# University of Nevada First Order Partial Derivatives Problem Set

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Hello, long time no see!Can you help me do these 21 questions. You can only insert the answers next to the equations.You do not need to write the process how you solved it, just answers please like last time.Thank you

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Problem Set 4
Find all first-order partial derivatives 𝑓𝑥 , 𝑓𝑦 for each of the following:
1. 𝑓(𝑥, 𝑦) = 4𝑥 2 + 2𝑥𝑦 − 8𝑦 2
2. 𝑓(𝑥, 𝑦) = 3𝑥 3 𝑦 3 − 2𝑥 2 + 3𝑦 + 40
3. 𝑓(𝑥, 𝑦) = 6𝑥 2 − 5𝑦 3 + 2𝑥 2 𝑦 3 + 100
Building off of 1-3, find second-order partial derivatives 𝑓𝑥𝑥 , 𝑓𝑦𝑦 for each of the following:
4. 𝑓(𝑥, 𝑦) = 4𝑥 2 + 2𝑥𝑦 − 8𝑦 2
5. 𝑓(𝑥, 𝑦) = 3𝑥 3 𝑦 3 − 2𝑥 2 + 3𝑦 + 40
6. 𝑓(𝑥, 𝑦) = 6𝑥 2 − 5𝑦 3 + 2𝑥 2 𝑦 3 + 100
7. Use the product rule to find f x and f y .
a. For 𝑓(𝑥, 𝑦) = (3𝑥 2 − 3𝑦)(2𝑥𝑦 + 4𝑦 2 ),
8. What is the general rule to find the values that optimize a two variable function?
9. What is the general rule to show that optimized values are a relative maximum?
10. What is the general rule to show that optimized values are a relative minimum?
11. A retail store estimated the following market response function giving sales (s) as a function
newspapers (y):
𝑠(𝑥) = 420𝑥 − 3𝑥 2 − 2𝑥𝑦 − 4𝑦 2 + 570𝑦 + 1725
Find the optimal x and y that will maximize the sales (show that it is indeed a maximum)
12.
Find the profit-maximizing prices and outputs for the multi-product monopolist with the
following demand and cost functions. Also find maximum profit.
𝑃𝑥 = 90 − 2𝑥 𝑃𝑦 = 180 − 4𝑦
𝐶(𝑥, 𝑦) = 3𝑥 2 + 𝑥𝑦 + 2𝑦 2 + 424
13. Assume that a firm produces sidewalk slabs at a constant average and marginal cost of \$3
each (assume no fixed costs). The weekly demand for slabs facing the firm is given by
Q=200 – 20P
Find the monopolist’s optimal price and quantity that would maximize profits.
14. Use Solver to find solutions to the following constrained optimization problems (include
printouts of the Solver models):
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑇𝐶(𝑥, 𝑦) = 6𝑥 2 + 14𝑦 2
𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑥 + 𝑦 ≥ 90
15. A retail store estimated the following market response function giving sales (s) as a function
newspapers (y):
s = 420 x − 2 x 2 − 3xy − 5 y 2 + 640 y + 1725
Find the optimal advertising plan given an advertising budget of \$180 per period,
and assuming the cost is \$1 per ad for circulars and \$4 per ad for newspapers. Also,
use the Lagrange multiplier to estimate the effect on sales of a \$1 increase in the
16. Suppose a firm has a production function (called a “Cobb-Douglas” production function)
given by
𝑄 = 2.5𝐾 0.6 𝐿0.4
Assume the budget is \$960, the price of capital (K) is 𝑃𝐾 = \$6, and the price of
labor (L) is 𝑃𝐿 = \$2. Determine the amounts labor and capital that maximize
output. Estimate the benefit of having an additional dollar in the budget.
17. A consulting firm for a manufacturing company arrived at the following Cobb-Douglas
production function for a particular product:
𝑄(𝑥, 𝑦) = 40𝑥 0.7 𝑦 0.3
where x is the number of units of labor and y is the number of units of capital required
to produce Q units of the product. Each unit of labor costs \$40 and each unit of
capital costs \$80. If \$400,000 is budgeted for production of the product, determine
how this amount should be allocated to maximize production, and find the maximum
production.
18. A store sells two brands of color print film. The store pays \$2 for each roll of brand A film
and \$3 for each roll of brand B film. A consulting firm has estimated the following daily
demand equations for these two competitive products:
𝑄𝐴 = 75 − 40𝑃𝐴 + 25𝑃𝐵
𝑄𝐵 = 80 + 15𝑃𝐴 − 30𝑃𝐵
where PA is the selling price for brand A and PB is the selling price for brand B. Use
calculus to determine how the store should price each brand of film to maximize daily
profits. What is the maximum daily profit from these two products? (Show your work.)
(Hint: write an expression that gives profit as a function of the two prices, and maximize
the function.)
19. Use calculus to find the first order conditions. Use solver then to solve the first order
conditions. Show your work for the first order conditions and include the relevant portion of
excel.
𝑓(𝑥, 𝑦) = −22𝑥 2 + 22𝑥𝑦 − 11𝑦 2 + 110𝑥 − 40𝑦 − 23
20. Use calculus to find the profit maximizing level of (a) output, (b) price, and (c) profit for a
two-product firm facing the following demand and cost functions (show your work):
𝑃1 = 475 − 6𝑄1
𝑃2 = 450 − 8𝑄_2
𝑇𝐶(𝑄1 , 𝑄2 ) = 4𝑄12 + 5𝑄1 𝑄2 + 3𝑄22 + 700
where quantities are expressed on a weekly basis.
21. Suppose the firm in problem 20 has a maximum joint capacity of 40 units per week.
a. Use Solver to find the profit maximizing level of output, price, and profit given this
production constraint.
b. Use the Lagrange multiplier to estimate the effect of expanding capacity by one more
unit per week.

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Tags:
product rule

partial derivatives

critical point

optimized values

market response

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