Syllabus and Course Guide
for
Precalculus – Fall ‘09

Syllabus – Tentative Schedule - Links - Tests - Grades

precalc

Instructor: Geoff Hagopian
Office: Math 12 
Office Hours: MWF: 10 – 11 and TR: 1 – 2
Email: ghagopian@collegeofthedesert.edu
Web Page: http://geofhagopian.net
Telephone: (760) 776-7223

Main Text:  Precalculus, by Stewart, Redlin and Watson or the Custom Ed., isbn 0495280909
Note that all the topics we will cover this semester can also be found (the exact same writing) in the text, Precalculus, Mathematics for Calculus, 5th ed., by Stewart, Redlin and Watson, isbn 0-534-49277-0

Catalog Description:
This course is the second in a two semester sequence preparing students for Calculus.  In this course, students will extend the concept of a function to polynomial, rational, exponential and logarithmic functions as well as studying analytic trigonometry.  Topics include analysis of equations and word problems involving polynomial, rational, exponential and logarithmic functions, trigonometric identities, inverse trigonometric functions, and solving trigonometric equations..

The course objectives, as listed in the course outline of record, are the following:

  1. Demonstrate an understanding of the real number system by identifying the sets that make up the real numbers and the basic properties of addition, multiplication, equality and inequalities.
  2. Demonstrate an understanding of rational exponents, roots and their corresponding properties by evaluating, and simplifying algebraic expressions involving these symbols. 
  3. Apply the basic properties of the real numbers and exponents to add, subtract, multiply, divide and factor algebraic expressions including expressions involving fractions and roots.
  4. Apply the properties of the real numbers and exponents to solve one variable algebraic equations including linear, quadratic, rational, and root equations.
  5. Apply the properties of the real numbers to solve one variable inequalities including linear, quadratic and rational inequalities.
  6. Translate word problems into one variable equations or inequalities and solve the problem.
  7. Find and use the basic characteristics of a polynomial function such as end behavior and zeros to construct a graph and to find an equation for a polynomial function.
  8. Perform arithmetic with the complex numbers and use the complex numbers to completely solve a quadratic equation.
  9. Solve polynomial equations using factoring, polynomial division and basic results such as the Remainder and Factor theorems and the Fundamental Theorem of Algebra.
  10. Model and solve word problems using polynomial functions.
  11. Find and use the basic characteristics of a rational function such as domain, end behavior, intercepts, and asymptotes to construct a graph and to find an equation for a rational function.
  12. Model and solve word problems using rational functions.
  13. Find and use the basic characteristics of an exponential function such as domain, concavity, intercepts, asymptotes, and transformations, to construct a graph and to find an equation for an exponential function.
  14. Use a constant growth (decay) factor or rate to write an equation for an exponential function.
  15. Demonstrate an understanding of logarithms and their basic properties by evaluating and simplifying algebraic expressions involving logarithms
  16. Find and use the basic characteristics of a logarithmic function such as domain, concavity, intercepts and asymptotes to construct a graph and to find an equation for a logarithmic function.
  17. Solve logarithmic and exponential equations algebraically and estimate solutions graphically.
  18. Model and solve word problems (especially involving growth and decay) using exponential and logarithmic functions.
  19. Demonstrate an understanding of the sum and difference identities, the double angle identities and the half angle identities by using them to deduce other identities.
  20. Demonstrate an understanding of the inverse trigonometric functions and their inverse properties by evaluating and simplifying expressions involving these functions.
  21. Graph the basic 6 inverse trig functions using basic characteristics such as domain, intercepts and asymptotes.
  22. Estimate the solutions to trigonometric equations using graphing.
  23. Solve trigonometric equations using algebra.
  24. Model and solve word problems involving periodic behavior using the trigonometric functions.
  25. Analyze independently and set up application problems, thus applying problem solving technique to new situations.  Demonstrate the ability to anticipate and check their proposed solutions.
  26. Communicate effectively with the instructor and mathematical community using proper terminology verbally as well as proper written notation.

 


Tutoring:
Math/Science Study Center in Math 4 --computers and tutors.
MESA club in SA9--study guides a tutoring staff.

Technology:
You are required to have access to a graphing calculator of some sort. 

Prerequisites:
Trigonometry (MATH 5) with a grade of “C” or better, indicating familiarity with:

  1. Apply facts about angles, parallel lines and triangles to deduce further results about a geometric figure.
  2. Prove when two triangles are congruent or similar.
  3. Justify the lengths of sides in an isosceles right triangle and in a 30 – 60 – 90 triangle.
  4. Deduce the lengths of sides in quadrilaterals such as trapezoids and rectangles using basic definitions, Pythagorean Theorem, perimeter and/or area.
  5. Calculate the measure of a central angle in a circle using the measure of the intercepted arc and calculate the areas of geometric figures involving circles.
  6. Apply facts about plane geometric figures to deduce the surface area and volume of three dimensional geometric figures.
  7. Demonstrate an understanding of the concept of a function by identifying and describing a function graphically, numerically and algebraically.
  8. Calculate the domain and range for a function expressed as a graph or an equation.  From a graph, estimate the intervals where a function is increasing, decreasing and/or has a maximum or minimum value.
  9. Use and interpret function notation to find “inputs” and “outputs” from the graph, table and/or an equation describing a function
  10. From an equation, graph or table, calculate average rates of change by using a difference quotient or by using slopes of secant lines.  Analyze average rates of change to determine the concavity of a graph.
  11. Demonstrate an understanding of the six basic transformations of functions by graphing translated functions including the quadratic functions.
  12. Represent a word problem (especially a geometric problem) with a function.
  13. Determine when a function has an inverse (one to one functions) and find the inverse function graphically or algebraically.
  14. Form new functions through addition, subtraction, multiplication, division and composition.
  15. Recognize classical and analytic definitions of the trigonometric functions.
  16. Solve triangles using right triangle trigonometry, the law of sines and the law of cosines.
  17. Convert from radian to degree measure and vice-versa.
  18. Graph the 6 trigonometric functions and demonstrate the ability to change parameters and predict corresponding graphic behavior.
  19. Use trigonometric functions to model periodic behavior.
  20. Use the basic Pythagorean identities to deduce further identities.
  21. Recognize the basic features of the graphs of the conic sections (including parabolas, ellipses, circles and hyperbolas) and use those features to graph shifted conics.
  22. Graph a function defined by parametric equations.
  23. Represent quantities such as velocity and force with vectors using both a geometric and analytic description.
  24. Apply vectors and the properties of vectors to solving problems involving force and navigation.
  25. Analyze independently and set up application problems, thus applying problem solving technique to new situations.  Demonstrate the ability to anticipate and check their proposed solutions.
  26. Communicate effectively with the instructor and mathematical community using proper terminology verbally as well as proper written notation.

Homework
We’ll use the ILRN.COM homework system which is detailed in lecture.  

Tests and Grading
Most of your grade points will be determined by 5 chapter tests, whose dates are indicated in the tentative schedule.  The homework assignments are the crucial touchstone that will guide daily class discussions.  Everyone should come prepared to lead and/or follow a discussion on the topic for each scheduled meeting.  To be successful, you’ll want to have test scores whose weighted average exceeds 70% (C), 80% (B) or 90% (A) where pre-final exam points are awarded by the following weighting scheme:
5% attendance
15% homework
80% chapter tests and final exam

Overview
Precalculus is a course designed to do just what it suggests: prepare you for a first course in calculus.  This means learning many definitions and properties of basic functions and methods of solving equations, but it also—perhaps most importantly—means learning how to solve problems.  The basic outline for general problem solving devised by Polya is a four step program:
1. Understand the problem
2. Devise a plan for solving the problem
3. Carry out the plan
4. Look back
This outline is fleshed out on the next page.

Polya’s Four Step Program for Problem Solving

  1. UNDERSTANDING THE PROBLEM
    • Do you have good definitions for all the words in the problem statement?.
    • What is the unknown?  That is, what does the problem want to be produced? What are the given data? What condition(s) must be satisfied?
    • Is it possible to satisfy the conditions? Are the condition(s) sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?
    • Draw a figure or make a diagram to help conceptualize what is going on. Introduce suitable notation.
    • Could you restate the problem in an equivalent way that makes more sense to you?
  1. DEVISING A PLAN

Polya mentions (1957) that there are many reasonable ways to solve problems. The skill at choosing an effective strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:

  • Guess and check
  • Make and orderly list
  • Eliminate possibilities
  • Use symmetry
  • Consider special cases
  • Look for a pattern
  • Draw a picture
  • Solve a simpler problem
  • Use a model
  • Work backward
  • Use a formula
  • Be ingenious
  • Solve an equation
  • Use direct reasoning

Often times an algebra problem is best solved using the algebraic method:

    • Introduce a variable to represent the unknown.
    • Write related quantities in terms of this variable.
    • Set up an equation using the variable.
    • Solve the equation.
  1. CARRYING OUT THE PLAN
    • Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?   If your plan isn’t working, you may need to go back to step 2 and devise a better plan.
  2. Looking Back
    • Can you check the result?   Can you derive the solution differently?
    • Can you use the result, or the method, for some other problem?
    • Can you generalize your solution to a larger class of problems?

The Importance of Looking Back

Looking back may be the most important part of problem solving and is the best opportunity to learn from the problem. The phase was identified by Polya with admonitions to examine the solution by such activities as checking the result, checking the argument, deriving the result differently, using the result, or the method, for some other problem, reinterpreting the problem, interpreting the result, or stating a new problem to solve.

Teachers and researchers report, however, that developing the disposition to look back is very hard to accomplish with students. Some researchers have found little evidence of looking back among students--even when it is stressed by instruction. One teacher put it succinctly: "In schools, there is no looking back." This likely stems from a culture of mathematics education that holds “answer getting” as the paramount objective.  Also, pressure to cover a prescribed course syllabus; the absence of tests that measure processes and student frustration contribute to the tendency not to reflect on what a problem means in a larger context.

The importance of looking back should outweigh these difficulties.. It is often what you learn after you have solved the problem that really counts.