Problem 41 - College Algebra and Calculus: An Applied Approach

Question

The demand function for a product is modeled by \(p=5000\left(1-\frac{4}{4+e^{-0.002 x}}\right)\) Find the price of the product if the quantity demanded is (a) \(x=100\) units and (b) \(x=500\) units. What is the limit of the price as \(x\) increases without bound?

Step-by-Step Solution

Answer

The price of the product for \( x=100 \) units and \( x=500 \) units will be calculated by substituting these values of \( x \) in the given function and solving the equation. The price of the product as \( x \) increases without bound is equal to the limit of the function as \( x \) approaches infinity.

Step 1: Substitute \( x = 100 \) in the function

The first requirement is to find the price when \( x=100 \). Substitute \( x=100 \) in the given function to get the corresponding price.

Step 2: Substitute \( x = 500 \) in the function

The second requirement is to find the price when \( x=500 \). Substitute \( x=500 \) in the given function to get the corresponding price.

Step 3: Calculate the limit as \( x \) increases without bound

Finally, take the limit of the given function as \( x \) approaches infinity. Here you have to use the limit properties correctly.

Key Concepts

Price Calculation

In economics, understanding how the price of a product changes with varying demand is vital. This task involves using a demand function, which is a mathematical representation to calculate the price based on demand. For the provided problem, the demand function is:

\[ p = 5000 \left( 1 - \frac{4}{4 + e^{-0.002 x}} \right) \]

To determine the price for specific quantities, we substitute these values directly into the function:

• For \( x = 100 \), replace \( x \) in the formula and compute the result to find the price when 100 units are demanded. This process will involve exponential calculation, which we'll address in the next section.
• Similarly, substitute \( x = 500 \) to find the price for 500 units. The function dynamically adjusts the price based on demand through this substitution.

This computation reveals how prices adapt as market demand changes based on the function given.

Exponential Function

An exponential function is a mathematical function depicting exponential growth or decay. Here, we're dealing with the term \( e^{-0.002x} \). The constant "\( e \)" is Euler's number, approximately 2.71828, a fundamental base of natural logarithms. The exponent in our function contains \((-0.002x)\) which affects how quickly the function changes as \( x \) increases or decreases.

• When substituting specific values of \( x \), like 100 or 500, calculate \( e^{-0.002x} \) to see its impact. A larger \( x \) results in a smaller exponential term, since \( e^{-0.002x} \) becomes a fraction approaching zero.
• This expression influences how the denominator in the demand function changes, leading directly to alterations in the price.

Understanding this exponential behavior helps predict the pricing effects as demand varies.

Limit at Infinity

Taking the limit of a function as a variable approaches infinity helps us understand its behavior over time. For our demand function, we want to see what happens to the price \( p \) as \( x \to \infty \).

The term \( e^{-0.002x} \) approaches zero as \( x \) becomes larger. Consequently, the denominator \( 4 + e^{-0.002x} \) gets closer to 4.

• As \( e^{-0.002x} \to 0 \), the fraction \( \frac{4}{4 + e^{-0.002x}} \to 1 \) since the denominator becomes simply 4.
• Substituting back into the demand function, we see \( 1 - 1 = 0 \). Thus, the expression simplifies to \( 5000 \times 0 = 0 \).

The conclusion is that the price \( p \) approaches 0 as \( x \) becomes infinitely large, meaning the product becomes free when the demand is unlimited. Understanding limits is crucial in predicting long-term market behavior.