Cost and Revenue Functions from Marginal Functions

Recovering total cost from marginal cost

The marginal cost $MC(x)=\frac{dC}{dx}$ is the extra cost of producing one more unit at output level $x$. Since integration reverses differentiation, the total cost function is recovered by integrating:

where the constant $K$ is found using the condition $C(0)=\text{Fixed Cost}$ — the cost incurred even at zero output (rent, fixed salaries, etc.).

Recovering total revenue from marginal revenue

Similarly, the marginal revenue $MR(x)=\frac{dR}{dx}$ integrates to the total revenue function:

but here $K=0$ always, since no output means no revenue: $R(0)=0$.