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Application Of Derivatives

Question

The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.005x³ - 0.02 x² + 30 x +5000. Find the marginal cost when 3 units are produced, whereby marginal cost we mean the instantaneous rate of total cost at any level of output.

Solution

Given,

C(x) = 0.05x³ - 0.02 x² + 30 x +5000

Marginal cost (C_M),

$\frac{d}{dx} (C (x)) = 0.005 x 3 x^{2} - 0.02 x 2 x + 30 ∵ x = 3 C_{M} = 0.005 x 3 x 9 - 0.02 x 2 x 3 + 30 = 0.135 - 0.12 = 30.015$