Problem 76 - Calculus and its applications

Question

Utility. Utility is a type of function that occurs in economics. When a consumer receives \(x\) units of a product, a certain amount of pleasure, or utility, \(U,\) is derived. Suppose the utility related to the number of tickets \(x\) sold for a ride at a county fair is $$ U(x)=80 \sqrt{\frac{2 x+1}{3 x+4}} $$ Find the rate at which utility changes with respect to number of tickets bought.

Step-by-Step Solution

Answer

Differentiate the given utility function to find the rate of change of utility with respect to the number of tickets sold, resulting in a formula for \( U'(x) \).

Step 1: Understand the Utility Function

The given utility function is \( U(x) = 80 \sqrt{\frac{2x + 1}{3x + 4}} \). This function represents the utility or enjoyment derived from selling \( x \) tickets for a ride at the fair.

Step 2: Differentiate the Utility Function

To find the rate of change of utility with respect to the number of tickets, differentiate \( U(x) \) with respect to \( x \). Use the chain rule and quotient rule for differentiation where necessary.

Step 3: Apply the Quotient Rule

Let \( f(x) = 2x + 1 \) and \( g(x) = 3x + 4 \). The derivative of \( h(x) = \frac{f(x)}{g(x)} \) is \( h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \). Calculate \( f'(x) = 2 \) and \( g'(x) = 3 \). Substitute these into the derivative formula.

Step 4: Differentiate the Square Root

The function is \( 80 \sqrt{h(x)} \). Apply the chain rule: the derivative of \( \sqrt{h(x)} \) with respect to \( x \) is \( \frac{1}{2\sqrt{h(x)}} \cdot h'(x) \). Multiply this by the constant \( 80 \).

Step 5: Combine Derivatives

Substitute \( h'(x) \) from Step 3 into the expression from Step 4 and simplify to find the derivative \( U'(x) \), which is the rate at which utility changes with the number of tickets \( x \).

Step 6: Simplify the Expression

After substitution, simplify the resulting expression to get a clear formula for \( U'(x) \). This will be the desired rate of change of utility with respect to the number of tickets.

Key Concepts

Utility Function in Economics

In economics, the utility function is a mathematical expression used to represent the satisfaction or enjoyment a consumer derives from consuming a certain quantity of goods or services. The whole idea is to model consumer preferences so that you can predict economic behavior.
By analyzing utility functions, economists can better understand how changes in price or income affect demand. In this context, we are considering the utility derived from buying tickets for a ride at a fair. The given utility function is expressed as:

• \[ U(x) = 80 \sqrt{\frac{2x + 1}{3x + 4}} \]

Here, \( U(x) \) indicates the utility derived from selling \( x \) tickets. Functions like this allow us to quantify qualitative factors like pleasure or satisfaction in mathematical terms.

Rate of Change

The rate of change in a function is a measure of how one quantity changes with respect to another quantity. In this case, we're talking about how the utility changes as the number of tickets sold increases.
It's crucial to evaluate the rate of change to understand how much additional utility or satisfaction each additional ticket sold brings.
When the utility function is differentiated with respect to \( x \) (the number of tickets), we find the rate at which utility is changing as more tickets are sold.

Chain Rule

The chain rule is an essential concept in calculus used to differentiate composite functions. A composite function is a function made up of two or more functions. For example, if we have a function \( z \) that can be expressed as a function of \( y \), which itself is a function of \( x \), we use the chain rule to find \( \frac{dz}{dx} \).
In our case, the utility function involves a square root of a function. Thus, when differentiating, we use the chain rule to break it into simpler steps.

• First, we handle the square root, which results in an intermediate derivative.
• Next, identify the inner function within the square root and differentiate it separately using the quotient rule.

This approach simplifies the process of finding how the outer function's rate of change connects to the inner function's rate.

Quotient Rule

The quotient rule in calculus is used to find the derivative of composite functions where one function is divided by another. The formula for the quotient rule is as follows:

• For \( h(x) = \frac{f(x)}{g(x)}, \) the derivative \( h'(x) \) is given by:
• \[ h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \]
• Apply it by identifying the top function \( f(x) \) and the bottom function \( g(x) \).

In our exercise, \( f(x) = 2x + 1 \) and \( g(x) = 3x + 4 \). Derivative calculations render \( f'(x) = 2 \) and \( g'(x) = 3 \).
Using the quotient rule simplifies the differentiation process of the function inside the utility function allowing us to evaluate changes in utility effectively.