1. Farmer Jones, and his wife, Dr. Jones, decide to build a fence in their field, to keep the sheep safe. Since Dr. Jones is a mathematician, she suggests building fences described by y=2x^2 and y=x^2+8. Farmer Jones thinks this would be much harder than just building an enclosure with straight sides, but he wants to please his wife. What is the area of the enclosed region?
2. Sketch the region enclosed by y=e^4x, y=e^9x, and x=1. Find the area of the region.
3. Find the volume of the solid obtained by rotating the region bounded by y=6x^2, x=1, x=4 and y=0, about the xx-axis.
4. Find the volume of the solid formed by rotating the region enclosed by
y=e^3x+3, y=0, x=0, x=0.7
about the x-axis.
5. Suppose you deposit $3000 at 3% interest compounded continuously. Find the average value of your account during the first 2 years.
6. If a cup of coffee has temperature 98°C in a room where the ambient air temperature is 20°C, then, according to Newton’s Law of Cooling, the temperature of the coffee after t minutes is T(t)=20+78e−t/50. What is the average temperature of the coffee during the first 28 minutes?
7. Given: (x is number of items)
Demand function: d(x)=3920√x
Supply function: s(x)=5√x
Find the equilibrium quantity: items
Find the consumers surplus at the equilibrium quantity: $
8. Given: (x is number of items)
Demand function: d(x)=4205√x
Supply function: s(x)=5√x
Find the equilibrium quantity: items
Find the producer surplus at the equilibrium quantity: $
9. Given: (x is number of items)
Demand function: d(x)=300−0.3x
Supply function: s(x)=0.5x
Find the equilibrium quantity:
Find the consumers surplus at the equilibrium quantity:
10. Given: (x is number of items)
Demand function: d(x)=200−0.6x
Supply function: s(x)=0.2x
Find the equilibrium quantity:
Find the producers surplus at the equilibrium quantity:
11. Given: (x is number of items)
Demand function: d(x)=784−0.4x^2
Supply function: s(x)=0.6x^2
Find the equilibrium quantity:
Find the consumers surplus at the equilibrium quantity:
12. Given: (x is number of items)
Demand function: d(x)=588.7−0.3x^2
Supply function: s(x)=0.4x^2
Find the equilibrium quantity:
Find the producers surplus at the equilibrium quantity:
13. Suppose the demand function for a product is given by the function:
D(q)=−0.016q+54.4
Find the Consumer’s Surplus corresponding to q=650 units.
(Do no rounding of results until the very end of your calculations. At that point, round to the nearest tenth, if necessary. It may help you to sketch the demand curve, which crosses the horizontal at q=3,400)
Answer: dollars
14. Find the accumulated present value of an investment over a 10 year period if there is a continuous money flow of $10,000 per year and the interest rate is 1.3% compounded continuously.
15. A company is considering expanding their production capabilities with a new machine that costs $52,000 and has a projected lifespan of 7 years. They estimate the increased production will provide a constant $8,000 per year of additional income. Money can earn 1.1% per year, compounded continuously. Should the company buy the machine?
a. Select an answer Yes, the present value of the machine is greater than the cost by No, the present value of the machine is less than the cost by
b. $ over the life of the machine
16. Find the present value of a continuous income stream F(t)=20+7t, where t is in years and F is in thousands of dollars per year, for 30 years, if money can earn 2.5% annual interest, compounded continuously.
Present value = thousand dollars.
17. Given f(x,y)=−2x^2+4xy^6+3y^5, find
fxx(x,y) =
fxy(x,y) =
18. Find the critical point of the function f(x,y)=−4−7x−5x^2+6y−2y^2
19. Suppose that f(x,y)=x^4+y^4−xy
Then the minimum is
20. Find and classify the critical points of z=(x^2−8x)(y^2−6y)
Local maximums:
Local minimums:
Saddle points:
For each classification, enter a list of ordered pairs (x, y) where the max/min/saddle occurs. If there are no points for a classification, enter DNE.
21. A company manufactures 2 models of MP3 players. Let x represent the number (in millions) of the first model made, and let y represent the number (in millions) of the second model made.
The company’s revenue can be modeled by the equation
R(x,y)=50x+110y−4x^2−2y^2−xy
Find the marginal revenue equations
Rx(x,y)=
Ry(x,y)=
We can achieve maximum revenue when both partial derivatives are equal to zero. Set Rx=0and Ry=0 and solve as a system of equations to the find the production levels that will maximize revenue.
Revenue will be maximized when:
x =
y =
21. A chemical manufacturing plant can produce z units of chemical Z given p units of chemical P and r units of chemical R, where:
z=70p^0.9r^0.1
Chemical P costs $100 a unit and chemical R costs $700 a unit. The company wants to produce as many units of chemical Z as possible with a total budget of $140,000.
A) How many units each chemical (P and R) should be “purchased” to maximize production of chemical Z subject to the budgetary constraint?
Units of chemical P, p=
Units of chemical R, r =
B) What is the maximum number of units of chemical Z under the given budgetary conditions? (Round your answer to the nearest whole unit.)
Max production, z= units
22. An open-top rectangular box is being constructed to hold a volume of 300 in^3. The base of the box is made from a material costing 8 cents/in^2. The front of the box must be decorated, and will cost 9 cents/in^2. The remainder of the sides will cost 2 cents/in^2.
Find the dimensions that will minimize the cost of constructing this box.
Front width: in.
Depth: in.
Height: in.
