Homework Set 3
Multivariable Functions
PART 1: Suggested Individual Work
The Individual portion of homework is primarily for preparation for the Final Exam. These problems are not due and should not be submitted. Do them on your own and come to class with questions about particular problems. We will deal with those during our break out times.
Problems from Forgotten Calculus
Here a list of practice problems from Forgotten Calculus that you can try if you feel you need more practice with basic computational type of problems.
Unit #
/Problem #’s
28 / 1 / 3 / 4 / 5 / 67 / 8 / 9 / 10
Suggested Supplemental Individual Problems:
This is a list of problems that are more applied and can be done if you need more time and practice with this kind of problem.
1)Evaluating basic multivariable functions.
a. Evaluate the function at the points (3,3) and (-2,-5).
b. Evaluate the function at the points (10,1) and (26,1).
2)A company sells cameras and film to be used in the camera. Let x and y be, respectively, the number of cameras sold per day and the number of rolls of film sold per day and p1 and p2 the respective prices. Suppose and .
a.What is the Revenue Function ? What is the total revenue when 110 cameras and 150 rolls of film are sold?
b.Suppose the daily cost function for this company is What is the value of C(110,150)? Interpret your numerical answer?
c.What is the profit function P(x,y)?
d.What is the value and meaning of P(110,150)?
3)Douglas Cocks (1981) determined a Cobb-Douglas production function for Eli Lilly and Company. He found that , where A is a positive constant, P is a measure of physical output, L is a measure of labor input, and K a measure of physical input.
a.What is the marginal productivity of labor function?
b.What is the marginal productivity of capital function?
c.Compute and interpret its meaning.
4)For each of the following functions, find both first-order partial derivatives. Then evaluate the partial derivatives at the indicated points. Don’t forget: When you take partials, all the normal derivative rules (chain, product, quotient, etc.) apply.
a.at (2,3) .
b.at (3,4) .
5)For each of the following functions, find all four of the second order partial derivatives and verify in each case that .
a.
b.
c.
6)Suppose a Cobb-Douglas production function is given by
a.What is the marginal productivity of labor at L = 13 and K = 16? Interpret the result.
b.What is the marginal productivity of capital at L = 13 and K = 16? Interpret the result.
7)In 1978, Jeffrey Cullen found an engineering production function in the natural gas transmission industry to be
where Q is the output in cubic feet of natural gas, H is the station horsepower, d is the inside diameter of the transmission line in inches, and L is the length of the pipeline.
a.Find the three first-order partial derivatives of Q; QH, Qd, and QL. Write each answer with only positive exponents and simplify all expressions completely (with a calculator, if necessary).
b.Find the value of and interpret its meaning.
8)Find the critical points and determine whether each point is a relative max, relative min, or saddle point by using the second derivative test.
a.
b.
c.
9)A firm has two plants that manufacture the same item. Let x and y be the amounts produced by the respective plants with respective cost functions
and .
a.What allocation of production in the two plants will minimize the firm’s costs?
b.What is the minimum cost?
10)Use the Method of Lagrange Multipliers to solve the following. Parts (a) and (b) are for practice only. You only have to show work for part (c).
a.Maximize subject to the constraint
b.Minimize subject to the constraint
c.Maximize subject to the constraint
11)Suppose a CobbDouglas production function is given by , where x is the number of units of labor and y is the number of units of capital, and P is the number of units of a certain product that is produced. If each unit of labor costs $100 and each unit of capital costs $200, and the total expense for both is limited to $1,000,000, then.
a.What is the constraining equation in the form ?
b.Using the Method of Lagrange Multipliers, what is the number of units of labor and capital needed to maximize production?
c.What is the maximum number of units that can be produced?
12)A firm has three separate plants producing the same product and has an agreement to sell 1000 units of this product. Let x, y, and z be the number of items produced at the three plants which have the following separate cost functions:
a.What allocation of production among the three plants that will minimize the firm’s total cost.
b.What is the minimum cost of production?
13)Consider the functionsubject to the constraint . Using level/contour curves in Graphmatica, at what point does the function take on a max or min? Your answer should be in the form of (x,y,z).
PART 2: Required Team Mini-Projects
The following problems are to be done and submitted in the teams you were assigned to for the course. All members of the group should attempt all parts of all problems and the team should meet in and/or out of class to compare solutions and then decide what the best write-up should be. For each problem, one person on the team should be designated as the person who prepares the solution in Microsoft Word. This responsibility should be rotated equally throughout the quarter. All problems are due in print form by the end of class on the due date specified.
Please start each new problem on a new page using page breaks where appropriate.
Please type up computations and answers after each part on the following problems. Include this cover page with your work.
Group Number: Group Names:
Last Name / First NameResults and Comments
Problem 1:
CR+ / CR / CR / NCProblem 2:
CR+ / CR / CR / NCProblem 3:
CR+ / CR / CR / NCProblem 4:
CR+ / CR / CR / NCName of person who typed this problem up:
1.TONY POPLIN, THE FARMER
Please write up Problem #9 from Unit 28 in Forgotten Calculus.
a. Interpret :
Computations…Your interpretation…
b.Find
c.Find
d.Compute
Computations…Final result/value…=
e.Compute
Computations…Final result/value… =
f.Find
g.Compute
Computations…Final result/value… =
h.Use marginal analysis to report on the professor’s wife’s suggestion:
Computations…Final result and interpretation…
i.Use marginal analysis to report the professor’s youngest son’s suggestion:
Computations…Final result and interpretation…
j.What about the oldest son’s suggestion?
Computations…Final result and interpretation…
k.Explain why part (j.) cannot use “marginal analysis.” Give a plausible explanation for why the results of parts (i) and (j) do not add up to this result.
Your explanation…Name of person who typed this problem up:
2. MANUFACTURING PROFITS
A firm manufactures and sells two products, A and B, which sell for $15 and $10 each respectively. The total daily cost of producing x units of A and y units of B is
a.What is the daily revenue function ? Find the total daily revenue when 160 units of A and 50 units of B are sold.
Computations…The revenue function is
Final numerical result and interpretation…
b.What is the daily profit function ? Find the total daily profit when 160 units of A and 50 units of B are sold.
Computations…The profit function is
Final numerical result and interpretation…
c.Find and , then clearly interpret the values of and .
Computations…Interpretations…
d.What is the number of each product that will maximize their daily profit? Show all work. Use only algebra and calculus to do this problem. Graphmatica will not help here.
Computations…Final numerical results and interpretation…
e.What is the actual daily maximum profit?
Computations…Final numerical results and interpretation…
f.Compare your daily profit results in part (d) to those in part (b). Explain how producing and selling a smaller amount of units, as found in part (c) would produce a larger profit than was realized when selling a larger amount of units, as found in part (b).
Your comparison and explanation…g.Draw enough contour curves of the profit function to begin to get an idea of what the profit function surface might look like. (Use an initial grid range of x = 0 to 250 and y = 0 to 200 and adjust from there.) Do your best to try to describe the surface. Show the graph of contour curves. Explain how the graph of contour curves confirms your numerical result for the allocation of each kind of unit required to produce maximum profit.
Graph…Your description of the surface…
Your explanation of the graph and the numerical result connection…
Name of person who typed this problem up:
3.COBB DOUGLAS PRODUCTION FUNCTION WITH CONTRAINT
A company has hired you to help them maximize the daily production for a small facility that produces a single kind of unit. The daily production function for the facility is given by , where P is the number of units produced by the facility, L is the number of dollars spent on labor, and K is the daily amount spent to run and maintain the equipment (think of this as capital investment) in dollars. They have indicated that the total daily amount spent on L and K together cannot be more than $1000.
a.Use only contour curves to find the allocation of labor and capital that will maximize this facility’s daily production. Your solution should be laid out and explained clearly so that it is obvious what logic you are using to get to your final result.
Graph…Your explanation of the process and result…
b.Use only calculus and algebraic work to find the allocation of labor and capital that will maximize this facility’s daily production. Your solution should be laid out and explained clearly so that it is obvious what logic you are using to get to your final result. Make sure this closely matches the result you get in part (a). All algebraic and calculus work should be shown in steps.
Computations…Final numerical results and interpretation…
c.Because they are unsure how to weight their labor and capital allocations, they are currently spending $500 per day on each. In addition to the daily capital/equipment costs described in the original statement of the problem, there are other costs to produce x units given bydollars. They sell these items for $6.25 each to their customers. Based on your results in parts (a) and (b) of this problem and the information just provided, write a memo to the company with your recommendations. Your memo should include your basic findings, and a meaningful breakdown of what they should do and how it will affect their production and profits (assuming that they can sell all units produced). Your breakdown should include not only all pertinent dollar/unit amounts but also the percent increases in labor/capital allocations, production amounts, and profits that would result from the changes you propose. You may want to fill in the given table to help present your findings.
Your memo…Current / Optimal / % Change
L / 500
K / 500
Production
Revenue
Total Costs
Profit
Name of person who typed this problem up:
4. PARTIAL DERIVATIVES AND ADVERTISING BUDGETS
The economists Dorfman and Steiner (1954) used the function to describe the quantity of an item demanded when the total advertising budget is B and P is the price of a unit of advertising. (If it helps, think of P as the price to run one ad on a radio station and think of B as the total budget you’ve been given to run all the radio ads possible. For the purposes of this problem, we will assume P is the same no matter when you run the ad on the air.) The assumptions that Dorman and Steiner make are as follows:
and
a.Remembering what it means to take a partial derivative, clearly but carefully explain what it means in this situation to say that . Remember to take into account that some variables are being allowed to change while others are being held fixed as constants when you take partial derivatives.
Your explanation…b.Now explain why the assumption is reasonable in these circumstances? Clearly explain.
Your explanation…c.Remembering what it means to take a partial derivative, clearly but carefully explain what it means in this situation to say that . Remember to take into account that some variables are being allowed to change while others are being held fixed as constants when you take partial derivatives.
Your explanation…d.Now explain why the assumption is reasonable in these circumstances? Clearly explain.
Your explanation…Part 3: Selected Answers to Supplemental Individual Problems
1)
a.
b.
2)
a.$55,200 per day
b.It costs $20,100 to produce 100 cameras and 200 rolls of film per day.
c.
d.The company profits $60,300 when 100 cameras and 200 rolls of film are produced per day.
3)
a.
b.
4)
5)
a.b. c.
6)
a.When 12 units of Labor and 15 units of Capital are utilized, the next additional unit of labor will result in .407 additional units being
b.When 12 units of Labor and 15 units of Capital are utilized, the next additional unit of capital will result in .081 units being produced.
7)
8)
c.(0,0) is a saddle point; (1,1) is a minimum point.
9)Costs are minimized when plant one produces 200 units of x and plant two produces 100 units of y.
10)
a.At (-3,-1)
b.At (1,1)
c.At (-1,-3)
11)
a.
b.2000 units of labor and 4000 units of capital will maximize production.
c.The maximum number of units that can be produced is 348,216.
12)
a.Costs are minimized when x=200, y=100 and z=700.
b.The minimum cost of production is $10,600.
13)
a.
There is a relative minimum at x=8.5854,y=3.2195 and z=204.805.