Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that after 1 year you have \( 1000\) in the bank. If the interest was compounded continuously at \(5 \%\), how much money did you put in the bank one year ago? This is called the present value.

Step-by-Step Solution

Answer

You initially placed approximately $951.23 in the bank.

Step 1: Identify the formula for continuously compounded interest

The formula for continuously compounded interest is given by the equation \( A = Pe^{rt} \), where \( A \) is the amount of money accumulated after time \( t \), \( P \) is the principal amount (the initial amount of money), \( r \) is the interest rate, and \( e \) is the base of the natural logarithm.

Step 2: Substitute the known values into the equation

In this problem, we know \( A = 1000 \), \( r = 0.05 \), and \( t = 1 \). Substitute these values into the equation to get: \[1000 = Pe^{0.05 \times 1}.\]

Step 3: Solve for the principal \( P \)

To find \( P \), we need to isolate it in the equation. First, simplify the equation: \[1000 = Pe^{0.05}.\]Now, divide both sides of the equation by \( e^{0.05} \): \[P = \frac{1000}{e^{0.05}}.\]

Step 4: Calculate \( e^{0.05} \) and find \( P \)

Calculate \( e^{0.05} \) using a calculator to find it is approximately \( 1.051271 \). Then calculate \( P \):\[P = \frac{1000}{1.051271} \approx 951.23.\] So, \( P = 951.23 \).

Key Concepts

Continuously Compounded Interest

Continuously compounded interest is a concept that helps you understand how your money can grow over time. Unlike other interest calculation methods, continuously compounded interest assumes that interest is added to the principal infinite times per year. This means your money grows without interruption. The basic formula is \( A = Pe^{rt} \), where:

• \( A \) is the final amount of money accumulated after time \( t \).
• \( P \) is the initial principal balance.
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
• \( r \) represents the interest rate, and \( t \) is the time in years.

This formula is essential in financial mathematics as it shows how investments grow exponentially over time. By continuously compounding, your principal earns interest every moment, making it a powerful tool for maximizing growth.

Exponential Functions

Exponential functions are crucial in understanding growth patterns, especially in finance. The general form of an exponential function is \( f(x) = ab^{x} \), where:

• \( a \) is a constant term.
• \( b \) is the base of the exponential, which determines the growth rate.
• \( x \) is the exponent, often representing time.

In the case of continuously compounded interest, we use \( e \) instead of a general base, transforming the formula into \( A = Pe^{rt} \). This transformation makes it a classic exponential growth function. With exponential functions, the quantity grows by a constant percent rate, resulting in a curve that rises sharply after a certain point. Such models are not only important in finance but also in natural processes like population growth and radioactive decay.

Present Value

The present value (PV) is a fundamental concept in finance that helps you determine how much a future sum of money is worth right now. The idea is to discount the future value using the concept of interest. To find the present value when interest is compounded continuously, you rearrange the continuously compounded interest formula \( A = Pe^{rt} \) to solve for \( P \):

• \( P = \frac{A}{e^{rt}} \)

By applying this formula, you can see how much you should invest today to achieve a desired future amount, given a certain interest rate and time period. Understanding present value is vital because it helps in making informed decisions about investments, savings, and knowing the worth of future cash flows today.

Mathematical Modeling

Mathematical modeling involves using mathematical equations and concepts to represent real-world systems and predict future outcomes. In the context of finance, it enables us to simulate scenarios and understand complex interactions. The formula \( A = Pe^{rt} \) is a perfect example of a mathematical model used to describe continuously compounded interest.The beauty of mathematical modeling lies in its ability to provide quantifiable insights and predictions. It allows individuals and businesses to model potential investments and make data-driven decisions. Whether for large corporations or individual investors, mathematical modeling translates intricate financial concepts into understandable and actionable insights, crucial for strategic planning and risk management.