Final Review Sheet for Math 121, Spring 2009

Subjects we covered:

1. types of numbers, basic operations
   • powers -- negative, fractional
2. basic algebraic manipulations
   • long division of polynomials
3. linear equations: solve
4. quadratic equations: solve
   • the quadratic formula
   • completing the square
   • substitute, group, factor
5. complex numbers
   • complex solutions of quadratic equations
6. solving inequalities
   • interval notation
   • with absolute value
7. applied problems with linear equations, quadratic equations, inequalities:
   • mixtures
   • constant rate (speed)
8. basic geometry
   • distance formula
   • midpoint formula
   • slopes of parallel and perpendicular lines
9. graphing equations
   • intercepts (finding them)
   • symmetries, with respect to:
     • the x-axis
     • the y-axis
     • the origin
   • circles
10. functions
    • the funtional notation f(x)
    • (f+g)(x), (f-g)(x), (f×g)(x), (f/g)(x)
    • the VLT
    • domain, range
    • local max/min
    • increasing/decreasing
    • symmetry: even/odd
    • graphing -- transformations (shift U/D/L/R, stretch/squeeze)
    • library of examples:
      • powers
      • absolute value
      • exponentials
      • logarithms
      • piecewise-defined functions
11. quadratic functions
    • opens up or down
    • x- and y-intercepts (finding)
    • max/min/vertex
    • inequalities
    • applications (models with quadratic functions, finding max/min)
12. polynomials
    • definitions, basic form
    • roots, x-intercepts, zeroes, multiplicity
    • graphing: zoomed behavior near zeroes, smooth, continuous
    • the remainder theorem
    • the factor theorem
    • the maximum number of zeroes
13. rational functions
    • definitions, basic form
    • x-intercepts (zeroes)
    • VAs
    • HAs
    • OAs
14. log_b x
    • definition (inverse of exponential!), basic computation
    • properties (multiplcation, power and change of base rules)
    • solving equations (also with exponentials)
15. exponential growth/decay problems
    • growth of bacterial cultures, bank accounts
    • radioactive decay
16. composing functions
    • working with f_°g
    • inverses -- definition, graphing, computing, the HLT

Possible ways to study:

1. Go over the bold items above, and make sure you have some ideas about the theoretical content -- so, you recall:
   • the definitions
   • the theorems
   • common examples
   • pictures we usually drew
   • algorithms (recipes, procedures) we followed to analyze situations (to solve problems)
2. Go over problems we've done
   • in class
   • on tests (see links on the HW/schedule web page)
   • on quizzes (see links on the HW/schedule web page)
   • in the on-line HW (it's all open for revisiting)
   • in the paper HW (turn some of it in, why not?)
   in several ways:
   • first, skim many problems and make sure you have a fairly clear idea of what your problem-solving strategy will be (and this has to be active knowledge on your part -- it is not good enough to be able to nod and understand when your professor or tutor explains how to do a problem, you have to be able to produce from yourself the solution strategy without hints ... that will be the situation on the final [and at any future time you actually use this material]
   • do this 'skim and strategize' for at least a few problems from every type of problem we did in every topic we covered (see the list above on this page)
   • After making the broad survey of the whole course, pick at least one, moderately challenging problem from each topic to do carefully and completely, in order to check that you can actually carry through your problem-solving strategy when it counts.
   • Rinse and repeat... no, repeat the above survey and selective in-depth practice as much as possible.
3. General problem-solving strategies, for when you don't know how to get started:
   • Choose a variable (some variables) relevant to the problem and write clearly what they represent
   • Make a picture, if at all possible or relevant.
   • Clearly state all given information in terms of your variable(s).
   • Clearly state the goal in terms of your variable(s).
   • State any other relevant formulæ or results (theorems) which are relevant to this situation, in terms of your variable(s).
   • As you write out steps of a solution, make each step clear and complete, with a (word or two of) explanation and all equations or symbolic expressions clear, legal, and correct.