Problem 17 - Calculus

Question

After investing \(\$ 1000\) at an annual interest rate of \(2 \%\) compounded continuously for \(t\) years, your balance is \(S B,\) where \(B=f(t) .\) What are the units of \(d B / d t ?\) What is the financial interpretation of \(d B / d t ?\)

Step-by-Step Solution

Answer

The units are dollars per year; it represents the growth rate of the investment balance over time.

Step 1: Understand the Problem

We have an initial investment of $1000 at a continuous compound interest rate of 2%. The final balance after \(t\) years is given by \(B = f(t)\). We need to find the units of \(\frac{dB}{dt}\) and its financial interpretation.

Step 2: Formula for Continuous Compounding

The formula for continuous compounding is given by \( B = P e^{rt} \), where \( P \) is the principal amount, \( r \) is the interest rate, and \( t \) is the time in years. Substituting the given values, we get \( B = 1000 e^{0.02t} \).

Step 3: Differentiate the Function

Differentiate \( B = 1000 e^{0.02t} \) with respect to \( t \) to find \( \frac{dB}{dt} \). This involves using the chain rule: \( \frac{dB}{dt} = 1000 \cdot e^{0.02t} \cdot 0.02 = 20 e^{0.02t} \).

Step 4: Determine Units of \(\frac{dB}{dt}\)

Since \( B \) has units of dollars \((\\))\( and \( t \) has units of years, the derivative \(\frac{dB}{dt}\) will have units of dollars per year \)(\\(/\)\text{year})$.

Step 5: Financial Interpretation of \(\frac{dB}{dt}\)

The derivative \(\frac{dB}{dt}\) represents the rate of change of the balance \(B\) with respect to time \(t\). In financial terms, it describes how quickly the balance grows over time due to continuous compounding.

Key Concepts

Continuous Compound Interest

When we talk about continuous compound interest, it refers to a method where the interest earned is constantly reinvested to earn even more interest. This concept of compounding continuously is a powerful way to grow an investment over time. The formula used for calculating continuous compound interest is given by:

• \( B = P e^{rt} \)

In this formula:

• \( B \) represents the future balance of your investment.
• \( P \) is the principal amount, which is the initial amount of money invested.
• \( e \) is a mathematical constant approximately equal to 2.71828, which is the base of the natural logarithm.
• \( r \) is the annual interest rate (expressed as a decimal).
• \( t \) is the time for which the money is invested, usually in years.

By using continuous compounding, the investment grows at an exponential rate, meaning the growth accelerates over time. Whenever you hear about continuous compounding, think of it as a way for money to earn interest on top of interest all the time.

Differentiation

Differentiation is a fundamental concept in calculus that is used to find the rate at which a function is changing at any given point. In simpler terms, it helps us measure how much a quantity changes as its input changes. For example, if you have a function \( f(t) = 1000 e^{0.02t} \), which represents the balance of an account over time, differentiating this function with respect to time, \( t \), gives us the rate of change of the balance over that time.The process of differentiation involves finding a derivative, which is the mathematical term for this rate of change. The notation \( \frac{dB}{dt} \) represents the derivative of \( B = f(t) \) with respect to \( t \). In our example, the differentiation leads to:

• The derivative \( \frac{dB}{dt} = 20 e^{0.02t} \), which indicates how quickly the balance grows each year.

The units of \( \frac{dB}{dt} \) are dollars per year, showing that differentiation can not only find the rate of increase but also provides the units, helping to interpret the meaning in a real-world context.

Chain Rule

The chain rule is a valuable tool in calculus for finding the derivative of a composition of functions. Essentially, it helps us differentiate functions that are nested within one another. When dealing with a function like \( B = 1000 e^{0.02t} \), we're looking at a composition where the outer function is a multiple of \( e^x \) and the inner function is \( 0.02t \).To apply the chain rule, we follow these steps:

• First, differentiate the outer function, leaving the inner function untouched. For \( e^{0.02t} \), the derivative of \( e^x \) is still \( e^x \).
• Next, multiply by the derivative of the inner function \( 0.02t \), which is simply 0.02.

Putting this together, the complete differentiation using the chain rule looks like this:

• \( \frac{d}{dt}(1000 e^{0.02t}) = 1000 \cdot e^{0.02t} \cdot 0.02 = 20 e^{0.02t} \)

The chain rule is particularly useful when functions become complex and layered. It streamlines the differentiation process, making it possible to manage intricate calculations with confidence.