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Question:

The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.05x³ - 0.02x² + 30x + 5000. Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output.

Solution:

Since it is given that marginal cost is the instantaneous rate of change of total cost, we have

→dC/dx

→dC/dx = 3 × 0.05 × x² - 2 × 0.02 × x + 30 = 0.15x² - 0.04x + 30

Given in the question, the marginal cost of 3 units is to be calculated, so x = 3.

→Marginal Cost = 0.15(3)² - 0.04(3) + 30 = 1.35 - 0.12 + 30 = 31.23