Problem 33 - Calculus Early Transcendentals: Pearson New International Edition

Question

It will be shown later for small \(x\) that \(\ln (1+x) \approx x .\) Use this fact to show that the doubling time for money invested at \(p\) percent compounded annually is about \(70 / p\) years.

Step-by-Step Solution

Answer

The doubling time is approximately \( \frac{70}{p} \) years, using \( \ln(1+x) \approx x \) and \( \ln(2) \approx 0.693 \).

Step 1: Understanding the Doubling Time Formula

The doubling time for money invested at a compound interest is determined by the formula \( t = \frac{\ln(2)}{\ln(1+r)} \), where \( r \) is the annual interest rate expressed as a decimal (i.e., \( r = \frac{p}{100} \)). We need to analyze this formula using the given approximation \( \ln(1+x) \approx x \).

Step 2: Applying the Approximation to Simple Interest

By applying the approximation \( \ln(1+r) \approx r \) for small values of \( r \), we simplify the doubling time formula to \( t \approx \frac{\ln(2)}{r} \).

Step 3: Calculating with Approximate Values

We approximate \( \ln(2) \approx 0.693 \). Thus, the doubling time becomes \( t \approx \frac{0.693}{r} \).

Step 4: Expressing Doubling Time in Terms of Percentage Rate

Substitute \( r = \frac{p}{100} \) into the equation \( t \approx \frac{0.693}{r} \). Therefore, \( t \approx \frac{0.693 \times 100}{p} = \frac{69.3}{p} \). For simplicity, it is often rounded to \( \frac{70}{p} \).

Key Concepts

Compound Interest

Compound interest is a powerful concept in finance that helps your money grow over time. When you invest money into an account with compound interest, you not only earn money on your original investment (the principal) but also earn interest on the interest that accumulates. This effect can cause your money to grow exponentially over a long period.

• Compound interest is calculated using the formula: \[ A = P(1 + r)^t \] where:
  • \( A \) is the amount of money accumulated after n years, including interest.
  • \( P \) is the principal amount (the initial amount of money).
  • \( r \) is the annual interest rate (as a decimal).
  • \( t \) is the time the money is invested for, in years.

To understand how compound interest helps in doubling the investment, consider the concept of doubling time, which refers to the time required for an investment to grow to twice its initial size. By using approximations, we can simplify the calculations and get a quick estimate of the doubling time based on the interest rate.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a logarithm to the base \( e \), where \( e \approx 2.71828 \). It is widely used in various fields such as physics, finance, and growth calculations due to its natural occurrence in mathematical expressions for continuous growth or decay.Understanding natural logarithms is crucial for determining the doubling time of investments in the context of compound interest. For any positive value \( x \), the natural logarithm \( \ln(x) \) tells us the power to which \( e \) must be raised to obtain \( x \).Additionally, if you have an equation involving products and powers, logarithms can help simplify those calculations, making them more manageable. For instance, when estimating how long it takes for an investment to double, the natural logarithm is used for more precise computations.

Interest Rate

The interest rate is the percentage at which interest is charged or paid. In the context of investments and compound interest, it signifies how much money you earn annually from your principal.

• The interest rate is typically expressed as a percentage; for instance, an interest rate of 5% per year means you earn 5% of your principal as interest every year.
• When calculating compound interest, it's crucial to express the interest rate as a decimal in calculations, as shown in formulas like \( t = \frac{\ln(2)}{\ln(1+r)} \). A 5% interest rate becomes \( r = 0.05 \).

The size of the interest rate plays a pivotal role in determining how quickly your investment will grow. With the help of logarithmic approximations, small interest rates can provide a quick estimate of the time it takes for your investment to double.

Logarithmic Approximation

Logarithmic approximation is a mathematical technique used to simplify complex logarithmic expressions. When dealing with small values of \( x \), the approximation \( \ln(1+x) \approx x \) can make calculations much easier and quicker.This comes in handy when estimating the doubling time of an investment. By applying the approximation \( \ln(1+r) \approx r \) for small interest rates \( r \), we simplify the calculation of the doubling time to \( t \approx \frac{\ln(2)}{r} \). This equation helps in providing a rough estimate of the doubling time without the need to use a calculator for logarithms.The approximation technique works well when the interest rate \( r \) is small (e.g., less than 10%), making it a useful tool for quick calculations in everyday scenarios.