The total cost $C (x)$ in Rupees, 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

As given, marginal cost is the instantaneous rate of change of total cost at any level of output.
Let $M . C$ be marginal cost, so

$M . C = \frac{d C}{d x}$

It is given $C (x) = 0.005 x^{3} - 0.02 x^{2} + 30 x + 5000$

$M . C = \frac{d}{d x} (0.005 x^{3} - 0.02 x^{2} + 30 x + 5000)$

$\Rightarrow M . C = 0.005 (3 x^{2}) - 0.02 (2 x) + 30$

When $3$ units are produced, putting $x = 3$ ,

we get

$\Rightarrow M . C = 0.005 (27) - 0.02 (6) + 30$

$\Rightarrow M . C = 0.015 + 30 = 30.015$

Hence, the required marginal cost is $R s .30 .02$ (nearly).

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