[Notes & HW Answers] 4.5 Applications to Marginality

[Notes & HW Answers] 4.5 Application to Marginality.pdf

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[Prepwork 4.5]

Q1. What is the relationship between (total) cost and marginal cost? 

A: Marginal cost is the derivative of total cost

Q2. Suppose C and R are the cost and revenue functions for a particular product.

Check all the statements below that are true.
A. The critical points of the profit function happen when 𝑀𝑅=𝑀𝐶 or one of these derivatives does not exist.
B. The profit is maximized when the difference between 𝑀𝑅MR and 𝑀𝐶MC is greatest.
C. Any time 𝑀𝑅=𝑀𝐶MR=MC, the profit is maximized.

Q3. Find the quantity 𝑞 that maximizes profit if the total revenue, 𝑅(𝑞), and total cost, 𝐶(𝑞) are given in dollars by 𝑅(𝑞)=5𝑞−0.002𝑞^2 and 𝐶(𝑞)=200+2.2𝑞, where 0≤𝑞≤2500 units.

A: q = 700

? They refer to the same thing  Marginal cost is the minimal possible cost There is no simple relationship between the two Total cost is the derivative of marginal cost 

y = f (x) such that f'(x) = 10x^9*e^x+x^10*e^x?

• A. x10ex+10

 

[HW 4.5]

Q1. The figure below shows cost (blue) and revenue (black) functions over the domain 0<𝑞<45 (in thousands of units).

A: For what production levels is the profit function positive? (18,38)
(Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10) .)

For what production levels is the profit function negative? (0,18),(38,45)
(Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10) .)

Estimate the production at which profit is maximized. 𝑞≈ 30 thousands of units.

Q2. The revenue from selling 𝑞 items is 𝑅(𝑞)=500𝑞−𝑞^2, and the total cost is 𝐶(𝑞)=100+14𝑞. Write a function that gives the total profit earned, and find the quantity which maximizes the profit.

A: Profit 𝜋(𝑞)= 500q−q^2−100−14q

Quantity maximizing profit 𝑞= 243

Q3. The table below shows cost, 𝐶(𝑞), and revenue, 𝑅(𝑞).

𝑞	0	500	1000	1500	2000	2500	3000
𝑅(𝑞)	0	1875	3750	5625	7500	9375	11250
𝐶(𝑞)	2750	4077.5	4741.25	5257.5	5773.75	6953.75	10125

A: At approximately what production level, 𝑞, is profit maximized?
𝑞≈ 2500 units

What is the price of the product?
price = 3.75 dollars

What are the fixed costs?
fixed costs = 2750 dollars

Q4. The average cost per item to produce 𝑞 items is given by

𝑎(𝑞)=0.01𝑞^2−1.2𝑞+43, for𝑞>0.

What is the total cost, 𝐶(𝑞), of producing 𝑞 goods?
𝐶(𝑞)= q(0.01q^2−1.2q+43)

What is the minimum marginal cost?
minimum MC = −5

At what production level is the average cost a minimum?
𝑞= 60
What is the lowest average cost?
minimum average cost = 7

Compute the marginal cost at 𝑞=60.
𝑀𝐶(60)= 7