Algebra/Trig Review
This review was originally written for my Calculus I class
but it should be accessible to anyone needing a review in some basic algebra
and trig topics. The review contains the
occasional comment about how a topic will/can be used in a calculus class. If you aren’t in a calculus class you can
ignore these comments. I don’t cover all
the topics that you would see in a typical Algebra or Trig class, I’ve mostly
covered those that I feel would be most useful for a student in a Calculus
class although I have included a couple that are not really required for a
Calculus class. These extra topics were
included simply because they do come up on occasion and I felt like including
them. There are also, in all likelihood,
a few Algebra/Trig topics that do arise occasionally in a Calculus class that I
didn’t include.
Because this review was originally written for my Calculus
students to use as a test of their algebra and/or trig skills it is generally
in the form of a problem set. The
solution to the first problem in a set contains detailed information on how to
solve that particular type of problem.
The remaining solutions are also fairly detailed and may contain further
required information that wasn’t given in the first problem, but they probably
won’t contain explicit instructions or reasons for performing a certain step in
the solution process. It was my
intention in writing the solutions to make them detailed enough that someone
needing to learn a particular topic should be able to pick the topic up from
the solutions to the problems. I hope
that I’ve accomplished this.
So, why did I even bother to write this?
The ability to do basic algebra is absolutely vital to
successfully passing a calculus class.
As you progress through a calculus class you will see that almost every
calculus problem involves a fair amount of algebra. In fact, in many calculus problems, 90% or
more of the problem is algebra.
So, while you may understand the basic calculus concepts, if
you can’t do the algebra you won’t be able to do the problems. If you can’t do the problems you will find it
very difficult to pass the course.
Likewise you will find that many topics in a calculus class
require you to be able to basic trigonometry.
In quite a few problems you will be asked to work with trig functions,
evaluate trig functions and solve trig equations. Without the ability to do basic trig you will
have a hard time doing these problems.
Good algebra and trig skills will also be required in
Calculus II or Calculus III. So, if you
don’t have good algebra or trig skills you will find it very difficult to
complete this sequence of courses.
Most of the following set of problems illustrates the kinds
of algebra and trig skills that you will need in order to successfully complete
any calculus course here at Lamar
University. The algebra and trig in these problems fall
into three categories :
- Easier
than the typical calculus problem,
- similar
to a typical calculus problem, and
- harder
than a typical calculus problem.
Which category each problem falls into will depend on the
instructor you have. In my calculus
course you will find that most of these problems falling into the first two
categories.
Depending on your instructor, the last few sections (Inverse Trig Functions through Solving Logarithm
Equations) may be covered to one degree or another in your class. However, even if your instructor does cover
this material you will find it useful to have gone over these sections. In my course I spend the first couple of days
covering the basics of exponential and
logarithm functions since I tend to use them on a regular basis.
This problem set is not designed to discourage you, but
instead to make sure you have the background that is required in order to pass
this course. If you have trouble with
the material on this worksheet (especially the Exponents - Solving
Trig Equations sections) you will find that you will also have a
great deal of trouble passing a calculus course.
Please be aware that this problem set is NOT designed to be
a substitute for an algebra or trig course.
As I have already mentioned I do not cover all the topics that are
typically covered in an Algebra or Trig course.
The most of the topics covered here are those that I feel are important
topics that you MUST have in order to successfully complete a calculus course
(in particular my Calculus course). You
may find that there are other algebra or trig skills that are also required for
you to be successful in this course that are not covered in this review. You may also find that your instructor will
not require all the skills that are listed here on this review.
Here is a brief listing and quick explanation of each topic
covered in this review.
Algebra
Exponents A brief review of the basic exponent
properties.
Absolute Value A couple of quick problems to remind you of
how absolute value works.
Radicals A review of radicals and some of their
properties.
Rationalizing
A review of a topic that doesn’t always get
covered all that well in an algebra class, but is required occasionally in a
Calculus class.
Functions Function notation and function evaluation.
Multiplying Polynomials A couple of polynomial multiplication problems
illustrating common mistakes in a Calculus class.
Factoring Some basic factoring.
Simplifying Rational Expressions The ability to simplify rational expressions
can be vital in some Calculus problems.
Graphing and Common Graphs Here are some common functions and how to
graph them. The functions include
parabolas, circles, ellipses, and hyperbolas.
Solving Equations, Part I Solving single variable equations, including
the quadratic formula.
Solving Equations, Part II Solving multiple variable equations.
Solving Systems of Equations Solving systems of equations and some interpretations
of the solution.
Solving Inequalities Solving polynomial and rational inequalities.
Absolute Value Equations and Inequalities
Solving equations and inequalities that
involve absolute value.
Trigonometry
Trig Function Evaluation How to use the unit circle to find the value
of trig functions at some basic angles.
Graphs of Trig Functions The graphs of the trig functions and some nice
properties that can be seen from the graphs.
Trig Formulas
Some important trig formulas that you will
find useful in a Calculus course.
Solving Trig Equations Techniques for solving equations involving
trig functions.
Inverse Trig Functions The basics of inverse trig functions.
Exponentials /
Logarithms
Basic Exponential Functions Exponential functions, evaluation of
exponential functions and some basic properties.
Basic Logarithm Functions Logarithm functions, evaluation of logarithms.
Logarithm Properties These are important enough to merit their own
section.
Simplifying Logarithms The basics for simplifying logarithms.
Solving Exponential Equations Techniques for solving equations containing
exponential functions.
Solving Logarithm Equations Techniques for solving equations containing
logarithm functions.