Application of Calculus - Cost, Revenue, and Profit Functions

Total Cost Function $C(x)$: This represents the total expense incurred by a company to produce $x$ units of a commodity. It consists of Fixed Cost ($FC$), which remains constant regardless of production level, and Variable Cost ($VC$), which changes with the number of units. Visually, the Total Cost curve typically starts at a point on the y-axis representing the fixed cost and rises as $x$ increases.

Average Cost ($AC$): This is the cost per unit of production, calculated as $AC = \frac{C(x)}{x}$. Visually, the Average Cost curve is generally U-shaped; it initially declines as fixed costs are spread over more units (economies of scale) and eventually rises due to inefficiencies at very high production levels.

Marginal Cost ($MC$): This represents the instantaneous rate of change of the total cost with respect to the number of units produced. Mathematically, $MC = \frac{dC}{dx}$. In a graph, the Marginal Cost at any point $x$ is the slope of the tangent to the Total Cost curve at that point.

Revenue Function $R(x)$: This is the total amount received from the sale of $x$ units of a product. If $p$ is the price per unit (demand function), then $R(x) = p \cdot x$. Visually, the Revenue curve often starts at the origin $(0,0)$ because zero sales result in zero revenue.

Marginal Revenue ($MR$): This is the rate of change of total revenue with respect to the number of units sold, given by $MR = \frac{dR}{dx}$. It represents the approximate revenue generated by selling one additional unit. Graphically, it is the slope of the Total Revenue curve.

Profit Function $P(x)$: Profit is the difference between total revenue and total cost, expressed as $P(x) = R(x) - C(x)$. The Break-even point occurs when $R(x) = C(x)$ or $P(x) = 0$. Visually, these are the points where the Revenue and Cost curves intersect.

Profit Maximization: A firm achieves maximum profit at a production level where Marginal Revenue equals Marginal Cost ($MR = MC$) and the second derivative of the profit function is negative ($\frac{d^2P}{dx^2} < 0$). Visually, this corresponds to the highest peak on the Profit curve graph.