15.MathReview

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15.Math-Review
Review 1
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Algebra

• Example After careful study our marketing
  team has estimated that the demand for knobs is
  related to the price as q 400 -10p . And,
  considering all the different producers of knobs
  the supply is estimated as q 150 15p .
• Find the markets equilibrium to estimate what
  should be the market price of knobs and the
  volume of sales.

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Algebra

• Example After a more careful study our
  marketing team has refined the estimates for the
  demand and supply to the following non-linear
  relations
• demand q e 9.1 p -0.10
• supply q e 2.3 p 1.5.
• Find the markets equilibrium to estimate what
  should be the market price of knobs and the
  volume of sales.

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Differentiation

• To differentiate is a trade.

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Stationary Points

• Example
• Consider the function defined over all x0, f(x)
  x - ln(x).
• Find any local or global minimum or maximum
  points. What type are they?

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Optimization

• Example Due to the interaction of supply and
  demand, we are able to affect p the price of door
  knobs with the quantity q of door knobs produced
  according to the following linear model
• p 100 - 0.1q
• Consider now variable operative costs 20q
• Maximize profit, with the consideration that the
  production level has to be at least 450 units due
  to contracts with clients.

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LP

• Example Write the constraints associated with
  the solution space shown

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3
2
1
-1
5
-1
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LP

• Example Graphically solve the following LP.
  Repeat replacing x 5 by x ? 5.

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LP

• Example Our company now produces two types of
  knobs. We can produce at most 300 knobs. The
  market limits daily sales of the first and second
  types to 150 and 200 knobs. Assume that the
  profit per knob is 8 for type 1 and 5 for type
  2. Try to maximize your profit.

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LP

• Example Our company can advertise our knobs by
  using local radio and TV stations. Our budget
  limits the advertisement expenditures to 1000 a
  month. Each minute of radio advertisement costs
  5 and each minute of TV advertisement costs
  100. Our company would like to use the radio at
  least twice as much as the TV. Past experience
  show that each minute of TV advertisement will
  usually generate 25 times as many sales as each
  minute of radio advertisement. Determine the
  optimum allocation of the monthly budget to radio
  and TV advertisements in order to maximize the
  estimated generation of sales.

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Equality Constrained Optimization

• Example Suppose we have the following model to
  explain q, the quantity of knobs produced
  qL0.3K0.9, where
• L Labor, and has a cost of 1 per unit of
  labor.
• K Capital, and has a cost of 2 per unit of
  capital.
• Interpret the model. Is it reasonable? (not the
  units, please)
• Find the mix of labor and capital that will
  produce q100 at minimum cost.

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Probability

• Example Suppose that of 100 MBA students in the
  first-year class, 20 of them have two years of
  work experience, 30 have three years, 15 have
  four years, and 35 have five years or more.
  Suppose that we select one of these 100 students
  at random.
• What is the probability that this student has at
  least four years of work experience?
• Suppose that you are told that this student has
  at least three years of work experience. What is
  the (conditional) probability that this student
  has at least four years of work experience?

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Probability

• Example It is a relatively rare event that a
  new television show becomes a long-term success.
  A new television show that is introduced during
  the regular season has a 10 chance of becoming a
  success. A new television show that is
  introduced as a mid-season replacement has only a
  5 chance of becoming a success. Approximately
  60 of all new television shows are introduced
  during the regular season. What is the
  probability that a randomly selected new
  television show will become a success.

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Tough examples to kill time

• Application of derivative LHopital rule.

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Tough examples to kill time

• Example
• Let us consider the function
• Obtain a sketch of this function using all the
  information about stationary points you can
  obtain.