Total cost from marginal cost

Statement

The marginal cost is $C'(q)=25+30q-9q^{2}$ (in €) and the fixed cost is $55\ \text{€}$. Find the total cost function $C(q)$.

Solution

Total cost is the antiderivative of marginal cost: $C(q)=\int (25+30q-9q^{2})\,dq=25q+15q^{2}-3q^{3}+c.$ At $q=0$ the cost equals the fixed cost: $C(0)=c=55$. Hence $\boxed{C(q)=25q+15q^{2}-3q^{3}+55}.$