Application of Differentiation to Business

Calculus offers essential techniques for businesses seeking to optimize pricing strategies and revenue. In this case, a bakery wants to determine the ideal price and daily sales volume to maximize revenue. By modeling how changes in price affect demand and revenue, the bakery can apply calculus to make data-driven decisions.

The demand function relates the price per cupcake to the number of cupcakes sold and captures how lower prices increase sales. Based on market data, the demand function can be represented as:

Where p(x) is the price per cupcake when x cupcakes are sold per day

The revenue function, which calculates total revenue, is represented as:

To maximize revenue, the bakery obtains the marginal revenue function, the derivative of the revenue function:

By solving for when marginal revenue equals zero, the bakery finds the optimal sales level of:

At this sales level, the price is approximately $2.33 per cupcake.

This example demonstrates how calculus links pricing, demand, and revenue into a cohesive optimization process, enabling businesses to make informed decisions that maximize financial outcomes using mathematical modeling.