Problem 40 - Calculus and its applications

Question

The total financial liability, in billions of dollars, of U.S. households can be modeled by \(L(t)=1547(1.083)^{t},\) where \(t\) is the number of years after \(1980 .\) The graph of this function follows. a) Using this model, predict the total financial liability of U.S. households in 2012 . b) Find \(L^{\prime}(25)\). c) Interpret the meaning of \(L^{\prime}(25)\).

Step-by-Step Solution

Answer

In 2012, liability is \( L(32) \) and \( L^{\prime}(25) \) measures the rate of change in 2005.

Step 1: Understand the function

The given function is \( L(t) = 1547(1.083)^t \), where \( t \) is the number of years after 1980. This function models the total financial liability of U.S. households in billions of dollars.

Step 2: Determine the value of \( t \) for 2012

To find \( t \) for the year 2012, calculate the number of years after 1980 that corresponds to 2012: \( t = 2012 - 1980 = 32 \).

Step 3: Predict the liability in 2012

Substitute \( t = 32 \) into the equation: \[ L(32) = 1547(1.083)^{32} \]. Calculate this expression to find the predicted liability in 2012.

Step 4: Differentiate the function to find \( L^{\prime}(t) \)

Differentiate the function \( L(t) = 1547(1.083)^t \) with respect to \( t \). The derivative is \( L^{\prime}(t) = 1547 \ln(1.083) \cdot (1.083)^t \).

Step 5: Evaluate \( L^{\prime}(25) \)

Substitute \( t = 25 \) into \( L^{\prime}(t) \): \[ L^{\prime}(25) = 1547 \ln(1.083) \cdot (1.083)^{25} \]. Calculate this value to find \( L^{\prime}(25) \).

Step 6: Interpret \( L^{\prime}(25) \)

The derivative \( L^{\prime}(25) \) represents the rate of change of the financial liability in the year 2005 (25 years after 1980). It indicates how rapidly the financial liability is increasing at that time in billions of dollars per year.

Key Concepts

Differentiation in Calculus

Differentiation is a core concept in calculus that involves finding the rate at which a function is changing at any point on its curve. It forms the basis for understanding how functions behave, allowing us to predict and analyze trends.

In the exercise you've seen, the function provided is an exponential growth model: \( L(t) = 1547(1.083)^t \). Here, differentiation helps us find the derivative of the function, \( L'(t) \), which represents the rate of change of financial liability with respect to time.

Let's break it down further:

• **Exponentials and their Derivatives:** The derivative of an exponential function \( a^t \) with base \( a \) is \( a^t \ln(a) \).
• **Constant Multipliers:** If a function is multiplied by a constant, like 1547 in our function, its derivative will be the constant multiplied by the derivative of the function.

Differentiation allows us to understand how quantity changes, which is particularly important in predicting future values in financial contexts.

Financial Modeling in Education

Financial modeling is a vital educational tool that equips learners with the skills to predict financial behavior using mathematical functions. This helps in developing a realistic understanding of complex financial phenomena.

In our exercise, the function \( L(t) = 1547(1.083)^t \) is a mathematical model for financial liability. Here’s how financial models play a significant role in education:

• **Predictive Analysis:** Models allow students to simulate financial scenarios, estimating data-driven predictions based on historical trends.
• **Understanding Growth:** In this case, an exponential function demonstrates compounding growth, which reflects real-world financial situations like compound interest.
• **Critical Thinking:** Students learn to interpret the implications of their models, encouraging analytical thinking.

Educational models help students bridge the gap between theoretical concepts and practical real-world financial issues.

Interpretation of Derivatives

The interpretation of derivatives goes beyond mere calculation. It provides insights into the real-world implications of mathematical models. By understanding derivatives, students can comprehend the specifics of changing rates and their significance.

In the context of the given exercise, interpreting \( L'(25) \) tells us about the rate of change of financial liability in 2005. Let’s consider important aspects:

• **Rate of Change:** Derivatives symbolize how rapidly a quantity is increasing or decreasing, measured here in billions of dollars per year.
• **Economic Insights:** Understanding \( L'(25) \) helps in grasping the acceleration of financial liability, offering insights into economic growth or potential issues.
• **Practical Application:** Such interpretations allow students to envisage the real-world consequences of financial changes, crucial for economic planning and decision-making.

Grasping the interpretation of derivatives enhances analytical skills and assists in meaningful evaluations of statistical models.