Problem 29 - Precalculus Mathematics for Calculus

Question

Compound Interest An investment of \(\$ 7,000\) is deposited into an account in which interest is compounded continuously. Complete the table by filling in the amounts to which the investment grows at the indicated times or interest rates. $$r=3 \%$$ $$\begin{array}{|c|c|} \hline \begin{array}{c} \text { Time } \\ \text { (years) } \end{array} & \text { Amount } \\ \hline 1 & \\ 2 & \\ 3 & \\ 4 & \\ 5 & \\ 6 & \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Use the continuous compound interest formula to compute the amount for each year and fill in the table.

Step 1: Understanding the Formula

To solve this problem, we need to use the formula for continuous compound interest: \( A = Pe^{rt} \), where \( A \) is the amount of money accumulated after n years, including interest. \( P \) is the principal amount (\$7,000), \( r \) is the annual interest rate (expressed as a decimal, so 3% becomes 0.03), \( t \) is the time in years, and \( e \) is the base of the natural logarithm (approximately 2.718).

Step 2: Calculate for Year 1

For \( t = 1 \), substitute into the formula: \( A = 7000 \times e^{0.03 \times 1} \). Calculate \( e^{0.03} \) and then multiply by 7000 to find the amount after 1 year.

Step 3: Calculate for Year 2

For \( t = 2 \), substitute into the formula: \( A = 7000 \times e^{0.03 \times 2} \). Compute \( e^{0.06} \) and multiply by 7000 to find the amount after 2 years.

Step 4: Calculate for Year 3

For \( t = 3 \), use the formula: \( A = 7000 \times e^{0.03 \times 3} \). Calculate \( e^{0.09} \) and then multiply by 7000 to find the amount after 3 years.

Step 5: Calculate for Year 4

For \( t = 4 \), substitute into the formula: \( A = 7000 \times e^{0.03 \times 4} \). Find \( e^{0.12} \) and multiply by 7000 to determine the amount after 4 years.

Step 6: Calculate for Year 5

For \( t = 5 \), the formula becomes: \( A = 7000 \times e^{0.03 \times 5} \). Calculate \( e^{0.15} \) and multiply by 7000 to get the amount after 5 years.

Step 7: Calculate for Year 6

For \( t = 6 \), use the formula: \( A = 7000 \times e^{0.03 \times 6} \). Compute \( e^{0.18} \) and multiply by 7000 to find the amount after 6 years.

Step 8: Complete the Table

Fill in the table with the values calculated for each year using the steps above.

Key Concepts

Exponential Growth

In the context of finance and investment, exponential growth occurs when the amount of money grows at a consistent rate over time. This is common in situations where interest is compounded continuously. The formula for calculating the exponential growth of an investment using continuous compounding is given by \( A = Pe^{rt} \). In this formula:

• \( A \) represents the final amount after a certain period.
• \( P \) is the principal amount or the initial investment.
• \( r \) is the interest rate expressed as a decimal.
• \( t \) is the time the money is invested for, in years.
• \( e \) is a mathematical constant approximately equal to 2.718.

Exponential growth in investments can lead to substantial increases over time. This occurs because the investment grows by a constant percentage, leading to larger absolute increases as time passes. The power of \( e \) in the formula ensures that the growth is continuous and smooth, which is a hallmark of exponential growth.
This type of growth contrasts with linear growth, where values increase at a constant absolute rate.

Interest Rates

Interest rates are crucial in determining how much an investment will grow over time. In the context of continuous compound interest, the rate is expressed as a decimal in the formula \( A = Pe^{rt} \). For example, an interest rate of 3% is written as 0.03 in this formula.

• The interest rate reflects how much additional money the principal will earn over a specified period, typically annually.
• A higher interest rate means faster growth of the investment, while a lower rate results in slower growth.
• Continuous compounding means that the interest is being calculated and added to the principal at every moment, making the effective annual rate slightly higher than the stated nominal rate.

Understanding interest rates is essential because they directly impact the overall return on an investment. By comparing different interest rates, investors can make informed decisions about where to allocate their resources to maximize growth and returns.

Principal Amount

The principal amount is the initial sum of money invested or loaned, before any interest is applied. In the formula \( A = Pe^{rt} \), the principal is denoted by \( P \). In our exercise, the principal amount is \( \$7,000 \), which is the starting point for calculating the accumulated value of the investment over time.

• The principal is essentially the base of the investment from which all interest calculations begin.
• Without the principal, there would be no foundation on which the interest can build.
• The magnitude of the principal amount affects the total future value of the investment; investing a larger principal will result in a higher final amount, assuming the same interest rate and time period.

It's important for investors to clearly understand and track the principal amount as it reflects the original value of their investment apart from the interest gains over time.

Natural Logarithm

The natural logarithm is a concept from mathematics that plays a significant role in the continuous compounding formula \( A = Pe^{rt} \). The base of the natural logarithm is the constant \( e \), which is approximately equal to 2.718. This constant arises naturally in various growth and decay processes.
Understanding \( e \) and its use is critical:

• The natural logarithm is important in finance because it underpins the exponential functions used in calculating continuously compounded interest.
• Using \( e \) in calculations allows for modeling the growth processes that occur continuously and smoothly over time.
• In finance, natural logarithms help in understanding the time value of money and the effects of compounding returns.

When solving for the time \( t \) in the compound interest formula, natural logarithms are often used to solve equations involving interiors of \( e \). Understanding these calculations is vital for deeper insights into financial modeling and predicting future values of investments.