Transcript 15.Math-Review

15.Math-Review
Monday 8/14/00
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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
Famous Sum:
n
 x1  x2 
 i  1 2 
i 1
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i
 xn
n 
n(n  1)
2
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General Mathematical Rules
Multiplication
Basics
(ab)c  a(bc),
a1  a,
Squares:
ab  ba,
if a  0 a(a 1 )  a 1a  1
( 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
i 1

n
i
a i b n i
Product Sign:
n
x
i
 x1 x2
xn
i 1
Distributive Property:
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a(b  c)  ab  ac
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General Mathematical Rules
Fractions
Addition:
a c ad  bc
 
b d
bd
a b ab
 
c c
c
Product:
a
c
d
b
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a b ab


c d cd
ab a

bd d
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General Mathematical Rules
Powers
a times
Interpretation:
x  xx
a
General rules:
x a x b  x a b ,
x 1
x,
x 0  1,
x1  x,
x a y a  ( xy ) a ,
( x a )b  x ab ,
a
1
 ,
x
what if a  (0,1) ??
xa
1
  ,
 x
xa
a b

x
xb
Series:
n
a
i
 1 a  a 
i
 1  a  a2 
2
i 0

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a
i 0
1  a n1
a 
1 a
1

, if a  1
1 a
n
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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
 log b 1  0,
(where e  2.71828...),
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
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-c/a
x
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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.
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The Linear Equation
 System of linear equations
2x – 5y = 12
3x + 4y = 20
(1)
(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
3x + 4y = 20
(1)
(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
y
When a>0
y
When a<0
c
r1
r2
r1
x
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r2
x
r1
x
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The Quadratic Equation
 Completing squares:
2
2


b
b
b
y  ax 2  bx  c  a  x 2  x  2    c
a
4a  4a

2
b 
b

 a x     c
2a 
4a

2
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
h ,
2a
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b2
k c
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  ?
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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:
3x 2 8 x  4  0
Formula
b  b 2  4ac
r1 ,r2 
2a
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x2  x 6  0
Example:
x2  x 6  0
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  510q  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.
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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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-8
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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)  (1  ) f (b)
f(a)
f(a)
f(b)
f (a  (1  )b)
a
b
x
a
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f(b)

1 
b
x
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