Problem 60 - Brief Applied Calculus

Question

A study estimated how a person's social status (rated on a scale where 100 indicates the status of a college graduate) depended on years of education. Based on this study, with \(e\) years of education, a person's status is \(S(e)=0.22(e+4)^{2.1}\). Find \(S^{\prime}(12)\) and interpret your answer.

Step-by-Step Solution

Answer

For \(e = 12\), \(S'(12) \approx 10.46\), meaning status increases by about 10.46 units per additional year of education.

Step 1: Understand the Problem

We need to find the derivative of the social status function, \(S(e) = 0.22(e+4)^{2.1}\), and evaluate it at \(e = 12\). This will give us the rate of change of social status with respect to years of education when \(e = 12\).

Step 2: Differentiate the Function

To find \(S'(e)\), differentiate \(S(e) = 0.22(e+4)^{2.1}\) with respect to \(e\). Use the chain rule, which states \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\). Here, \(f(u) = 0.22u^{2.1}\) and \(g(e) = e+4\), so \(S'(e) = 0.22 \cdot 2.1 \cdot (e+4)^{1.1} \cdot 1\.\)

Step 3: Simplify the Derivative

Simplify the expression obtained in the previous step: \(S'(e) = 0.462 \cdot (e+4)^{1.1}\).

Step 4: Evaluate the Derivative at e = 12

Substitute \(e = 12\) into the derivative: \(S'(12) = 0.462 \cdot (12+4)^{1.1}\).

Step 5: Compute the Numerical Result

Calculate \((16)^{1.1}\) using a calculator, then multiply by \(0.462\): \((16)^{1.1} \approx 22.63\). Therefore, \(S'(12) \approx 0.462 \times 22.63 \approx 10.46\.\)

Step 6: Interpret the Result

The derivative \(S'(12) \approx 10.46\) suggests that for someone with 12 years of education, their social status increases by approximately 10.46 units on the status scale for each additional year of education.

Key Concepts

Chain Rule in Differentiation

When you're trying to differentiate functions, especially those composed of other functions, the chain rule is extremely useful. It's like a tool that helps us unravel the layers of complicated functions.

Imagine you have a function within a function, such as the social status function given here, where the structure is \( S(e) = 0.22(e+4)^{2.1} \). This is a classic example where the chain rule applies because you have something inside those parentheses \((e+4)\) raised to a power.

• First, recognize that the outside function is the power itself \((u^{2.1})\), and the inner function is \((e+4)\).
• To apply the chain rule, find the derivative of that outer function with respect to the inner function. Then multiply it by the derivative of the inner function.
• For the example, you take the derivative of \(0.22u^{2.1} \) with respect to \(u\), which gives \(0.22 \cdot 2.1 \cdot u^{1.1}\).
• Then multiply by the derivative of the inner function \(g(e) = e+4\), which is simply \(1\).

This method efficiently breaks down the process, making it easier to find the derivative of complex expressions.

Understanding Rate of Change

In mathematics, the rate of change tells you how much one quantity changes with respect to another. It's like asking, "If I change one variable, how will it affect another?"

In the context of the social status function, when we calculate \(S'(12)\), we're finding the rate of change of social status with respect to years of education. This rate reflects how sensitive the social status is to changes in education.

• A high rate, like \(10.46\) in this scenario, means that each additional year of education significantly increases social status.
• This gives valuable insight into the impact of education on social status. It shows how pivotal each additional year of learning can be.
• Understanding the rate of change lets educators, policymakers, or social scientists gauge the importance of education in influencing social equity.

It's a practical indicator revealing the dynamic relationship between education and social perception.

Social Status Function Interpretation

The social status function in this problem evaluates how education affects an individual's perceived status. In simple terms, it's a mathematical way to quantify the societal value placed on education.

Formally expressed as \(S(e) = 0.22(e+4)^{2.1}\), this function helps interpret complex societal structures through a mathematical lens.

• Each component of the function has a purpose. The expression \((e+4)\) reflects the cumulative effect of years of education.
• The exponent \(2.1\) demonstrates how each additional year has a growing effect, not just a linear one.
• By multiplying with \(0.22\), the scale is adjusted to reflect the relative importance or impact of this equation on someone's social status.

Ultimately, when used accurately, such a function allows people to analyze and possibly predict the degree of change in social status linked to educational changes. It becomes a useful tool for understanding and bridging gaps in social statuses across different groups.