15060 Data_ Models and Decisions by fionan

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									15.Math-Review




Monday 8/14/00

                 1
    General Mathematical Rules
 Addition
      Basics:    (a  b)  c  a  (b  c), a  b  b  a,
                   a  0  a, a  (  a )  0
      Summation Sign:
                          n

                         x
                         i 1
                                i    x1  x2       xn

      Famous Sum:        n
                                                           n(n  1)
                          i  1 2 
                         i 1
                                                  n 
                                                              2



15.Math-Review                                                        2
    General Mathematical Rules

  Multiplication
       Basics     (ab)c  a(bc),       ab  ba,
                   a1  a,      if a  0 a(a 1 )  a 1  1
                                                      a
       Squares:
                   ( a  b) 2       a 2  2ab  b 2 ,
                   ( a  b) 2       a 2  2ab  b 2 ,
                   (a  b)(a  b)  a 2  b 2
       Cubes:
                   (a  b)3  a3  3a 2b1  3a1b2  b3 ,
                   (a  b)3  a3  3a 2b1  3a1b2  b3

15.Math-Review                                                3
    General Mathematical Rules
Multiplication
     General Binomial Product:

                                                        
                                                  n
                                  ( a  b)  
                                             n          n
                                                             a i b n i
                                                        i
                                                 i 1
     Product Sign:
                                   n

                                  x
                                  i 1
                                         i    x1 x2        xn


     Distributive Property:
                                  a(b  c)  ab  ac


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    General Mathematical Rules
Fractions
     Addition:
                  a c ad  bc    a b ab
                                 
                  b d   bd       c c  c
     Product:
                  a
                        a b ab   ab a
                  c
                  d
                                 
                  b     c d cd   bd d



15.Math-Review                             5
    General Mathematical Rules
Powers                                        a times
     Interpretation:
                                      x  xx
                                        a
                                                         x,       what if a  (0,1) ??
     General rules:
                                            x 0  1,                   x1  x,
                 x a x b  x a b ,         x a y a  ( xy )a ,        ( x a )b  x ab ,
                                                              a
                         1                          1                 xa
                 x 1    ,                 xa      ,                   x a b
                         x                           x                xb
     Series:
                            n
                                                          1  a n 1
                           a
                           i 0
                                  i
                                       1 a  a 
                                               2
                                                     a  n

                                                           1 a
                            
                                                          1
15.Math-Review
                            ai  1  a  a 2 
                           i 0
                                                     
                                                         1 a
                                                              ,    if a  1
                                                                                           6
    General Mathematical Rules
Logarithms
     Interpretation:
          The inverse of the power function.        a x  c  x  log a c
     General rules and notation:
                          log e x  ln x       (where e  2.71828...),

                          log b 1  0, log b b  1
                                     log c a
                          log b a 
                                     log c b
                          log b cd  log b c  log b d
                          log b c n  n log b c
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    General Mathematical Rules
Exercises:
     We know that project X will give an expected yearly return of $20 M
      for the next 10 years. What is the expected PV (Present Value) of
      project X if we use a discount factor of 5%?



     How long until an investment that has a 6% yearly return yields at least
      a 20% return?




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                 The Linear Equation
 Definition:
                     y( x)  y  ax  c

 Graphical interpretation:
                            y




                                              a

                                          1
                                 -c/a             x
                            c




15.Math-Review                                        9
                 The Linear Equation
        Example: Assume you have $300. If each unit of stock in Disney
    Corporation costs $20, write an expression for the amount of money you
    have as a function of the number of stocks you buy. Graph this function.




        Example: In 1984, 20 monkeys lived in Village Kwame. There were
    10 coconut trees in the village at that time. Today, the village supports a
    community of 45 monkeys and 20 coconut trees. Find an expression
    (assume this to be linear) for, and graph the relationship between the
    number of monkeys and coconut trees.



15.Math-Review                                                                 10
                 The Linear Equation
 System of linear equations
                   2x – 5y = 12           (1)
                   3x + 4y = 20           (2)
 Things you can do to these equalities:
             (a)   add (1) to (2) to get:
                               5x – y = 32

             (b)   subtract (1) from (2) to get:
                               x + 9y = 8

             (c)   multiply (1) by a factor, say, 4
                              8x – 20y = 48
 All these operations generate relations that hold if (1) and (2) hold.

15.Math-Review                                                             11
                 The Linear Equation
 Example: Find the pair (x,y) that satisfies the system of equations:
                  2x – 5y = 12        (1)
                  3x + 4y = 20        (2)
   Now graph the above two equations.

 Example: Solve, algebraically and graphically,
                            2x + 3y = 7
                            4x + 6y = 12


 Example: Solve, algebraically and graphically,
                             5x + 2y = 10
                            20x + 8y = 40


15.Math-Review                                                           12
                 The Linear Equation

        Exercise: A furniture manufacturer has exactly 260 pounds of plastic
    and 240 pounds of wood available each week for the production of two
    products: X and Y. Each unit of X produced requires 20 pounds of plastic
    and 15 pounds of wood. Each unit of Y requires 10 pounds of plastic and
    12 pounds of wood. How many of each product should be produced each
    week to use exactly the available amount of plastic and wood?




15.Math-Review                                                             13
            The Quadratic Equation
 Definition:
                          y ( x)  y  ax 2  bx  c
 Graphical interpretation:
                                                           Can have only 1 or no root.
  y        When a>0          y    When a<0             y




  c
      r1       r2                 r1         r2
                      x                           x          r1                 x



15.Math-Review                                                                      14
          The Quadratic Equation

     Completing squares:
                                        2 b    b  b
                                                  2    2

                 y  ax 2  bx  c  a  x  x  2    c
                                           a   4a  4a
                                                    2
                                             b 
                                                         2
                                                   b
                                       a x     c
                                             2a  4a

         Another form of the quadratic equation:
                             y  k  a ( x  h) 2
         The point (h,k) is at the vertex of the parabola. In this case:
                              b                 b2
                           h ,          k c
                              2a                4a
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          The Quadratic Equation
     Example: Find the alternate form of the following quadratic
      equations, by completing squares, and their extreme point.

                          x2  x 6 ?
                          3x 2 8 x  4  ?




15.Math-Review                                                      16
          The Quadratic Equation
 Solving for the roots
      We want to find x such that ax2+bx+c=0. This can
       be done by:
           Factoring.
                 Finding r1 and r2 such that ax2+bx+c = (x- r1)(x- r2)
                                                Example:
                                                             x2  x 6  0
                                                             3x 2 8 x  4  0
           Formula

                            b  b 2  4ac
                 r1 , r2                      Example:      x2  x 6  0
                                 2a
                                                             3x 2 8 x  4  0
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          The Quadratic Equation

         Exercise: Knob C.O. makes door knobs. The company has estimated
    that their revenues as a function of the quantity produced follows the
    following expression:
                              f (q)  q 2  510 q  5000
 where q represents thousands of knobs, and f (q), represents thousand of
  dollars.
      If the operative costs for the company are 20M, what is the range in which the
       company has to operate?
      What is the operative level that will give the best return?




15.Math-Review                                                                      18
                            Functions
  Definition:
      For 2 sets, the domain and the range, a function associates for
       every element of the domain exactly one element of the range.
      Examples:
           Given a box of apples, if for every apple we obtain its weight we
            have a function. This maps the set of apples into the real numbers.

           Domain=range=all real numbers.
                  For every x, we get f(x)=5.
                  For every x, we get f(x)=3x-2.
                  For every x, we get f(x)=3 x +sin(3x)



15.Math-Review                                                                    19
                          Functions
  Types of functions
      Linear functions
      Quadratic functions
      Exponential functions:  f(x) = ax
        Example: Graph f(x) = 2x , and f(x) = 1-2-x.

           Example: I have put my life savings of $25 into a 10-year CD
             with a continuously compounded rate of 5% per year. Note
             that my wealth after t years is given by w = 25e5t. Graph this
             expression to get an idea how my money grows.



15.Math-Review                                                                20
                          Functions
  Types of functions
      Logarithmic functions
                              f(x) = log(x)
      Lets finally see what this ‘log’ function looks like:
                                            8


                                            6


                                            4

                                            2


                                            0
                    -8                 -3        2     7
                                            -2
                         f(x)=exp(x)
                         f(x)=ln(x)         -4

                                            -6

15.Math-Review                              -8                 21
         Convexity and Concavity
  Given a function f(x), a line passing through f(a) and f(b)
   is given by:
                  y( )  y  f (a)  (1  ) f (b),  a real number.

  Definition:
      f(x) is convex in the interval [a,b] if

                 f (a)  (1  ) f (b)  f (a  (1  )b),  [0,1].
      f(x) is concave in the interval [a,b] if
                 f (a)  (1  ) f (b)  f (a  (1  )b),  [0,1].

          Another definition is f(x) is concave if -f(x) is convex
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           Convexity and Concavity
          These ideas graphically:


      y        y  f (a)  (1  ) f (b)
                   ( f (a)  f (b))  f (b)


   f(a)                                                    f (a)  (1  ) f (b)
                   f(b)                         f(a)

                                                           f (a  (1  )b)        f(b)

  a            b                 x



                                                 a                                  b      x
                                                                   1 

15.Math-Review                                                                                 23

								
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