15.Math-Review

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15.Math-Review
Monday 8/14/00
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General Mathematical Rules
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General Mathematical Rules
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General Mathematical Rules

• Multiplication
• General Binomial Product

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General Mathematical Rules

• Fractions
• Addition

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General Mathematical Rules

• Powers
• Interpretation

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General Mathematical Rules

• Logarithms
• Interpretation
• The inverse of the power function.

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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
• 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.

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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
• Graphical interpretation

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The Quadratic Equation

• Completing squares

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The Quadratic Equation

• Example Find the alternate form of the
  following quadratic equations, by completing
  squares, and their extreme point.

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The Quadratic Equation

• Solving for the roots
• We want to find x such that ax2bxc0. This
  can be done by
• Factoring.
• Finding r1 and r2 such that ax2bxc (x-
  r1)(x- r2)

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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

• 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.
• Domainrangeall 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

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Convexity and Concavity

• Given a function f(x), a line passing through
  f(a) and f(b) is given by

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Convexity and Concavity

• These ideas graphically

y
f(a)
f(b)
x
a
b