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 the 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.