Problem 41 - Brief Applied Calculus

Question

National Debt The national debt of a South American country \(t\) years from now is predicted to be \(D(t)=65+9 t^{4 / 3}\) billion dollars. Find \(D^{\prime}(8)\) and \(D^{\prime \prime}(8)\) and interpret your answers.

Step-by-Step Solution

Answer

The national debt increases at 24 billion dollars per year 8 years from now, and the rate of this increase itself is increasing by 1 billion dollars per year per year.

Step 1: Find the First Derivative

To find \(D'(t)\), the derivative of the national debt function \(D(t) = 65 + 9t^{4/3}\), we need to differentiate it with respect to \(t\). Using the power rule, which is \(\frac{d}{dt} [t^n] = nt^{n-1}\), we differentiate the term \(9t^{4/3}\). The derivative is \(12t^{1/3}\). Thus, \(D'(t) = 12t^{1/3}\).

Step 2: Evaluate the First Derivative at \(t = 8\)

Substitute \(t = 8\) into \(D'(t)\) to find \(D'(8)\). Thus, \(D'(8) = 12(8)^{1/3}\). Evaluate \(8^{1/3}\) which is 2. Hence, \(D'(8) = 12 \times 2 = 24\).​

Step 3: Find the Second Derivative

To find \(D''(t)\), differentiate \(D'(t) = 12t^{1/3}\) again with respect to \(t\). Using the power rule, differentiate \(12t^{1/3}\) to get \(4t^{-2/3}\). So, \(D''(t) = 4t^{-2/3}\).

Step 4: Evaluate the Second Derivative at \(t = 8\)

Substitute \(t = 8\) into \(D''(t)\) to find \(D''(8)\). Thus, \(D''(8) = 4(8)^{-2/3}\). Evaluate \(8^{-2/3}\), which is \(1/4\). So, \(D''(8) = 4 \times \frac{1}{4} = 1\).

Step 5: Interpret the Results

The value \(D'(8) = 24\) means that 8 years from now, the national debt is increasing at a rate of 24 billion dollars per year. The value \(D''(8) = 1\) indicates that the rate of increase in the national debt itself is increasing by 1 billion dollar per year per year 8 years from now.

Key Concepts

Power Rule

The power rule is a fundamental concept in calculus used to differentiate functions of the form \( t^n \), where \( n \) is any real number. Here’s how it works:

• The derivative of \( t^n \) is \( nt^{n-1} \).
• This rule simplifies finding derivatives, especially when the exponent \( n \) is not an integer.

For instance, in the function \( 9t^{4/3} \) from our exercise, you apply the power rule to find the derivative. Multiply the coefficient (9) by the exponent (4/3) to get 12, and then subtract 1 from the exponent to get \( t^{1/3} \). This gives us the derivative as \( 12t^{1/3} \). It's a straightforward process that allows you to handle complex exponents easily. Always remember, the power rule is your go-to tool for polynomial and similar expressions in calculus!
Understanding this rule is critical when working with different types of functions and helps you determine the rate of change efficiently.

First Derivative

The first derivative, denoted as \( D'(t) \), represents the rate of change of the national debt over time. Simply put, it indicates how fast the debt is increasing or decreasing at any given point. In our exercise, you start by taking the derivative of \( D(t) = 65 + 9t^{4/3} \). Using the power rule, the constant 65 has a derivative of 0, and the term \( 9t^{4/3} \) becomes \( 12t^{1/3} \).
When evaluated at \( t = 8 \), the first derivative \( D'(8) = 24 \) implies that, 8 years from now, the national debt is growing at 24 billion dollars per year. This insight is crucial for understanding trends and making predictions based on past and present data. By knowing the first derivative, stakeholders can assess whether the debt situation is improving or worsening over time. Remember, the first derivative is all about rates and trends.

Second Derivative

The second derivative, expressed as \( D''(t) \), provides insight into the acceleration of the debt growth or how the rate of change itself is changing. It’s essentially the derivative of the first derivative and can illustrate whether debt is speeding up or slowing down. For our function, \( D'(t) = 12t^{1/3} \), the power rule is applied again to compute \( D''(t) \), resulting in \( 4t^{-2/3} \).
Evaluating this at \( t = 8 \) gives \( D''(8) = 1 \), suggesting that the rate at which the national debt is increasing is accelerating by 1 billion dollars per year per year at that point. This can be a crucial indicator for policymakers, as it may require them to adjust financial strategies if the acceleration poses potential risks. In general, the second derivative gives a deeper look into the dynamics of change and can predict future patterns if current trends continue.