The rate of change of the unit price p (in dollars) of Apex...

The rate of change of the unit price $p$ (in dollars) of Apex women's boots is given by $$p^{\prime}(x)=\frac{-250 x}{\left(16+x^{2}\right)^{3 / 2}}$$,where $x$ is the quantity demanded daily in units of a hundred. Find the demand function for these boots if the quantity demanded daily is 300 pairs $(x=3)$ when the unit price is $\$ 50 /$ pair.

Key Concepts

Demand Function

A demand function describes the relationship between the unit price of a product and the quantity demanded. It is a key concept in economics that illustrates how changes in price affect consumer demand, often modeled using differential equations to capture the dynamic interaction between price and quantity over time.

Integration Techniques

Integration is the process of finding the antiderivative of a function, which is essential for solving differential equations. Techniques such as substitution are often used when integrating complex functions, particularly when the integrand involves composite functions or algebraic expressions raised to fractional powers.

Differential Equations

A differential equation is a mathematical equation that relates a function with its derivatives, representing how a quantity changes in relation to another quantity. It is a foundational concept in calculus and applied mathematics, used to model real-world phenomena such as rates of change in economics, physics, and biology.

Initial Value Problems

An initial value problem involves finding a specific solution to a differential equation that satisfies a given initial condition. This initial condition is used to determine the constant of integration that appears when performing the antiderivative or integration process, ensuring the solution accurately fits the conditions of the original problem.

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