User:Marty/UCSC Math 110 Fall 2008/Midterm study guide

Here are some tips for studying for the midterm:


 * Know what all the features are on the range topograph: rivers, lakes, wells, climbing, double-lakes, double-wells.
 * Know the connection between the discriminant and what features are found.
 * Know how to compute the discriminant using the $b^2 - 4ac$ formula, as well as from the topograph.

Make up your own problems! You can work with a partner and trade problems.


 * 1) Domain topographs:
 * 2) * Choose a primitive lax vector, such as $\pm (12,5)$.
 * 3) * Complete it to a lax basis.
 * 4) * Complete this to a lax superbasis.
 * 5) * Walk from this lax basis to home base.
 * 6) Quadratic Diophantine equations:
 * 7) * Make up a quadratic form, such as $Q(x,y) = 3x^2 + 5xy - 2y^2$.  (choose any three small integers as coefficients).
 * 8) * Topograph the quadratic form.
 * 9) * What features does the topograph have?
 * 10) * How many solutions does the equation $Q(x,y) = 7$ have?  (choose any small integer, such as $7$).
 * 11) * Write down a few of these solutions.
 * 12) Direct topographing:
 * 13) * Make up three integers $u,v,w$.  Put them "around a point" at home base in a topograph.
 * 14) * Find the quadratic form which yields those three integers.
 * 15) * Use the topograph to find the discriminant.
 * 16) * Use the $b^2 - 4ac$ formula to find the discriminant.
 * 17) Equivalence classes of quadratic forms:
 * 18) * Experiment with some different discriminants, positive, negative, and zero.
 * 19) * Choose a discriminant which is a multiple of $4$, or one more than a multiple of $4$.
 * 20) * Try to draw topographs with your chosen discriminant.
 * 21) * Make sure that the discriminant is what you choose, and the arithmetic progression rule is followed.
 * 22) * How many equivalence classes can you find?