraic
expressions that are quotients of
Chapter 1.5
Rational Expressions
polynomials.
P x
Q x
,
Q x 0
1
2
Rational Expressions
Example 1.5.1
A rational expression is in its
simplest form if the
6/28/2013
Subsets of
+ :
:
:
:
Solvable Equations
, 5, 4, 3, 2, 1
, 5, 4, 3, 2, 1,0
Is 2 = 6 solvable in ?
Is 2 = 1 solvable in ? in ? In ?
Is 2 = 3 solvable in ? in ? In ?
Multiplicative Inverses
Theorem
Existence of Multiplicative Inverses
For eve
6/21/2013
Definition
Empty Sets
Empty Set
Remarks:
Empty Set is a subset of any set.
- are sets having no elements
- denoted by or cfw_
, for any set .
The empty set is a subset of itself.
Example:
= y y is a country in Asia with no people
=
= cfw_pos
7/25/2013
Algebraic Expressions
Chapter 1.6
Equations and
Inequalities
Algebraic Expressions
x : number of hours you need
to finish a job
1
: amount of work you can do
x
in an hour.
Algebraic expressions are symbolic
forms of numbers.
h
r 2h
r
Algebraic
10/8/2015
Chapter 1.6
Equations and
Inequalities
2
1
Equations and Inequalities
Solution Set
Equation is a statement that two
algebraic expressions are equal.
The solution set of an equation
or inequality is the set of all real
Inequality is a statement t
SYSTEM of
EQUATIONS
SYSTEM OF EQUATIONS
1
Linear Equations in 2
Variables
A Linear Equation in two variables x and y is
an equation of the form
where a, b, and c are real numbers and a
and b are not both 0 at the same time.
Examples:
SYSTEM OF EQUATIONS
2
Equations
Involving Circular
Functions
Objective
solve equations
involving circular
functions
Solving Equations
Determine all arc lengths that will
satisfy a given equation.
Examples:
Solution set:
Solution set:
Fundamental Solution
Set
Set of solutions
Mathematics Division, IMSP, UPLB
9/4/15
Chapter 1
Section 1.2
The Real Number System
as a Number Field
Mathematics Division, IMSP, UPLB
1
Uses of Number
1.
Naming
3.
Counting
2.
4.
Mathematics Division, IMSP, UPLB
2
Definition
The Set of Real Numbers
Numb
9/4/15
From now on
MATH 17. COLLEGE ALGEBRA AND TRIGONOMETRY
The Algebra of
Numbers
Chapter 1
Mathematics Division, IMSP, UPLB
you learn math
by doing math
1
Mathematics Division, IMSP, UPLB
2
Mathematics Division, IMSP, UPLB
3
Chapter Outline
1. Sets, Se