Problem 7.3. Let f (x, y) = x6 + 3xy + y2 + y4.
(a) Show that f remains unchanged if you replace x by −x and y by −y. Hence,
if (x, y) is a critical point of f, so is (−x, −y). Thus, critical points other than
(0, 0) come in ± pairs.
140 7 Optimization in Several Variables
(b) Compute the partial derivatives of f . Show that solve applied directly to
the system fx = fy = 0 fails to locate any of the critical points except for (0, 0).
(c) Let’s compensate by eliminating one of the variables and then using solve
followed by double. First solve for y in terms of x in the equation fx = 0.
Substitute back into the formula for fy and then apply first solve and then
double. You should end up with three critical values of x, giving a total of
three critical points. Find the numerical values of their coordinates. (Be sure
you have set x and y to be real; otherwise you will also end up with many
irrelevant complex critical points.)
(d) Confirm the calculation of the critical points by graphing the equations fx =
0 and fy = 0 on the same set of axes (using fimplicit and hold on). You
should see exactly one additional pair of critical points (in the sense of (a)).
(e) Classify the three critical points using the second derivative test.
(f) Apply fminsearch to f with the starting values (1, 1) and (0, 0). Show
that in the first case you go to a minimum and that in the second case you stay
near the saddle point.

To please be done in excel: ABC Pty(Ltd) is a manufacturing company that has been listed on the Stock Exchange for the last 15 years. Approximately 35 percent of its sales are to government and 65 percent to private customers. The company has been growing erratically in recent years, but in real terms at a rate on average equal to that of the economy as a whole. Recent analyst’s reports suggest that the firm’s rate of growth might increase significantly in the near to mid future because of the government’s accelerated infrastructure investment program. However, ABC management believes that the analysts are overoptimistic in this regard and are worried about the effects of increased inflationary pressures. The company’s shares, which are largely institutionally held, are currently selling at 14 times earnings. The industry average PE ratio is 12. The company’s Return on Equity (ROE) was 17% p.a. compared to the industry average of 23%. The company’s most recent total dividend payout was $5m which represented a dividend cover of 2.5 times (the industry average is 2 times). The company has assets of $200 million and a debt to capital ratio of 20 percent (the industry average is 22 percent). ABC is budgeting for growth in the next 12m and needs an additional $25 million in capital over and above additions to retained earnings to support its projected level of business activties. a. Identify the stage of this company’s lifecycle and identify the normal sources of funding for such a stage. b. Calculate the company’s current Debt/Equity (D/E) ratio. What would it the new D/E ratio be if it raises debt to finance its growth?

QUESTION 21

Of the following fiscal programs, which has the biggest effect, per dollar, on aggregate demand?

QUESTION 22

One lesson of the Great Depression was that potential GDP could

QUESTION 23

Prior to the Great Depression, the principal fiscal policy was a balanced budget.

QUESTION 24

Raising taxes as an element of discretionary fiscal policy is intended to reduce aggregate demand, but it can also reduce aggregate supply if

QUESTION 25

Supply-side economics emphasized government policies to

1. Sharks are able to detect bioelectric fields of their prey. What other adaptations (morphological, behavioral, or physiological) do these sharks possess that make them excellent predators?
2. What are two features that put the octopus in the phylum Mollusca?
3. Which is more closely related to a human, an octopus or tunicate? Why?
4. What body form do sea jellies exhibit, medusa or polyp? What 3 or 4 components of the organism can you identify?
5. Why does a sea otter groom its fur so much?
6. How do they obtain their food preferences? How does it eat hard-shelled prey?
7. How do sea otters “stay warm”? How does this differ from the way in which seals stay warm?
8. . What type of coloration do penguins exhibit and how does this type of coloration aid them in their natural environment?
9. What other organisms at the aquarium have this type of coloration?
10. How is their body shape and morphology adapted for an aquatic habitat?
11. the giant sea bass, also known as the black sea bass. These fish were hunted to near extinction before laws were put in place to protect them. How does this fish move?
12. What do you think is this animal’s top predator as a full-sized adult?
13. What type of coloration do they exhibit?
14. What are two ways by which seals and sea lions differ anatomically?
15. Which set of flippers do sea lions use for propulsion? Seals?
16. How do you generally identify males vs. females?

An economist with a major bank wants to learn, quantitatively, how much spending on luxury goods and services can be explained based on consumers’ perception about the current state of the economy and what do they expect in the near future (6 months ahead).  Consumers, of all income and wealth classes, were surveyed.  Every year, 1500 consumers were interviewed.  The bank having all of the data from the 1500 consumers interviewed every year, computed the average level of consumer confidence (an index ranging from 0 to 100, 100 being absolutely optimistic) and computed the average dollar amount spent on luxuries annually.  Below is the data shown for the last 24 years.

Date                 X                     Y (in thousands of dollars)

1994                79.1                 55.6

1995                79                    54.8

1996                80.2                 55.4

1997                80.5                 55.9

1998                81.2                 56.4

1999                80.8                 57.3

2000                81.2                 57

2001                80.7                 57.5

2002                80.3                 56.9

2003                79.4                 55.8

2004                78.6                 56.1

2005                78.3                 55.7

2006                78.3                 55.7

2007                77.8                 55

2008                77.7                 54.4

2009                77.6                 54

2010                77.6                 56

2011                78.5                 56.7

2012                78.3                 56.3

2013                78.5                 57.2

2014                78.9                 57.8

2015                79.8                 58.7

2016                80.4                 59.3

2017                80.7                 59.9

Question:

1. Measure the strength of the linear association between consumers’ moods and the dollar amounts spent on luxury items.

An economist with a major bank wants to learn, quantitatively, how much spending on luxury goods and services can be explained based on consumers’ perception about the current state of the economy and what do they expect in the near future (6 months ahead).  Consumers, of all income and wealth classes, were surveyed.  Every year, 1500 consumers were interviewed.  The bank having all of the data from the 1500 consumers interviewed every year, computed the average level of consumer confidence (an index ranging from 0 to 100, 100 being absolutely optimistic) and computed the average dollar amount spent on luxuries annually.  Below is the data shown for the last 24 years.

Date                 X                     Y (in thousands of dollars)

1994                79.1                 55.6

1995                79                    54.8

1996                80.2                 55.4

1997                80.5                 55.9

1998                81.2                 56.4

1999                80.8                 57.3

2000                81.2                 57

2001                80.7                 57.5

2002                80.3                 56.9

2003                79.4                 55.8

2004                78.6                 56.1

2005                78.3                 55.7

2006                78.3                 55.7

2007                77.8                 55

2008                77.7                 54.4

2009                77.6                 54

2010                77.6                 56

2011                78.5                 56.7

2012                78.3                 56.3

2013                78.5                 57.2

2014                78.9                 57.8

2015                79.8                 58.7

2016                80.4                 59.3

2017                80.7                 59.9

Question:

1. Do you think that measuring the level of optimism is a good predictor for trying to forecast future spending on luxury items?  Explain why or why not.

Suppose a vertical pipe is to be used as part of a system to manually cycle nutrients upward from the floor of a lake. (Many lakes do this naturally, some do not. Green Lake, near Syracuse, NY, is one such lake.) A pump is to be installed on the lake floor at the base of the pipe. The base of the pipe will have a diameter of 9 cm. The nozzle of the pipe at the top will have a diameter of 4 cm. The lake is 59 m deep at the installation point. (Ignore any viscosity.)

f.)Write an equation that expresses the heat current H of conduction through the ice as a function of the thickness h of the ice sheet (of area A) already formed. Assume the air temperature is a constant -10 ?C, and that the temperature of the bottom of the ice sheet is 0 ?C.

(g) (1 point) Since H = dQ dt , use your answer to (f) to express the amount of heat dQ conducted through the ice sheet in time dt.

(i) (2 points) Express the amount of heat dQ that must be removed from the water at the bottom of the ice sheet to freeze the mass dm you found in (g).

(j) (2 points) Based on (g) and (i), set the expressions for dQ equal to each other to obtain a differential equation relating the heat that must be removed to freeze a new layer to the heat conducted through the ice sheet.

(k) (2 points) Separate variables, and integrate to find the thickness h of the ice sheet as a function of time t. (Note that h = 0 when t = 0

Four of your friends are planning a morning of boating on the Big Blue River. Two of them plan to start at the Rocky Ford Campground at 9:00 AM and drift down the river with the current in kayaks. The other two plan to start at the parking lot near the US-24 bridge, at the same time as the first pair, and travel upstream in a square-stern canoe with an outboard motor. You own a pickup truck with a high wheelbase, and you are friendly with all the farmers and property owners in the area, so they ask you to pick up them and their boats at the time and place where they will meet. You agree.

You research this section of the Big Blue River and find the following conditions that morning:

Depth: 12.8 feet

Average width: 200 feet

Flow: 13800 ft3/s

Water speed: 8.1 ft/s = 5.5 miles/hr

Water temperature: 78.5 degF

You check a map and find that distance between Rocky Ford Campground and the US-24 bridge parking lot is 4 miles “as the crow flies,” or 8 miles along the river. Your friends with the square-stern canoe say it can travel 2 miles on a still lake in 15 minutes.

(1) At what time should you pick up your friends?

(2) Where should you pick up your friends? (Express this answer in terms of distance downstream from Rocky Ford or upstream from US-24.)

Preston Concrete is a major supplier of concrete to residential and commercial builders in the Pacific Northwest. The company's general pricing policy is to set prices at $128 per cubic yard. Deliveries for 2019 were 380,000 cubic yards, and total costs were:

$4,218,000 of the estimated total yard operation costs were variable, and all of the administrative costs were fixed. In addition to the costs above, estimated fixed delivery costs were $190,000 for the year, and estimated variable delivery costs were $6.00 per mile and $42.50 per truck hour. The rate per mile reflects the fact that more miles result in more gas, oil, and maintenance. The rate per truck hour reflects the fact that trucks that are waiting at a jobsite are kept running (so the concrete mix won't solidify), and drivers continue to get paid during that time.

Near the end of 2019, Fairview Construction Company asked for a delivery of 4,600 cubic yards of concrete but was unwilling to pay the regular price; it was only willing to pay $82 per cubic yard. Preston estimated that the job would require 7,200 miles of driving and 290 truck hours. The housing market in the Pacific Northwest had slowed during recent months, leaving Preston with enough capacity to fill the order, but its sales manager was reluctant to commit to such a reduced price.

REQUIRED

If Preston accepted the offer, what would the profit or loss be (enter a loss as a negative number)?