Problem 62 - Precalculus : Building Concepts and Connections

Question

Applications In this set of exercises, you will use properties of functions to study real-world problems. Demand Function The demand for a product, in thousands of units, is given by \(d(x)=\frac{100}{x},\) where \(x\) is the price of the product, \((x>0) .\) Is this an increasing or a decreasing function? Explain.

Step-by-Step Solution

Answer

The function \(d(x)=\frac{100}{x}\) is a decreasing function as its derivative is negative (\(d'(x) = -\frac{100}{x^2}\)) for any \(x > 0\).

Step 1: Understanding the Function

The function given is \(d(x)=\frac{100}{x}\), representing demand as a function of price. Furthermore, it is given that the price, \(x\), is always positive (\(x > 0\)). The task is to determine whether the function increases or decreases as \(x\) increases.

Step 2: Deriving the Function

The derivative of the function needs to be calculated to determine if it's increasing or decreasing. The derivative of the function \(d(x)=\frac{100}{x}\) can be computed as \(d'(x) = -\frac{100}{x^2}\)

Step 3: Analyze the Derivative

As given, the value of \(x\) is always positive. So the denominator \(x^2\) of the derivative is always positive. The term -100 makes the entire derivative negative for any value of \(x > 0\). A negative derivative indicates that a function is decreasing.

Key Concepts

Derivative Analysis

When we talk about derivative analysis, we are referring to the process of studying how a function behaves, especially in terms of whether it increases or decreases. Derivatives are a fundamental tool in calculus, and they allow us to understand the rate at which a function changes. In the case of the demand function given by \(d(x)=\frac{100}{x}\), finding the derivative helps us identify the nature of the demand as price changes. The derivative of this function, \(d'(x) = -\frac{100}{x^2}\), is crucial here. Calculating the derivative involves applying differentiation techniques to understand how \(d(x)\) responds to small changes in \(x\). The derivative informs us about the slope of the tangent to the graph of \(d(x)\). When \(d'(x)\) is negative, it suggests that the slope is downward, indicating that \(d(x)\) decreases as \(x\) increases.

Decreasing Functions

A function is said to be decreasing on an interval if, as the input (x) increases, the output (the value of the function) decreases. In simpler terms, think of walking down a hill—the further you go, the lower you get. Mathematically, a function \(f(x)\) is decreasing if its derivative \(f'(x)\) is negative over the interval under consideration. For the demand function \(d(x) = \frac{100}{x}\), as the price \(x\) rises, the demand decreases. This relationship can be affirmed by the negative derivative \(d'(x) = -\frac{100}{x^2}\). This shows that with every increase in \(x\), the value of \(d(x)\) gets smaller, pointing towards a decreasing function. Understanding decreasing functions is helpful in many economic contexts, such as predicting what happens to sales when prices go up.

Real-World Applications

The concept of a demand function isn't just theoretical—it has real-world implications, especially in economics and business. Demand functions like \(d(x) = \frac{100}{x}\) are employed by companies to ascertain how consumer purchasing behavior changes with pricing.For instance, businesses use these functions to set pricing strategies that maximize revenue or profit. If a firm's goal is revenue maximization, understanding whether the demand is increasing or decreasing with pricing adjustments helps in determining the optimal price point. Additionally, these functions can guide marketing decisions by revealing potential changes in demand due to external factors, such as economic shifts or competitor actions.By utilizing concepts from derivative analysis and characteristics of decreasing functions, companies can make informed decisions to better align with consumer expectations and market conditions.