# 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

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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

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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

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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.

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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

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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?

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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
15.Math-Review                                                                   17
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?

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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)

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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

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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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