Page 52 - How To Solve Word Problems In Calculus
P. 52
Business and Economics
The marginal cost of producing an item is the rate at which its
cost changes with respect to the number of items produced.
Thus if C(x) is the cost of producing x items, the marginal cost
is C (x).
The marginal cost approximates the additional cost nec-
essary to produce one additional item. Thus C (x) is the ap-
proximate cost incurred to produce the (x + 1)st item.
Similarly, if R(x) and P(x) represent the revenue and
profit, respectively, in selling a quantity of x units, then R (x)
represents marginal revenue and P (x) the marginal profit.
EXAMPLE 14
Suppose the total cost of producing x items is given by the
function
3
2
C(x) = 0.001x + 0.025x + 3x + 5
Compute the marginal cost of producing the 51st item.
Solution
Since C (x) is the approximate cost incurred to produce
the (x + 1)st item we need to compute C (50).
2
C (x) = 0.003x + 0.05x + 3
2
C (50) = 0.003(50) + 0.05(50) + 3
= 7.50 + 2.50 + 3
= $13.00
The marginal cost is $13.00. For comparison, the exact cost to
produce the 51st item is
3
2
C(51) − C(50) = [0.001(51) + 0.025(51) + 3(51) + 5]
2
3
−[0.001(50) + 0.025(50) + 3(50) + 5]
= 355.676 − 342.50
= $13.176
39