Download Section 3.4 – Marginal Functions in Economics Marginal cost

Section 3.4 – Marginal Functions in Economics
Marginal cost- The actual cost incurred in producing an additional unit of a certain commodity given that a
plant is already at a certain level of operation.
The marginal cost is approximated by 𝐶 ′(𝑥 ) the rate of change of the total cost function evaluated at the
appropriate point. (The marginal cost function is the derivative of the total cost function).
NOTE: The word marginal is the same as saying derivative of.
To compute the marginal cost of the (𝑥 + 1)st unit:
Find
𝑪′(𝒙)
𝐶 ′(𝑥 ) measures the rate of change of the total cost function and provides a good approximation for the
actual cost to produce the (𝑥 + 1)st unit assuming that the 𝑥th unit has been produced.
EX2: The total weekly cost (in dollars) incurred by Lincoln Records in pressing x compact discs is given by the
following function. C(x) = 2000 + 2x - 0.0001x2 (0 x 6000)
(a) What is the actual cost incurred in producing the 941st and the 2161st disc? (Round your answers to the
nearest cent.)
(b) What is the marginal cost when x = 940 and 2160? (Round your answers to the nearest cent.)
Average cost function: the total production cost divided by the number of units produced
Suppose 𝐶 (𝑥 ) is a total cost function. Then the average cost function is
𝐶 ̅ (𝑥 ) =
𝐶(𝑥)
𝑥
EX 3: Custom Office makes a line of executive desks. It is estimated that the total cost for making x units of
their Senior Executive model is represented by the following function, where C(x) is measured in dollars/year.
C(x) = 94x + 280000
(a) Find the average cost function 𝐶̅ .
(b) Find the marginal average cost function 𝐶̅ ′
(c) What happens to 𝐶̅ (x) when x is very large?
Revenue function: the revenue realized by a company from the sale of x units of a certain commodity at a unit
price p
𝑅𝑒𝑣𝑒𝑛𝑢𝑒 = (𝑝𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡)(𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑢𝑛𝑖𝑡𝑠)
𝑅 (𝑥 ) = 𝑝𝑥
Marginal Revenue: the actual revenue realized from the sale of an additional unit of the commodity given that
the sale are already at a certain level. The marginal revenue is approximated by 𝑅′ (𝑥 ) which measures the rate
of change of the revenue function.
𝑅′(𝑥 ) measures the rate of change of the revenue function and provides a good approximation for the actual
revenue to be realized from the sale of the (𝑥 + 1)st unit assuming that the sales are already at the 𝑥th level.
EX4: Williams Commuter Air Service realizes a monthly revenue represented by the following function, where
R(x) is measured in dollars and the price charged per passenger is x dollars.
R(x) = 4800x - 120x2
(a) Find the marginal revenue R'(x).
(b) Compute the following values. R'(19), R'(20), R'(21)
(c) Based on the results of part (b), what price should the airline charge in order to maximize their revenue?
EX5: The management of Acrosonic plans to market the ElectroStat, an electrostatic speaker system. The
marketing department has determined that the demand for these speakers is represented by the following
function, where p denotes the speaker's unit price (in dollars) and x denotes the quantity demanded.
𝑝 = −0.01𝑥 + 870
0 ≤ 𝑥 ≤ 20000
Find the revenue function R.
EX6: The quantity of Sicard wristwatches demanded each month is related to the unit price by the following
equation, where p is measured in dollars and x in units of a thousand.
44
𝑝 = 0.05𝑥 2 +1
(0 ≤ x ≤ 20)
(a) Find the revenue function R.
(b) Find the marginal revenue function R'.
(c) Compute R'(6). (Round your answer to three decimal places.)
EX7: The weekly demand for the Pulsar 25 color LED television is represented by p, where p denotes the
wholesale unit price in dollars and x denotes the quantity demanded.
𝑝 = 540 − 0.1𝑥
0 ≤ 𝑥 ≤ 12000
The weekly total cost function associated with manufacturing the Pulsar 25 is given by C(x), where C(x)
denotes the total cost incurred in producing x sets.
𝐶 (𝑥 ) = 0.000007𝑥 3 − 0.02𝑥 2 + 250𝑥 + 87000
(a) Find the revenue function R. Find the profit function P.
(b) Find the marginal cost function C '. Find the marginal revenue function R '.
Find the marginal profit function P '.
(c) Compute the following values. (Round your answers to two decimal places.)
C '(2300)
R '(2300)
P '(2300)