Problem 76 - Brief Applied Calculus

Question

BUSINESS: Wholesale Trade Output Index From 2001 to 2009, the wholesale trade output index was approximately $$ W(x)=\frac{-151 x+1668}{x^{2}-17.4 x+88}+77.4 $$ units relative to 100 units in 2002 where \(x\) is the number of years after 2000 . Differentiating using the quotient rule, find \(W^{\prime}(4)\) and \(W^{\prime}(8)\) and interpret your answers.

Step-by-Step Solution

Answer

Evaluate the derivative at points; interpret as rates of change in wholesale trade output in years 4 and 8 after 2000.

Step 1: Review the Function and Quotient Rule

Given the function for the wholesale trade output index: \[ W(x) = \frac{-151x + 1668}{x^2 - 17.4x + 88} + 77.4 \]We need to differentiate the main part of the function using the quotient rule, which states that for \( \frac{u(x)}{v(x)} \), the derivative is:\[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} \] where \( u(x) = -151x + 1668 \) and \( v(x) = x^2 - 17.4x + 88 \).

Step 2: Compute \( u'(x) \) and \( v'(x) \)

First, compute the derivatives of the numerator and denominator functions. - \( u'(x) = -151 \)- \( v'(x) = 2x - 17.4 \)

Step 3: Differentiate Using the Quotient Rule

Apply the quotient rule to find \( \left( \frac{u}{v} \right)' \).\[W'(x) = \frac{(-151)(x^2 - 17.4x + 88) - (2x - 17.4)(-151x + 1668)}{(x^2 - 17.4x + 88)^2}\]

Step 4: Simplify the Derivative Expression

Expand and simplify the numerator:- The first term becomes: \((-151)(x^2 - 17.4x + 88) = -151x^2 + 2637.4x - 13288\). - The second term: \((2x - 17.4)(-151x + 1668) = -302x^2 + 3171.6x + 2635.2x - 29083.2\), combine to get: \(-302x^2 + 5806.8x - 29083.2\).Combine all similar terms into:\[ -151x^2 + 2637.4x - 13288 + 302x^2 - 5806.8x + 29083.2 \]

Step 5: Evaluate \(W'(4)\) and \(W'(8)\)

Substitute \(x = 4\) and \(x = 8\) into the simplified derivative expression. Calculate each step carefully using the results from the prior step for each \(x\). Provide the calculated results for both values of \(x\).Example:1. Substitute \(x = 4\) and simplify.2. Substitute \(x = 8\) and simplify.

Step 6: Interpret the Results

Calculate the numerical results for \(W'(4)\) and \(W'(8)\). Interpret the meanings:- \(W'(4)\) describes the rate of change of wholesale trade output 4 years after 2000.- \(W'(8)\) describes the rate of change 8 years after 2000.A positive result indicates that the trade output is increasing at that year, while a negative result indicates a decreasing trend.

Key Concepts

Quotient Rule

The Quotient Rule is a fundamental tool in calculus for finding the derivative of a ratio of two functions. This is especially useful when dealing with complex functions that form fractions. To apply it, you need two functions: a numerator \( u(x) \) and a denominator \( v(x) \). The rule states that the derivative of \( \frac{u}{v} \) is given by:

• \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} \)

Here, \( u' \) and \( v' \) represent the derivatives of \( u \) and \( v \) respectively. To compute this, you'll:

• Differentiate the numerator to get \( u' \).
• Differentiate the denominator to get \( v' \).
• Apply these into the formula to find the derivative of the quotient.

This approach is particularly essential in economic modelling, where functions often present as complex rational expressions.

Derivative Calculation

The process of computing derivatives involves several rules and methods, including the Quotient Rule. Understanding how to correctly apply these is crucial for analyzing the behavior of functions in contexts like economics. In our exercise, we used the Quotient Rule to differentiate the function representing wholesale trade output:For the function \( W(x) = \frac{-151x + 1668}{x^2 - 17.4x + 88} + 77.4 \), we focus primarily on differentiating the quotient and then incorporate the constant term into the final result. Break down the process as follows:1. **Identify \( u(x) \) and \( v(x) \):**- \( u(x) = -151x + 1668 \)- \( v(x) = x^2 - 17.4x + 88 \)2. **Compute Derivatives:**- \( u'(x) = -151 \)- \( v'(x) = 2x - 17.4 \)3. **Apply Quotient Rule:**- Derivative \( W'(x) = \frac{(-151)(x^2 - 17.4x + 88) - (2x - 17.4)(-151x + 1668)}{(x^2 - 17.4x + 88)^2} \)This derivative helps determine the rate at which changes occur, important for understanding economic dynamics.

Economic Modelling

Economic modelling involves using mathematical functions to represent real-world economic processes. In this context, the function \( W(x) \) models the wholesale trade output index, with the index values relative to 100 units in the year 2002. By differentiating this function, we gain insights into the dynamics of economic performance over time.Using derivatives, economists can:

• Analyze the rate of change or trend in output over specific time periods.
• Predict future economic behavior based on past trends.
• Identify periods of growth or decline in trade activities.

In our exercise, finding \( W'(4) \) and \( W'(8) \) allows us to interpret the rate at which the wholesale trade output is changing 4 and 8 years after 2000. A positive derivative indicates an increase in output, reflecting potential economic growth, whereas a negative one suggests a slow down or decrease, indicating possible economic challenges.