Problem 18 - Applied Calculus

Question

The amount of money, \(\$ B,\) in a bank account earning interest at a continuous rate, \(r,\) depends on the amount deposited, \(\$ P,\) and the time, \(t,\) it has been in the bank, where $$ B=P e^{r t} $$ Find \(\partial B / \partial t, \partial B / \partial r\) and \(\partial B / \partial P\) and interpret each in financial terms.

Step-by-Step Solution

Answer

\( \partial B / \partial t = Pr e^{rt}, \partial B / \partial r = Pt e^{rt}, \partial B / \partial P = e^{rt}. \)

Step 1: Differentiate with Respect to Time

First, we find \( \partial B / \partial t \) by differentiating \( B = Pe^{rt} \) with respect to \( t \). Using the chain rule, we have:\[\frac{\partial B}{\partial t} = P \cdot \frac{d}{dt}(e^{rt}) = P \cdot e^{rt} \cdot r\]Thus, \( \frac{\partial B}{\partial t} = Pr e^{rt} \).In financial terms, this represents the rate of change of the balance with respect to time, which corresponds to the amount of interest earned per unit of time.

Step 2: Differentiate with Respect to the Interest Rate

Next, we find \( \partial B / \partial r \) by differentiating \( B = Pe^{rt} \) with respect to \( r \). Applying the chain rule again gives:\[\frac{\partial B}{\partial r} = P \cdot \frac{d}{dr}(e^{rt}) = P \cdot e^{rt} \cdot t\]Thus, \( \frac{\partial B}{\partial r} = Pt e^{rt} \).In financial terms, this represents how the balance changes as the interest rate changes, indicating the sensitivity of the bank balance to fluctuation in interest rate.

Step 3: Differentiate with Respect to the Principal Amount

Finally, we find \( \partial B / \partial P \) by differentiating \( B = Pe^{rt} \) with respect to \( P \). Since \( P \) is a constant factor:\[\frac{\partial B}{\partial P} = e^{rt}\]Thus, \( \frac{\partial B}{\partial P} = e^{rt} \).In financial terms, this reflects the proportional impact of the initial amount of money deposited on the balance, showing how the bank balance scales with changes in the initial deposit.

Key Concepts

Continuous Compound Interest

Continuous compound interest refers to a method of calculating interest where the interest earned is constantly added to the principal, so the amount gets bigger exponentially. In simpler terms, your money grows at an ever-increasing rate since the interest is calculated on the existing balance, not just the original deposit.

Here’s a closer look at how it works:

• **Exponential Growth:** The formula used is \( B = P e^{rt} \), where \( B \) is the balance, \( P \) is the principal sum, \( r \) is the interest rate, and \( t \) is the time period.
• **Compounding Continuously:** Unlike simple interest, which is calculated only on the initial amount, continuous compounding adds the earned interest back into the account balance constantly.

This results in a faster increase in the balance than with other interest calculation methods. The beauty of continuous compound interest is in the exponential growth, where even small interest rates over time can lead to significant sums.

Differentiation

Differentiation in mathematics involves finding a derivative, which essentially measures how a function changes as its input changes. In the context of continuous compound interest, we use differentiation to understand how different factors affect the bank balance.

To differentiate the function \( B = Pe^{rt} \) with respect to different variables, we help reveal the importance of each factor in finance:

• **With Respect to Time (\( t \)):** The derivative \( \frac{\partial B}{\partial t} = Pr e^{rt} \) shows how the balance grows over time, indicating how much interest is accumulated over time given a fixed interest rate and principal.
• **With Respect to Interest Rate (\( r \)):** The derivative \( \frac{\partial B}{\partial r} = Pt e^{rt} \) highlights the sensitivity of the balance to changes in the interest rate, which is crucial for understanding risks in investments or savings.
• **With Respect to Principal (\( P \)):** The derivative \( \frac{\partial B}{\partial P} = e^{rt} \) illustrates how the initial deposit impacts the balance, demonstrating that increasing the principal results in a proportional increase in the balance.

Differentiation allows us to explore the role of each variable in financial growth, offering insights into how alterations in initial conditions affect financial outcomes.

Financial Interpretation

Understanding partial derivatives in a financial context provides valuable insights into how your money behaves in an interest-bearing account.

The process of interpreting the derivatives in financial terms allows us to draw practical insights:

• \( \frac{\partial B}{\partial t} \): Indicates the speed of growth in your balance over time. The bigger this value, the more your money grows each time period, representing the interest earned continuously.
• \( \frac{\partial B}{\partial r} \): Measures how sensitive your balance is to changes in interest rates. This is particularly important for investors as it shows how much a shift in interest rates can impact future gains or losses.
• \( \frac{\partial B}{\partial P} \): Demonstrates the effect of changing the original deposit on the balance. It's straightforward—it’s a linear relationship, so doubling your deposit will double your balance at any future time, assuming the interest rate and time period remain constant.

By analyzing these financial interpretations, one can make more informed decisions: from how much to deposit, to understanding interest rate impacts, all contributing to better financial planning and resource allocation.