Application of Differentiation to Business

A demand equation models the relationship between price and units sold using historical sales data and market studies. For the Smart TV example, the equation combined a known data point (200 units at $350) with a slope of negative one-half to express units sold as a function of price. This mathematical relationship allows businesses to substitute into the revenue formula and find the price that maximizes total sales income.

The revenue function calculates total sales income by multiplying price by units sold. In the Smart TV case, substituting the demand equation into the revenue formula creates a single-variable quadratic function. This function reveals how revenue changes with price, enabling businesses to identify the optimal price point that generates maximum financial returns.

Calculus identifies maximum revenue by differentiating the revenue function and setting the derivative equal to zero. This process finds critical numbers where the revenue curve reaches its highest point. For the Smart TV store, solving the derivative equation yielded an optimal price of $225, demonstrating how calculus translates real business problems into mathematical solutions.

Revenue behavior reflects competing forces: higher prices increase per-unit income but reduce customer demand. The demand equation captures this inverse relationship through its negative slope. As price rises from zero, revenue initially grows, reaches a maximum point, then declines as lost sales volume outweighs price gains, creating the characteristic curve shape.

Marginal revenue, the derivative of the revenue function, measures how revenue changes with each additional unit sold. Setting marginal revenue to zero identifies where revenue stops increasing, pinpointing the optimal sales level. In the bakery example, this calculation determined that approximately 2.33 dollars per cupcake maximizes daily revenue, linking pricing directly to demand.

Businesses gather market data on price-demand relationships, then construct demand and revenue functions specific to their products. By applying calculus to find where marginal revenue equals zero, companies identify optimal prices without trial-and-error. This data-driven approach transforms pricing from guesswork into mathematical precision, enabling informed decisions that maximize financial outcomes.

Building a demand equation requires at least one known price-quantity data point and the slope of the price-demand relationship. The Smart TV example used sales records showing 200 units sold at $350 and a market study indicating a slope of negative one-half. These elements allow you to apply point-slope form and express units sold as a function of price for revenue calculations.