Application of Calculus - Application in Commerce and Economics

Cost Functions and Marginal Cost: The Total Cost function $C(x)$ represents the total expenditure for producing $x$ units, consisting of Fixed Costs (constant) and Variable Costs. Marginal Cost ($MC$) is the instantaneous rate of change of total cost with respect to the number of units produced, calculated as $MC = \frac{dC}{dx}$. Visually, $MC$ represents the slope of the tangent to the Total Cost curve at any point $x$.

Revenue and Demand: Total Revenue $R(x)$ is the total amount received from selling $x$ units at price $p$, given by $R(x) = p \cdot x$. The demand function $p = f(x)$ shows the relationship between price and quantity. Graphically, the demand curve is usually downward-sloping, indicating that as price decreases, the quantity demanded increases.

Profit Maximization: Profit $P(x)$ is the difference between Total Revenue and Total Cost, $P(x) = R(x) - C(x)$. Profit is maximized at a production level $x$ where the Marginal Revenue ($MR$) equals Marginal Cost ($MC$), provided the second derivative of profit is negative ($P''(x) < 0$). On a graph, this occurs where the vertical distance between the revenue curve and the cost curve is greatest.

Average Cost and its Relationship with MC: Average Cost ($AC$) is the cost per unit, calculated as $AC = \frac{C(x)}{x}$. The $AC$ curve is typically U-shaped. A key geometric property is that the Marginal Cost ($MC$) curve always intersects the Average Cost ($AC$) curve at its lowest point (the minimum average cost).

Price Elasticity of Demand: This measures the responsiveness of the quantity demanded to a change in price. It is defined as $\eta = -\frac{p}{x} \cdot \frac{dx}{dp}$. If $|\eta| > 1$, demand is elastic (sensitive to price changes); if $|\eta| < 1$, demand is inelastic; and if $|\eta| = 1$, it has unit elasticity.

Break-even Point: This is the production level where Total Revenue equals Total Cost, resulting in zero profit ($P(x) = 0$). Visually, these are the points where the $R(x)$ and $C(x)$ graphs intersect. A company starts making a profit only after surpassing the first break-even point.

Marginal Revenue and Elasticity Relation: There is a direct relationship between $MR$, Price ($p$ or Average Revenue), and Elasticity ($\eta$), expressed as $MR = p(1 - \frac{1}{\eta})$. This shows that if demand is unit elastic ($\eta = 1$), then $MR = 0$, meaning total revenue is at its maximum.