1 15.math-review monday 8/14/00. 15.math-review2 general mathematical rules zaddition ybasics:...
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15.Math-Review
Monday 8/14/00Monday 8/14/00
15.Math-Review 2
General Mathematical Rules
( ) ( ), ,
0 , ( ) 0
a b c a b c a b b a
a a a a
Addition Basics:
1 21
n
i ni
x x x x
Summation Sign:
1
( 1) 1 2
2
n
i
n ni n
Famous Sum:
15.Math-Review 3
General Mathematical Rules
1 1
( ) ( ), ,
1 , if 0 ( ) 1a
ab c a bc ab ba
a a a a a a
Multiplication Basics
2 2 2
2 2 2
2 2
( ) 2 ,
( ) 2 ,
( )( )
a b a ab b
a b a ab b
a b a b a b
Squares:
3 3 2 1 1 2 3
3 3 2 1 1 2 3
( ) 3 3 ,
( ) 3 3
a b a a b a b b
a b a a b a b b
Cubes:
15.Math-Review 4
Multiplication General Binomial Product:
1
( )n
n i n i
i
n
ia b a b
General Mathematical Rules
1 21
n
i ni
x x x x
Product Sign:
acabcba )( Distributive Property:
15.Math-Review 5
Fractions Addition:
General Mathematical Rules
a b a b
c c c
a c ad bc
b d bd
ac
db
a b ab
c d cd a b a
b d d
Product:
15.Math-Review 6
General Mathematical Rules
Powers Interpretation:
times
what if (0,1) ?? , a
aax xx x
0 1
1
1, ,
, ( ) , ( ) ,
1 1, ,
a b a b a a a a b ab
a aa a b
b
x x x
x x x x y xy x x
xx x x
x x x
General rules:
12
0
11
1
nni n
i
aa a a a
a
2
0
11 , if 1
1i
i
a a a aa
Series:
15.Math-Review 7
General Mathematical Rules
Logarithms Interpretation:
The inverse of the power function. logxaa c x c
(where 2.71828...), log ln
log 1 0, log 1
log log
log
log log log
log log
e
b b
cb
c
b b b
nb b
ex x
b
aa
b
cd c d
c n c
General rules and notation:
15.Math-Review 8
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?
General Mathematical Rules
15.Math-Review 9
Definition:
Graphical interpretation:
The Linear Equation
( )y x y ax c
c
1
a
-c/a
y
x
15.Math-Review 10
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.
The Linear Equation
15.Math-Review 11
System of linear equations2x – 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.
The Linear Equation
15.Math-Review 12
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
The Linear Equation
15.Math-Review 13
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?
The Linear Equation
15.Math-Review 14
Definition:
Graphical interpretation:
The Quadratic Equation
2( )y x y ax bx c
y
x
When a<0
r2r1
y
x
When a>0
r2
c
r1
y
x
Can have only 1 or no root.
r1
15.Math-Review 15
Completing squares:
The Quadratic Equation
2 2
2
2
2 2
2
4 4
2 4
b b b
a a a
b b
a a
y ax bx c a x x c
a x c
2)( hxaky Another form of the quadratic equation:
a
bck
a
bh
4 ,
2
2
The point (h,k) is at the vertex of the parabola. In this case:
15.Math-Review 16
Example: Find the alternate form of the following quadratic equations, by completing squares, and their extreme point.
The Quadratic Equation
?483
?62
2
xx
xx
15.Math-Review 17
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)
The Quadratic Equation
0483
062
2
xx
xx Example:
a
acbbrr
2
4,
2
21
Formula
0483
062
2
xx
xx Example:
15.Math-Review 18
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:
The Quadratic Equation
5000510)( 2 qqqf 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 19
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)
Functions
15.Math-Review 20
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.
Functions
15.Math-Review 21
Types of functions Logarithmic functions
f(x) = log(x) Lets finally see what this ‘log’ function looks like:
Functions
-8
-6
-4
-2
0
2
4
6
8
-8 -3 2 7
f(x)=exp(x)
f(x)=ln(x)
15.Math-Review 22
Given a function f(x), a line passing through f(a) and f(b) is given by:
Convexity and Concavity
number. real a ),()1()()( bfafyy
.]1,0[ ),)1(()()1()( bafbfaf
.]1,0[ ),)1(()()1()( bafbfaf
Definition: f(x) is convex in the interval [a,b] if
f(x) is concave in the interval [a,b] if
Another definition is f(x) is concave if -f(x) is convex
15.Math-Review 23
These ideas graphically:
Convexity and Concavity
y
xba
f(a)
f(b)
)())()((
)()1()(
bfbfaf
bfafy
xba
f(a)
f(b)
)()1()( bfaf
))1(( baf
1