Application of Calculus - Marginal Cost and Marginal Revenue

Total Cost Function $C(x)$: This represents the total cost incurred in producing $x$ units of a commodity. It consists of two parts: Fixed Cost (FC), which is constant regardless of production levels, and Variable Cost (VC), which changes with the quantity produced. Visually, the Total Cost curve typically starts at a positive value on the vertical axis (the fixed cost) and slopes upward as $x$ increases.

Marginal Cost $MC$: This is the instantaneous rate of change of the total cost with respect to the number of units produced. Mathematically, it is the derivative of the cost function, $MC = \frac{dC}{dx}$. Visually, at any production level $x$, the marginal cost is represented by the slope of the tangent to the total cost curve at that point.

Average Cost $AC$: This is the cost per unit of production, calculated as $AC = \frac{C(x)}{x}$. Visually, the average cost at any point on the cost curve is the slope of the line segment connecting the origin to that point. The $AC$ curve is typically U-shaped, indicating that unit costs decrease due to economies of scale before eventually rising.

Relationship between $MC$ and $AC$: The Marginal Cost curve and Average Cost curve have a specific geometric relationship. When $MC < AC$, the average cost is decreasing. When $MC > AC$, the average cost is increasing. Consequently, the $MC$ curve intersects the $AC$ curve exactly at the minimum point of the $AC$ curve.

Total Revenue $R(x)$: This is the total amount received from selling $x$ units at a price $p$ per unit, given by $R(x) = p \cdot x$. If the price is constant, the revenue curve is a straight line passing through the origin. If the price depends on demand (where $p = f(x)$), the curve is often a downward-opening parabola.

Marginal Revenue $MR$: This is the rate of change of total revenue with respect to the quantity sold, calculated as $MR = \frac{dR}{dx}$. Visually, $MR$ is the slope of the tangent to the total revenue curve. In a competitive market where price is constant, the $MR$ curve is a horizontal line equal to the price $p$.

Profit Function $P(x)$: Profit is the difference between total revenue and total cost, $P(x) = R(x) - C(x)$. The break-even points occur where $R(x) = C(x)$, which are the points where the revenue and cost curves intersect on a graph, resulting in zero profit.

Marginal Average Cost $MAC$: This measures the rate of change of the average cost with respect to the output $x$. It is found by differentiating the average cost function: $MAC = \frac{d}{dx}(AC)$. If $MAC$ is negative, it indicates that the cost per unit is falling as production increases.