Graphing the Function using Python

Oct 13, 2025 by miodeya.com 106 views

Introduction

Continuous compounding is a fundamental concept in finance, where interest is compounded on an initial investment at regular intervals, resulting in exponential growth. In this article, we will delve into the mathematical analysis of continuous compounding, focusing on the future value of an investment. Specifically, we will examine the function ${S = 5000 e^{0.08t}}$ , where ${S}$ represents the future value of a $5000 investment after ${t}$ years at an 8% interest rate compounded continuously.

The Function: ${S = 5000 e^{0.08t}}$

The given function represents the future value of an investment, where ${S}$ is the amount of money accumulated after ${t}$ years, starting with an initial investment of $5000. The interest rate is 8%, and it is compounded continuously. The function can be broken down into three components:

Initial Investment: $5000, which is the initial amount invested.
Interest Rate: 8%, which is the rate at which the interest is compounded.
Time: ${t}$ , which represents the number of years the investment is held.

Graphing the Function

To visualize the growth of the investment, we will graph the function ${S = 5000 e^{0.08t}}$ for ${0 \leq t \leq 15}$ . This will provide a clear understanding of how the investment grows over time.

Graphing the Function using Python

import numpy as np
import matplotlib.pyplot as plt

# Define the function
def S(t):
    return 5000 * np.exp(0.08 * t)

# Generate t values
t = np.linspace(0, 15, 100)

# Calculate S values
S_values = S(t)

# Create the plot
plt.plot(t, S_values)
plt.xlabel('Time (years)')
plt.ylabel('Future Value ($)')
plt.title('Future Value of a $5000 Investment at 8% Interest Rate')
plt.grid(True)
plt.show()

Discussion

The graph of the function ${S = 5000 e^{0.08t}}$ reveals an exponential growth pattern, where the future value of the investment increases rapidly over time. This is a characteristic of continuous compounding, where the interest is compounded on an initial investment at regular intervals, resulting in exponential growth.

Conclusion

In conclusion, the function ${S = 5000 e^{0.08t}}$ represents the future value of a $5000 investment at an 8% interest rate compounded continuously. The graph of this function reveals an exponential growth pattern, where the future value of the investment increases rapidly over time. This analysis provides a clear understanding of the mathematical concept of continuous compounding and its application in finance.

Part (b)

(a) If $$5000 is invested for ${t}$ years at ${8\%}$ interest compounded continuously, the future value is given by ${S = 5000 e^{0.08t}}$ dollars. Find the value of ${t}$ when the future value is $$20,000.

Solution

To find the value of ${t}$ when the future value is $$20,000, we can set up the equation:

20,000 = 5000 e^{0.08t}

We can then solve for ${t}$ using logarithms:

\ln\left(\frac{20,000}{5000}\right) = 0.08t

\ln(4) = 0.08t

\frac{\ln(4)}{0.08} = t

t = \frac{\ln(4)}{0.08}

t \approx 34.66

Therefore, the value of ${t}$ when the future value is $$20,000 is approximately 34.66 years.

Part (c)

Solution

To find the value of ${t}$ when the future value is $$50,000, we can set up the equation:

50,000 = 5000 e^{0.08t}

We can then solve for ${t}$ using logarithms:

\ln\left(\frac{50,000}{5000}\right) = 0.08t

\ln(10) = 0.08t

\frac{\ln(10)}{0.08} = t

t = \frac{\ln(10)}{0.08}

t \approx 86.62

Therefore, the value of ${t}$ when the future value is $$50,000 is approximately 86.62 years.

Conclusion

$In conclusion, we have analyzed the function ${S = 5000 e^{0.08t}}$, which represents the future value of a $5000 investment at an 8% interest rate compounded continuously. We have found the value of ${t}$ when the future value is \$$20,000 and \$$50,000, and have demonstrated the use of logarithms to solve for ${t}$. This analysis provides a clear understanding of the mathematical concept of continuous compounding and its application in finance.<br/> **Continuous Compounding: A Q&A Guide**$

Introduction

Q: What is continuous compounding?

A: Continuous compounding is a method of calculating interest on an investment where the interest is compounded on an initial investment at regular intervals, resulting in exponential growth.

Q: How does continuous compounding work?

A: Continuous compounding works by calculating the interest on an investment at regular intervals, resulting in exponential growth. The formula for continuous compounding is:

S = P e^{rt}

Where:

${S}$ is the future value of the investment
${P}$ is the principal amount (initial investment)
${r}$ is the interest rate
${t}$ is the time period

Q: What is the formula for continuous compounding?

A: The formula for continuous compounding is:

S = P e^{rt}

Where:

${S}$ is the future value of the investment
${P}$ is the principal amount (initial investment)
${r}$ is the interest rate
${t}$ is the time period

Q: How do I calculate the future value of an investment using continuous compounding?

A: To calculate the future value of an investment using continuous compounding, you can use the formula:

S = P e^{rt}

Where:

${S}$ is the future value of the investment
${P}$ is the principal amount (initial investment)
${r}$ is the interest rate
${t}$ is the time period

For example, if you invest $5000 at an 8% interest rate compounded continuously for 10 years, the future value of the investment would be:

S = 5000 e^{0.08 \times 10}

S = 5000 e^{0.8}

S \approx 5000 \times 2.22554

S \approx 11127.70

Therefore, the future value of the investment would be approximately $11,127.70.

Q: What is the difference between continuous compounding and compound interest?

A: Continuous compounding and compound interest are both methods of calculating interest on an investment, but they differ in the frequency of compounding.

Continuous compounding involves compounding interest at regular intervals, resulting in exponential growth. Compound interest, on the other hand, involves compounding interest at fixed intervals, such as monthly or annually.

Q: When should I use continuous compounding?

A: You should use continuous compounding when you want to calculate the future value of an investment with exponential growth. This is typically used in situations where the interest rate is high and the time period is long.

Q: What are the advantages of continuous compounding?

A: The advantages of continuous compounding include:

Exponential growth: Continuous compounding results in exponential growth, which can lead to significant returns on investment.
Flexibility: Continuous compounding can be used to calculate the future value of an investment with any interest rate and time period.
Accuracy: Continuous compounding provides accurate results, as it takes into account the compounding of interest at regular intervals.

Q: What are the disadvantages of continuous compounding?

A: The disadvantages of continuous compounding include:

Complexity: Continuous compounding can be complex to calculate, especially for those without a strong mathematical background.
Assumptions: Continuous compounding assumes that the interest rate remains constant over the time period, which may not always be the case.
Risk: Continuous compounding involves risk, as the interest rate and time period can affect the results.

Conclusion

In conclusion, continuous compounding is a powerful tool for calculating the future value of an investment with exponential growth. By understanding the formula and how to use it, you can make informed decisions about your investments and achieve your financial goals. Remember to consider the advantages and disadvantages of continuous compounding and to use it wisely.