Review

Final Review

Math 99, Fall 2009

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This version: Wednesday, November 25, 2009, 5:31 PM

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Your Final Exam, worth 30% of your semester grade, will be as follows:

5% of your semester grade is determined by taking the COMPASS test in the Assessment Center, L912, outside of class time. Click on this link for the Math Department Memo which includes COMPASS details.
On Wednesday, December 9 you will take an exam similar to your midterm exam, and this counts for 25% of your semester grade. The first part will consist of ten multiple-choice questions, worth 40 points, and the rest will be non-multiple choice questions, worth 60 points. You may choose to take a break or not, just as on the midterm exam.

Missing the December 9 exam will give you a grade of zero. DO NOT miss the exam for any reason. You may only take the in-class exam if you have already completed the COMPASS test.

What you should bring:

When taking the COMPASS exam, bring your City Colleges ID or a driver's license. Bring a calculator—see note below and calculator memo.
Wednesday, December 9: You will not need your own paper. You may use pencil or pen. Scratch paper will be provided. You may bring your own blank graph paper, but graph paper will also be provided. A straightedge is suggested for drawing lines.
Unlike your midterm exam, your in-class multiple-choice questions will be machine graded. You must bring "#2" (soft lead) pencils for the in-class exam. Be sure you have a pencil eraser too.

Important note on calculators: Calculators (but not graphing calculators—see calculator memo) are allowed on the COMPASS test you will take in the Assessment Center, L912. In fact, I recommend you use one to check your work. Learn how to use your calculator. If you don't bring a calculator, there should be one available on the computer you use for the exam. Ask the Assessment Center staff for help. Of course, calculators are not allowed on the in-class final exam.

There is a math department review. This is a PDF File.

As always, please see me at class or during any of my office hours if you have questions.

Final grades: Please get your semester grades at http://my.ccc.edu/. Grades may be available by Friday, December 18. If you want a breakdown of how your grade was calculated (including your final exam grade), then send me e-mail after the final exam.

Your next math course: There are many math courses you may take immediately after Math 99: Math 118, Math 121, Math 125, Math 140, and Math 143. You may take one of these if either your Math 99 grade is C or higher, or if your COMPASS is high enough (Algebra score of 49 or more).

Note: If your grade was D or lower but your COMPASS score is high enough, you do not need to repeat Math 99. However, in such cases repeating Math 99 may nonetheless be advisable in order to succeed in your next math course.
For help in choosing your next math class, including deciding whether to repeat Math 99, please see me, the chair of the math department, or an advisor in Room 1112. Also see the chart of math courses.

Exam questions can come from any topic from the semester.

See our Midterm Review for a more detailed summary of topics covered on that exam, which covered Reviews C, D, E, F, G, Sections 3.4–3.6, 4.3–4.4, 7.3–7.7, and 9.1–9.2. Be sure to review these topics, since they were covered so long ago!
Since the midterm exam we covered Sections 8.1–8.3, 10.1, 10.3, and 10.5, Chapter 11, and Chapter 12. You must be able to complete the square for an expression, as on our Completing the Square handout. (Your textbook only does this in Section 12.5, not in Section 12.1).

Tips on studying:

Begin studying NOW.
Make a list of topics you think you need to review most, and don't try to review all of them at once.
It is very important that you can solve all of the problems from all of the quizzes. Don't just review the correct answers, but also choose similar problems from the text.
Reviewing old homework assignments can be useful, but even better is to practice by doing NEW problems like those in the homework. You may want to use the resources at the end of each chapter: the Chapter Summary, Review Exercises, Test, and Cumulative Review.

Tips on taking the test:

Come early and be alert. Your score will probably be worse if you are late, rushed, or tired. Get a good night sleep the night before and eat first.
Show your work, and spread out your work. Organization is the key to complex problems. Do not crowd your work or use two columns. Please ask for as much scratch paper as you want. For non-multiple choice problems, you may submit scratch paper if you want it graded, so if you run out of room on the printed exam (or make mistakes), do those problems on the scratch paper. You will make fewer errors if you show your work completely!
Simplify. For full credit, final answers must be simplified. Additionally, you should simplify as you solve problems, because this may prevent errors and it help avoid extra work. Simplification includes:

Fractions must be reduced.
Simple negative numbers shouldn't be in a numerator or denominator—take the negative sign "out front." (Ask me if you're not sure what I mean here.)
1x must be simplified to x, and –1x to –x, etc.
No negative exponents: convert to an expression with only positive exponents.
No fractional exponents: convert to radical (root) notation
Final simplified form for expressions involving radicals (roots):

Never write ²√ for square roots. (Leave out the small 2.)
Radicands (inside root sign) should be as small as possible.
Radicands should never include negative signs or fractions.
No fraction should have a radical (including the imaginary number i) in the denominator: rationalize all denominators.

Complex numbers should be in the form a + bi (real part then imaginary part), including rationalizing denominators.

Answer the question which was asked. Give units, if appropriate. If you introduced a variable in order to solve a word problem, then your final answer will not involve the variable you chose.

For topics from the first half of the semester, see our Midterm Review. Here is a summary of topics we have covered since then.

Relations, functions, function notation, domain and range, and the vertical line test (Sections 8.1 and 8.2)
Graphing functions (Section 8.3)

Recognizing constant, linear, and quadratic functions
Knowing the shape of six basic functions: x, x², x³, |x|, √x, 1/x
Finding x- and y- intercepts of the graph of a function
Determining intervals where a function is increasing or decreasing or constant

Solving inequalities and graphing solutions on a number line (Section 2.7)
Interval notation (and the "union" operation) (Section 10.1)
Understanding the operations of union and intersection (Section 10.1)
Graphing a linear inequality (Section 10.5)

Use dotted lines for < or > and solid line for ≤ or ≥.
Use a test point to know which side to shade.

Combining inequalities in one variable using "and" or "or," and giving a simplified answer as either an equation (or equations), in interval notation, or in a graph on the number line (Section 10.1)
Combining inequalities in two variables using "and" or "or," and giving a simplified answer as a shaded graph in the plane (Section 10.5)
Solving various absolute value equations: always begin by getting your equation into the form |X| = a or |X| = |Y|. (Remember that the first has no solutions if a < 0!) (Section 10.3)
Radicals (Chapter 11)

You must know about square roots, cube roots, and higher roots.
Simplify all radicals—see above simplification guidelines.
Even roots versus odd roots (e.g. when is a root not a real number? Odd roots are always real! But complex numbers can be used to give square roots of negative numbers.)
Use absolute values correctly! See shaded box on page 744.
Even roots can't be negative: know when to use absolute values for variables.
Pythagorean theorem
Rewriting fractional exponents using roots, and vice versa
Combining, manipulating and simplifying radicals

Look for "like radicals" when adding or subtracting only
You may prefer to think of radicals as variables in order to simplify (e.g. for 8–3√2, think 8–3t).

Rationalizing denominators, including using conjugates
Solving equations by squaring (or higher even powers) requires checking solutions. (On non-multiple choice questions, show work for full credit!). Also see below.

Complex numbers (Section 11.8)

Correctly using i, including using conjugates when necessary

Completing the Square: see handout or click on the link. You must be able to complete the square on an expression, not just an equation.
The Square Root Property says that x² = c is solved by x = √c or x = –√c. Checking is not required to use this. (Section 12.1)
Solving quadratic equations: you must know all these methods:

Solving by factoring (Section 6.7)

Easiest method, but doesn't always apply

Solving by completing the square (Handout and Section 12.1)

Always works; sometimes easy

Solving by quadratic formula: everyone MUST memorize this. (Section 12.2)

Always works; sometimes easy

Be familiar with what the discriminant b² – 4ac tells us about solutions of a quadratic equation. (Section 12.2; this is not a very important topic.)
Some equations which are not quadratic can be put into that form by using a substitution like x = t³ or y = m^½. Others can be converted to a quadratic equation by, for example, clearing denominators or factoring first. (Section 12.3)
Be able to use Completing the Square to graph any parabola (even when the leading coefficient is not 1). To graph y = a(x – h)² + k:

The vertex is at (h, k) (Careful of signs! Check by plugging in.)
Plug in x = 0 to get y-intercept (easier for y = ax² + bx + c)
Plug in y = 0 to get x-intercepts. There will be zero, one, or two. (Sometimes if two exist, finding them may require a calculator.)

Summary of checking answers:

Checking is required (and must be shown on non-multiple choice questions) when solving the following types of equations of one variable:

Equations with fractions when the variable is in the denominator. A full check is not required—just make sure the solution doesn't make a denominator zero. (Review Section 7.6.)
Equations in which you square both sides (or take a fourth power, etc.). Here a full check is required. (Why? Consider squaring √x = –3.)

Checking is not required when solving the following types of equations:

Using the square root property (Why? By taking x = ±√c we counted both solutions)
Taking an odd power of both sides (Why? Odd powers don't mess with signs.)

A full check of solutions is always a good idea. If you have time at the end of the final exam, I suggest checking all solutions. In this way you may be able to find and correct your own errors. However, only do optional checks after you finish the exam, so that you don't run out of time.

Since we haven't recently completed word problems, here are some reminders:

Define variables clearly, in complete sentences.
Answer what was asked; remember to include appropriate units!
Setting up problems:

Motion problems: (rate)(time) = distance
Dry mixture, e.g. (pounds)(dollars per pound)=dollars spent
Bank accounts: (principal)(interest rate) = dollars interest
Liquid mixture, e.g. (pints liquid)(percent dye)=pints dye

Revised Friday, May 1, 2009. E-mail corrections, suggestions to mmaltenfort@ccc.edu