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15.Math-Review

15.Math-Review. Review 1. 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 .

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15.Math-Review

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  1. 15.Math-Review Review 1

  2. 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 market’s equilibrium to estimate what should be the market price of knobs and the volume of sales.

  3. 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 market’s equilibrium to estimate what should be the market price of knobs and the volume of sales.

  4. Differentiation • To differentiate is a trade….

  5. Stationary Points • Example: • Consider the function defined over all x>0, f(x) = x - ln(x). • Find any local or global minimum or maximum points. What type are they?

  6. 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.

  7. LP • Example: Write the constraints associated with the solution space shown: 5 3 2 -1 1 5 -1

  8. LP • Example: Graphically solve the following LP. Repeat replacing x = 5 by x  5.

  9. 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.

  10. 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.

  11. Equality Constrained Optimization • Example: Suppose we have the following model to explain q, the quantity of knobs produced: q=L0.3K0.9, 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 q=100 at minimum cost.

  12. 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?

  13. 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.

  14. Use this rule to find a limit for f(x)=g(x)/h(x): Tough examples to kill time • Application of derivative: L’Hopital rule.

  15. Sketch the function Hint: for this we will need to know that the ex‘beats’ any polynomial for very large and very small x. 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.

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