MATH 1003C Highlights

MAT 1033C Highlights for Chapters 5

Be able to distinguish between a monomial and a polynomial . The key to this distinction
is that polynomials have more than one term connected by the addition and subtraction
operations. The only operation used in a single term is multiplication. Review in your
text how to find the degree and leading coefficient of a monomial in more than one
variable. Also review how to identify the degree and leading coefficient of a polynomial.
Know what like terms of a polynomial are and be able to combine them algebraically.
Polynomials can have exponents in them when they are written out. However, for an
expression to actually be a polynomial, you will need to make sure that the variables in
the polynomial have whole number exponents.

Know how to read and use function notation. For example, given that f(x) = x + 5 be
able to find f(-3) by plugging in -3 into the function to get a final answer of (-3) + 5 = 2.
To evaluate a function at a given number means to “plug” that number into the function
to get a number out. The number that you get out is your answer.

You should be able to add and subtract two or more polynomials and combine like terms
to yield another polynomial. Remember that to subtract two polynomials; you need to
distribute the subtraction sign throughout each term in the polynomial that you are
subtracting. Also, you will need to know how to multiply two polynomials together
using the FOIL method or by simply distributing a common term.

For sure, know how to factor quadratic polynomials symbolically. In addition, you need
to be comfortable with the factoring process using a table of values and/or a graph. You
do this by looking for the x-intercepts i.e. where the y-value is zero. Please review your
book for examples on how this is done both numerically and graphically. The main
purpose of factoring is to be able to solve equations using the zero-product property.
There are various types of factoring. One is to factoring a trinomial into a product of two
binomials. Other ones include factoring out the greatest common factor and factoring
using the difference of two squares. Your textbook shows you how to factor a sum or
difference of two cubes; be sure you memorize the formulas for both types.

Practice working with polynomial graphs in conjunction with word problems. In such
problems, a polynomial typically models some real word phenomena. Your job is to
interpret the model in context of the real world situation. The model may come in the
form of a graph or actual polynomial formula. Interpreting a model means to make sense
of the mathematical information that is given to you and to be able to draw conclusions
based on that information in complete sentences. Review the section in your textbook
dealing with polynomial modeling for some good examples to review. A good one to
review is the one that models the height of a falling object such as a ball. Be sure to
review other types as well.

Know how to use the feature WINDOW on your graphing calculator when graphing
polynomials. For example, if you see notation such as [0, 50, 1] realize that the first
number indicates the minimum value of the variable and the second indicates the
maximum value of the variable while the third number indicates the scale being used .

MAT 1033C Highlights for Chapter 6

Know the rules of exponents for multiplication and division. For example, you add
exponents when multiplying numbers in the same base and subtract exponents when
dividing numbers in the same base. You will need to know how to reduce fractions to
lowest terms . The key to this is to cancel out all the common factors in both the
numerator and the denominator. This process of canceling out common factors can be
thought of as “tearing down” a fraction. The fractions that you will need to reduce will
have multiple variables , exponents, and coefficients in them. Sometimes you may need
to factor the numerator and denominator before performing any cancellations.

You need to know how to add and subtract fractions using a common denominator
preferably the least such one. Again, these fractions will involve many variables,
coefficients as well as exponents. The key is to remember to “build up” fractions before
you add them so that they all have the same denominators.

Review the term reciprocal and know how to find it for various fractions. Be able to
multiply and divide fractions. Remember that dividing two fractions is the same as
multiplying the top fraction by the reciprocal of the denominator.

Know how to use graphs of functions to evaluate expressions such as f(-5). Also, know
how to use graphs to solve equations by seeing the x-coordinates where the two graphs
intersect. Usually each side of an equation is graphed on the same set of axes.

As for graphing “fraction like” functions by hand or with the aid of a graphing calculator,
be sure you can identify vertical asymptotes as well the domain.

Know how to simplify complex fractions. A complex fraction is one where both the
numerator and denominator are themselves fractions or the sum/difference of fractions.
Your textbook outlines two methods for simplifying complex fractions. Choose the
method that you feel most comfortable with and practice it.

You will also need to be able to solve single variable equations involving fractions. One
trick you may have learned is to cross-multiply in such situations. You should be able to
solve such equations algebraically and numerically . When you solve numerically, you
need to construct and show a table of values to support your work.

Like always, there will be word problems on the test . You will need to know how to set
up various types of word problems from information given to you in the word problem
itself. Review each type of word problem in your homework and do them until you feel
you can independently do them without any help. One type of word problem you should
review is the one where two people that work individually at different rates decide to
work together to finish a job and you are asked to find out how long it will take to finish
the job if both people work together. Sometimes function notation is used within a word
problem itself and you need to interpret this notation in context of the problem by writing
out a complete sentence.

Be able to pick out graphs of functions that best models real life situations.

MAT 1033C Highlights for Chapter 7

Be able to interpret a fractional exponent as a radical. For example 4 to the ½ power
means the same as the square root of 4. Feel comfortable going between fractional
exponent form and radical form and be able to evaluate examples of each type.

There are two additional rules of exponents in this chapter that you are responsible for.
The first deals with the situation where you have an exponent raised to another exponent.
In this situation, remember to multiple the two exponents together . The second situation
deals with negative exponents. Remember that a negative sign in an exponent indicates
to take the reciprocal. After you take the reciprocal, the negative sign is not longer there.

Know how to simplify radical expressions. A radical expression can be simplified when
the radicand has any factors with powers greater to or equal to the index of the radical
sign. For example, the square root of 8 is equal to 2 times the square root of 2. Be able
to do the same sort of simplification with variables in the radicand as well.

Know how to perform the 4 basic arithmetic operations when working with radicals.
You can add or subtract like radicals in the same way you add or subtract like terms
involving variables. When you multiply radical expressions you may need to convert to
fractional exponent form first. Then you can add exponents if you are multiplying in the
same base.

Remember that the square root of -1 is the imaginary unit called i. Know how to
compute square roots of negative numbers using this imaginary unit. A complex number
is a sum or difference of a real number with an imaginary one. You need to know how to
find the complex conjugate of a given complex number and use it to divide two complex
numbers. Always write complex numbers in standard form which is a + bi. You also
need to know how to perform the 4 basic arithmetic operations on complex numbers.
You add and subtract using like terms and multiply using the FOIL method. You divide
two complex numbers using the conjugate method.

Know how to interpret function notation in word problems that involve “radical type”
functions. For these types of word problems, the equations are pretty much set up for you
ahead of time. You just need to be able to correctly interpret and manipulate the function
formulas to answer the questions posed to you in the problem itself.

Be proficient at using your calculator in computing radicals and fractional exponents.
There is a common error to watch out for what entering fractional exponents in your
calculator. Remember to put the whole fractional exponent into a set of parenthesis after
you press the exponent key. You answer will be incorrect if you forget the parenthesis.

MAT 1033C Highlights for Chapter 8

Know how to solve quadratic equations where one side of the equation is zero using only
a table of values and be prepared to state, in a sentence, the reasoning behind your work.
Be able to solve quadratic equations where one side is zero using only a graph. The
graphs of quadratic functions have the shape of a parabola. A parabola may cross the x-axis
in two different places. These x-intercepts are the solutions to the corresponding
quadratic equation.

There are other qualitative features of parabolas besides their intercepts. These include
the vertex and axis of symmetry. You need to memorize the formula and procedure used
to find the vertex of any parabola. You should be able to locate the vertex of a parabola
using only a table of values. There is no need for the vertex formula in this situation.
You simply locate the smallest or largest output value in the table and check the
corresponding x-value.

The coefficient of the x squared term in a quadratic function is called the leading
coefficient (LC) and governs the orientation of the parabola. If the LC is positive then
the parabola opens upwards, and when the LC is negative then the parabola opens
downwards. Know the effects of different values the LC has upon the graph of a
parabola. For example, if the LC is a large positive number will the parabola get
narrower or wider? Why?

Review what the discriminate of a quadratic function is and how to compute it.
Remember that when the discriminate is positive the corresponding quadratic function
has two distinct real roots. What happens when the discriminate is negative or zero?
You need to be able to identify the discriminate as positive, negative, or zero based solely
on the graph of a parabola.

You need to memorize the quadratic formula verbatim and be able to apply it to various
quadratic equations. You also need to solve quadratic equations using other methods as
well such as factoring and the square root property. Remember that when you factor a
quadratic equation with one side being zero, you must set each factor equal to zero. This
is called the zero-product property of equations.

Given a quadratic function be prepared to sketch its graph without the aid of your
graphing calculator. This is typically done by assembling certain qualitative information
and knowing that the general shape of such a function is a parabola. The qualitative
information mentioned includes the vertex, axis of symmetry, any intercepts, as well as
the orientation of the parabola. This process works in reverse too. Specifically, you
should be able to find a quadratic function given only its graph and vertex or other
qualitative information.

Review the different forms of a parabola. There are two main forms. The first is called
the standard form and must be used before you apply the quadratic formula. The second
is called the vertex form. When a quadratic function is in vertex form you don’t need a
formula to locate the vertex. Why not?

Sometimes when using the quadratic formula you get “nice answers” such as whole
numbers or simple fractions. Other times the quadratic formula yields “messy” answers
that include irrational numbers. Be prepared to use your calculator in order to
approximate quadratic equations that have “messy” answers.

Like in previous tests, there will be word problems on this test as well. The main focus
for the word problems is the interpretation, within the context of the word problem, of the
qualitative features of quadratic functions that we mentioned previously. For example, an
x-intercept may have a very important and specific meaning within the context of a word
problem. Be prepared to flesh out the underlying meaning of this x-intercept in a
sentence or two. You should actively seek out such word problems in your text to
practice on. They are also given as examples embedded in each section.

The vertex has a special interpretation. You can interpret the vertex as either the
maximum or minimum of a certain quantity. For example, businesses are looking to
maximize profit while minimizing cost. Whatever units one decides to impose on the
variables x and y will largely determine the underlying meaning of the various qualitative
features of a parabola. Look for word problems to practice on that include a variety of
units being used. These units could include, but are not limited to, time, money, and
distance (in feet, miles etc.)

Beginning Algebra

COURSE DESCRIPTION:
This course includes the following topics: operations with signed numbers: addition,
subtraction, multiplication, and division with algebraic expressions ; factoring; techniques
for solving linear and fractional equations ; and an introduction to graphing and systems of
equations and functions .

PREREQUISITES: Grade of C or better in MAT 155 or COMPASS Pre-Algebra score 60-99
or SAT Math score 400-430 or ACT Math score 18-20.

PURPOSE:
Beginning Algebra is designed for students who have never had high school algebra, or
who have one year and did not do well or who completed the course several years ago.
All skills are learned with a focus on solving word problems, so that algebra becomes a
useful tool in their related fields.

REQUIRED MATERIALS:
1. Textbook: Beginning and Intermediate Algebra, Fifth Edition, Gustafson/Frisk, 2008.
2. Scientific Calculator

SUPPLEMENTARY MATERIALS:
1. Solution manual on closed reserve in all campus libraries.
2. Tutoring available through the Student Success and Technology Center.

ENTRY LEVEL COMPETENCIES:
1. Students should be proficient in whole numbers, fractions and decimal
computation.

2. Students should be able to solve word problems involving ratio , proportion and
percent .

3. Students should be able to solve basic geometry problems and convert
measurements.

4. Students should know how to operate with signed numbers.

***A Comprehensive Proficiency Exam for this course can be taken to receive credit. See
instructor, Mathematics Department Head or Student Services for details.

COURSE OUTLINE:

UNIT I: Real Numbers and their Basic Properties (Chapter One) (6 hours)

1. Subsets of Real Numbers (1.1)
2. Equality, Inequality Symbols, and Variables (1.1)
3. Absolute Value of a Number (1.1)
4. Exponents and Order of Operations (1.3)
5. Formulas from Geometry (1.3)
6. Operations on Real Numbers (1.4, 1.5)
7. Writing & Evaluating Algebraic Expressions (1.6)
8. Properties of Real Numbers (1.7)

UNIT II: Linear Equations and Inequalities (Chapter Two) (6 hours)

1. Solving Basic Equations (2.1)
2. Solving Equations by Addition and Subtraction (2.1)
3. Solving Equations by Multiplication and Division (2.1)
4. Applications Involving Percent (2.1)
5. Solving More Equations (2.2)
6. Simplifying Expressions in Solving Equations (2.3)
7. Identities and Contradictions (2.3)
8. Introduction to Problem Solving; Investment Problems (2.4)
9. Motion and Mixture Problems (2.5)
10. Solving Formulas (2.6)
11. Solving Inequalities (2.7)

UNIT III: Polynomials and Factoring

(Chapter Four) (8 hours)
1. Natural Number Exponents; Properties of Exponents (4.1)
2. Zero and Negative-Integer Exponents (4.2)
3. Scientific Notation (4.3)
4. Polynomials (4.4) [Omit polynomial functions and graphing]
5. Operations on Polynomials (4.5, 4.6, 4.7, 4.8)

(Chapter Five) (8 hours)
6. Factoring Out the Greatest Common Factor (5.1)
7. Factoring by Grouping (5.1)
8. Factoring the Difference of Two Squares (5.2)
9. Factoring Trinomials (5.3, 5.4)
10. Factoring Sum and Difference of Cubes (5.5)
10. Summary of Factoring (5.6)
11. Solving Equations by Factoring (5.7)
12. Problem Solving (5.8)

UNIT IV: Graphing Linear Equations (Chapters Three and Eight) (5 hours)

1. The Rectangular Coordinate System (3.1)
2. Graphing Linear Equations (3.2)
3. The Slope of a Nonvertical Line (8.2)
4. Writing Equations of Lines (8.3)

Differential Calculus Of One Variable

Course Description

The course is an introduction to differential calculus of one variable. The
course will focus on limits, derivatives, and their applications. At the conclusion
the course, you will be able to take the derivative of many functions
and use that information to graph functions and their tangent lines, find
local extrema, calculate polynomial approximations , calculate related rates,
and solve optimization problems.

In addition to learning the above techniques, a main goal of the course
is for you to understand the geometric interpretations of the concepts of
calculus . Particular emphasis is placed on learning how to convert “word
problems” to a mathematical language and then use the tools of differential
calculus to solve them.

Prerequisites: Pre-calculus including basic algebra and trigonometry . In
particular: basic notation for polynomials, the quadratic formula, factoring
polynomials and polynomial division, trigonometric functions and their interpretation,
the ability to graph and find the equation of a line , and familiarity
with similar triangles.

Text: The course textbook is Single Variable Calculus: Concepts and
Contexts, Third Edition (2005) by James Stewart. The book is on permanent
reserve in the Math Library (on the fourth floor of Building 380).
Although the textbook is not required for the course, it will be a helpful
supplement to the lectures. This text will be required for Math 20, if you
plan to take that course.

Homework

Homework will be due at 5pm every Wednesday . It may be turned in during
class, or placed in the appropriate hanging envelope outside of 380-381J.

You should solve the homework questions in order; multiple sheets should
be stapled together. Answers should be simplified when possible and kept
in exact form (i.e., do not give a decimal approximation unless specifically
asked).

Since solutions to the homework will be posted on Wednesday evening,
late homework is not accepted. However the lowest homework score will be
dropped. The goal of dropping the lowest homework score is to discourage
requests for extensions on the homework while providing you with some
flexibility during the term. I would be happy to make arrangements for
extenuating situations.

Quizzes

Every Friday (except for exam weeks), there will be a short quiz during the
last few minutes of class. These quizzes are designed to help solidify material
from the previous week. There will be no makeup quizzes given, but the
lowest score will be dropped. It will often be helpful to review the homework
solutions from the previous week before the Friday quiz!

Exams

There will be two midterm exams and one final exam. The problems on the
exams will be similar to examples done in class and homework problems . If
you have any conflicts with the scheduled exam times, please let me know as
soon as possible and in any event by January 21.

First Midterm
Thursday, January 29 7:00-9:00pm
Second Midterm
Thursday, February 26 7:00-9:00pm
Final Exam
Monday, March 16 7:00-10:00pm

Please note that the final exam is given during the group exam time, and
not during the time slot that would ordinarily be associated with the class.

University Calendar

There are university holidays (no class or office hours) on January 19 and
February 16.

The drop deadline is February 1, the change of grading basis deadline is
February 16, and the course withdrawal deadline is March 1. Please talk to
me before dropping the class or changing to pass/fail!

Grading

Although the lowest homework and quiz scores will be dropped, all should
be submitted. The scores from homework and exams will be weighted as
follows. The two midterms will be weighted equally.

Homework: 16%
Quizzes: 7%
Midterms: 44%
Final Exam: 33%

Please note that the exam scores are the most important factor in the
course grade .

Calculators

Calculators will not be allowed on the exams. Although calculators may be
useful for the homework, they will never be required. Even if you get the
correct answer to a homework problem via a calculator , you will not receive
full credit unless you include all intermediate work leading to the answer.