2 Optimization

Use fminsearch to solve the following unconstrained optimization problem. min x∈R4 f(x) = (x1 + 10x2)^2 + 5(x3 − x4)^2 + (x2 − 2x3)^4 + 10(x1 − x4)^4

Use the following as initial guess x0 = [3 -1 0 1] (x0 us a column not a row)

What is the minimizer you find, and what is the value of the objective function f(x) at that point?

Also, report the number of iterations taken to converge.

Numerical Analysis: Apply Newton’s method to find the roots of polynomial P(x) = x^3 + 3x^2 − 2x + 1. Find the convergence rate.

A spring-mass system has a spring constant of 3 Nm. A mass of 2 kg is attached to the spring, and the motion takes place in a viscous fluid that offers a resistance numerically equal to the magnitude of the instantaneous velocity. If the system is driven by an external force of 9cos(3t)−6sin(3t) N,determine the steady-state response in the form Rcos(ωt−δ).

The Regal Cycle Company manufactures three types of bicycles—a dirt bike, a mountain bike, and a racing bike. Data on sales and expenses for the past quarter follow: Total Dirt Bikes Mountain Bikes Racing Bikes Sales $ 927,000 $ 269,000 $ 406,000 $ 252,000 Variable manufacturing and selling expenses 473,000 117,000 204,000 152,000 Contribution margin 454,000 152,000 202,000 100,000 Fixed expenses: Advertising, traceable 69,600 8,200 40,500 20,900 Depreciation of special equipment 44,000 20,500 7,500 16,000 Salaries of product-line managers 113,900 40,100 38,700 35,100 Allocated common fixed expenses* 185,400 53,800 81,200 50,400 Total fixed expenses 412,900 122,600 167,900 122,400 Net operating income (loss) $ 41,100 $ 29,400 $ 34,100 $ (22,400) *Allocated on the basis of sales dollars. Management is concerned about the continued losses shown by the racing bikes and wants a recommendation as to whether or not the line should be discontinued. The special equipment used to produce racing bikes has no resale value and does not wear out. Required: 1. What is the financial advantage (disadvantage) per quarter of discontinuing the Racing Bikes? 2. Should the production and sale of racing bikes be discontinued? 3. Prepare a properly formatted segmented income statement that would be more useful to management in assessing the long-run profitability of the various product lines.

A concrete walk 3 feet wide is placed around the outside of a circular pond with diameter 15 feet.

What is the total cost of the walk if the cost per square foot is $1.31

Course: Math Modeling

A tank contains 8 L (liters) of water in which is dissolved 32 g (grams) of salt. A solution containing 2 g/L of the brine flows into the tank at a rate of 9 L/min, and the well-stirred mixture flows out at a rate of 5 L/min.

    A) Find an equation to model this system using A to represent the amount of salt in the tank and t to represent minutes.

    B) Find the amount of salt in the tank after 4 minutes.

Find pictures of the five Platonic solids. Imagine you are an ant that likes to walk along the edges of solids. For which of these solids is it possible to walk along each edge exactly once, ending at your starting point? Give justification, and the name(s) of the solid(s) solids where such a walk is possible.

Explain why diagonalization is the same as changing to a basis of eigenvectors, computing the linear transformation with respect to this basis, and then changing back to the standard basis. In addition, create your own example (with specific numbers) that illustrates your explanation. You should write in complete sentences, though you can certainly have equations and symbols as part of your sentences (and probably should).

solve the given system ofdifferential equation by elimination.

x’-2x-y=1

x+y’-4y=0

1. Prove that every complete lattice has a unique maximal element.

2. (ii) Give an example of an infinite chain complete poset with no unique maximal element.

3. (iii) Prove that any closed interval on R ([a, b]) with the usual order (≤) is a complete lattice (you may assume the properties of R that you assume in Calculus class).

4. (iv) Say that a poset is almost chain complete if every nonempty chain has an l.u.b. Give an example of an almost chain complete poset with no minimal element.

A container with mass m kg is dropped by a helicopter from height h km at time t=0, with zero velocity. from the outset, its fall is controlled by gravity and the force of air resitance, f(v)= -kv, where v is the current velocity of the container. in t seconds after the drop, a parachute opens, resulting in an increase of air resistance up to f(v) = -kv. determine the time t at which the container touches the ground. and its velocity at this moment. if m = 200 kg, h = 2000 m, t = 20 s, k = 10 kg/s, and k = 400 kg/s

In each of Problems 16 through 25, find all eigenvalues and eigenvectors of the given matrix. 16) A= ( 1st row 5 −1 2nd row 3 1) 23) A= (1st row 3 2 2, 2nd row 1 4 1 , 3rd row -2 -4 -1)

The following table represents a project with four activities. All times are in weeks.

A. According to the data in the table. What is the critical path?

B. According to the data in the table, what is the minimum expected completion time for the project?

C. According to the table there are four activities in the project. Assume the normal distribution is appropriate to use to determine the probability of finishing by a particular time. If you wished to find the probability of finishing the project in 21 weeks or fewer. It would be necessary to find the variance and then the standard deviation to be used with the normal distribution. What variance and standard deviation would be used?

D. According to the table, There are four activities in the project. Assume the normal distribution is appropriate to use to determine the probability of finishing by a particular time. What is the probability that the project is finished in 20 weeks or fewer.

You are the account manager in charge of Internet advertising at Impact Sales. Spending on Internet advertising has increased steadily over the last few years and you predict the growth will continue.

) Do you think a linear model is appropriate for this data? Explain clearly.

f) Find a quadratic model for this data since 2000.

g) Which model better fits your data? Support your answer with a scatter plot of the data and both models on the same set of axis.

h) Using the quadratic model what is the average rate of change between 2002 and 2004?

i) As the account manager what would you budget for Internet advertising in 2008?

TABLE:

Year Internet Advertising Spending (Billions)

2001 0

2002 0.3

2003 0.8

2004 1.9

2005 3

2006 4.3

2007 5.8