The total cost $C (x)$ associated with the production of $x$ units of an item is given by $C (x) = 0.005 x^{3} - 0.02 x^{2} + 30 x + 5000$ .
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.

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Solution

Given that
$C (x) = 0.005 x^{3} - 0.02 x^{2} + 30 x + 5000$
The marginal cost is the instantaneous rate of change of total cost at any level of output.
$∴$ Marginal Cost at any $x$ level of output = $C^{'} (x)$
$C^{'} (x) = 0.015 x^{2} - 0.04 x + 30$
Marginal Cost when $3$ units are produced,
= $C^{'} (x) a t x = 3$
= $C^{'} (3)$
= $0.015 (3)^{2} - 0.04 (3) + 30$
= $0.135 - 0.12 + 30 = 30.015$

Marginal Cost when $3$ units are produced is $30.015$ .

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