PREREQUISITES
=============

For math 75 - you should know (or at least, be able to regenerate
after some review) precalculus mathematics things like this:

                                         2
  * how to solve a quadratic equation  Ax  + Bx + C = 0:
                       2
           -B +- sqrt(B  - 4 A C)
      x =  -----------------------
                     2A

(WARNING I have written +- to mean + OR - here; really the + should be
written ABOVE the - but "computerese")
equivalent expression (know why it is equivalent?):
                   2C
      x =  ----------------------
                       2
           -B +- sqrt(B  - 4 A C)

  * how to solve systems of linear equations (Gaussian elimination)

  * What is "mathematical induction"?  Can you prove that

       N
     -----
     \
      >      k   =  (N+1) N / 2   ?
     /
     -----
      k=0

  * how to manipulate fractions and algebraic expressions & equations,
  e.g.
         A     C    A D + C B
        --- + --- = ---------
         B     D       B D

         A     C    A D - C B
        --- - --- = ---------
         B     D       B D

         A    C    A C
        ---  --- = ---
         B    D    B D

          A       C     A D
        (---) / (---) = ---
          B       D     B C

                        2   2
          (x-y)(x+y) = x - y

  * how to draw a plot (graph) of f(x)

  * How to do geometry with Cartesian coordinates, e.g.
                                2          2                 2         2
      Distance   = sqrt( (x -x )  + (y -y )  ) = sqrt( DeltaX  + DeltaY  )   
              12           1  2       1  2

                           x  x   +   y  y
                            1  2       1  2            
      cos(Angle  ) = -------------------------------
               12           2    2          2    2
                     sqrt( x  + y )  sqrt( x  + y  )
                            1    1          2    2

      rotate the XY coordinate system (anticlockwise) by angle t 
            [ cos(t)  sin(t)] [X]  <-- 2x2 matrix times 2-vector
            [-sin(t)  cos(t)] [Y]

  * about trig functions and solving triangles, esp. right triangles
                  sin(a+b) = sin(a) cos(b) + sin(b) cos(a)
    |\            cos(a+b) = cos(a) cos(b) - sin(a) sin(b)
    | \h
   b|  \                                           2        2
    |  t\                       sin(t)       sin(t) + cos(t)  = 1 
    +---(*  <--  b/a = tan(t) = ------
      a                         cos(t)    b/h=sin(t)    a/h=cos(t)

for a triangle with sides A,B,C, opposed angles a,b,c:

                               2    2    2
             Law of cosines:  C  = A  + B - 2AB cos c

                            sin a   sin b   sin c
             Law of sines:  ----- = ----- = -----
                              A       B       C

  * about the binomial formula for powers, and about Pascal's triangle
               2    2         2
          (x+y)  = x + 2xy + y

               3    3        2      2      3
          (x+y)  = x  + 3 x y  + 3 x  y + y

               n   n        n-1
          (x+y) = x  + n x y     + ... 

                     n   k  n-k            n-1      n
              ... + ( ) x  y    + ... + n x    y + y  .
                     k

         1               n! = 1 . 2 . 3 ... n
        1 1
       1 2 1                       n       n!
      1 3 3 1      (n choose k) = ( ) = ---------
     1 4 6 4 1                     k    (n-k)! k!

  * exponentials and logarithms

              L
     If  y = B     then  L = log  y.  And vice versa.
                                B  
          m    m  m     0          u-w   u   w
     (A B)  = A  B.    B = 1.     B   = B / B.

      u+w   u  w       -u        u     -1          1/2               u  w    uw
     B   = B  B  .    B   = 1 / B  .  B  = 1/B.   B    = sqrt(B).  (B  )  = B.

     log(x y) = log x + log y .    log  x = ( log  x ) / log  B.
                                      B          A          A
                                        p
     log(x/y) = log(x) - log(y).   log(X ) = p log(X).   log(1)=0.

  * math common sense - how to check your results & avoid
    answers which are clearly insane.