Explain using words and graphs of how the supply curve is derived (where do they come from)?

In economics, the supply curve is derived from firms' profit maximization decision. In this decisions, firms choose the quantity of output to maximize total profit, given prices of output and inputs, and the production technology. By changing the price of the output, the firm will respond by choosing different quantities. By tracing these changes, we can derive the optimal quantity produced at different price levels, i.e., the supply curve.

Graphically, at a given price, the optimal amount of output is such that the marginal revenue of last unit produced is equal to the marginal cost, which is presented in the following diagram:

please see the attached file for the complete solution