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Marginal cost and unit cost

The marginal cost is the derivative of the cost function. It says roughly what it costs to make one more unit. The unit cost is the whole cost divided by the number of units, that is the average so far. The two are not the same. The unit cost is smallest where it equals the marginal cost. In the example below that happens exactly at 20 units.

Symbols

Example

$C(x) = 0.5x^2 + 10x + 200$. $C'(x) = x + 10$, so $C'(20) = 30$ NOK.

$A(20) = \dfrac{0.5 \cdot 20^2 + 10 \cdot 20 + 200}{20} = 30$ NOK.

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Part of Mathematics S1: Differentiation and optimisation.