Cost function and short run supply Suppose there is a number of identical firms producing diamonds. Let the total cost function for each firm be given by  $C(q, w)$, where q is the firm’s output level and w is the wage rate of diamond cutters.

I'm supposed to solve for a firm's (short-run) supply curve, given a wage rate $w$, and the industry's (total) supply curve).

I'm very confused with the relationship between the cost function and the supply curve. For part a this question, I get the marginal cost from taking the derivative of the cost function -- is the marginal cost also the supply curve? We know the cost function of the firm, but how can I arrive at the industry's supply curve?
Just like a Production function, in Economics, a Cost function is not an "engineering relation", but a "best response", reflecting minimum cost to produce a given quantity $q$.  

Cost-minimizing firms are at first assumed not to determine their scale of operations -what they decide upon is the demand for production inputs,given the inputs' prices.  

Still, by duality, the profit-maximizing condition should also hold -these are for-profit private companies, not, say, some public utility without a profit motive. If the firms are price-takers in the diamonds market, this relation is well-known, and provides the answer to a) (remember that there are functions, and then there are inverse functions).  

As for b), I see in the question that the suppliers are assumed identical.