Unlocking Exponential Growth: Mastering Continuous Compounding

Algebra 2 Grades High School 2:38 Video
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Lesson Description

Explore the power of continuous compounding using the perk formula (A = Pe^(rt)). Learn how to calculate the time it takes for an investment to double, using natural logarithms and practical examples.

Video Resource

Time to Double Compounding Continuously

Mario's Math Tutoring

Duration: 2:38
Watch on YouTube

Key Concepts

  • Continuous Compounding
  • The perk formula (A = Pe^(rt))
  • Natural Logarithms
  • Exponential Functions

Learning Objectives

  • Students will be able to apply the formula for continuous compounding (A = Pe^(rt)) to solve problems.
  • Students will be able to use natural logarithms to solve for the time it takes for an investment to double when compounded continuously.
  • Students will be able to convert percentages to decimals for use in the continuous compounding formula.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of compound interest and how it differs from simple interest. Briefly introduce the idea of continuous compounding as the limit of compounding interest infinitely often. Engage students by asking about real-world scenarios where continuous compounding might be relevant (e.g., certain investment accounts).
  • The Perk Formula (5 mins)
    Introduce the formula A = Pe^(rt). Explain each variable: A (amount after time t), P (principal), e (Euler's number, approximately 2.71828), r (interest rate as a decimal), and t (time in years). Emphasize that 'e' is a mathematical constant, just like pi. Relate this formula back to other exponential functions they have studied. Show the portion of the video from 0:12-0:40.
  • Example Problem: Time to Double (10 mins)
    Walk through a detailed example problem similar to the one in the video (starting with a principal, interest rate, and solving for the time it takes to double the investment). Use the video example from 0:41-1:32. Emphasize the steps: setting up the equation, dividing both sides by the principal, and isolating the exponential term. Ask students to predict the next step at each stage.
  • Using Natural Logs (10 mins)
    Explain the concept of natural logarithms and their relationship to exponential functions with base 'e'. Demonstrate how to take the natural log of both sides of the equation to isolate 't' (as shown from 1:32 to the end of the video). Review the properties of logarithms. Show students how to use a calculator to find the natural log of a number. Then, have students complete similar problem to the example from the video.
  • Practice Problems (10 mins)
    Provide students with additional practice problems involving different principal amounts, interest rates, and target amounts. Encourage them to work independently or in pairs. Circulate to provide assistance and answer questions.
  • Wrap-up and Review (5 mins)
    Summarize the key concepts of the lesson. Review the formula for continuous compounding and the steps involved in solving for the time it takes to double an investment. Answer any remaining questions. Briefly preview upcoming topics related to exponential functions and logarithms.

Interactive Exercises

  • Compound Interest Calculator Simulation
    Use an online compound interest calculator to demonstrate the effect of increasing the compounding frequency (e.g., annually, monthly, daily, continuously) on the final amount. Have students observe how the amount approaches a limit as the compounding frequency increases.
  • Group Problem Solving
    Divide students into small groups and assign each group a different investment scenario. Have them work together to calculate the time it takes to reach a specific financial goal using continuous compounding. Each group presents their solution and process to the class.

Discussion Questions

  • Why is it important to convert the interest rate into a decimal before using it in the continuous compounding formula?
  • What is the significance of the number 'e' in the continuous compounding formula, and where does it come from?
  • How does continuous compounding differ from compounding interest annually, semi-annually, or quarterly?
  • In what real-world financial scenarios might you encounter continuous compounding?

Skills Developed

  • Problem-solving
  • Algebraic manipulation
  • Application of logarithmic properties
  • Using calculators for financial calculations

Multiple Choice Questions

Question 1:

What does 'P' represent in the formula A = Pe^(rt)?

Correct Answer: Principal

Question 2:

What is the approximate value of 'e' (Euler's number)?

Correct Answer: 2.718

Question 3:

In the formula A = Pe^(rt), 'r' must be expressed as a:

Correct Answer: Decimal

Question 4:

Which function is used to 'undo' or cancel out an exponential function with base 'e'?

Correct Answer: Natural logarithm

Question 5:

If an investment doubles, the value of A/P is:

Correct Answer: 2

Question 6:

To convert an interest rate of 7% to a decimal, you would:

Correct Answer: Divide by 100

Question 7:

What is the purpose of using logarithms when solving for 't' in the continuous compounding formula?

Correct Answer: To isolate the variable 't' from the exponent

Question 8:

Which of the following equations represents the setup for finding the time it takes to triple an investment using continuous compounding?

Correct Answer: 3 = e^(rt)

Question 9:

What happens to the amount earned as the compounding frequency increases toward continuous?

Correct Answer: Approaches a limit

Question 10:

If the natural log of 2 is approximately 0.693, what value would you divide by to solve for t in the doubling problem?

Correct Answer: Interest rate (r)

Fill in the Blank Questions

Question 1:

The formula for continuous compounding is A = ____ e^(rt).

Correct Answer: P

Question 2:

The natural base 'e' is approximately equal to ____.

Correct Answer: 2.718

Question 3:

To find the time it takes to double an investment, you set A equal to 2 times the ____.

Correct Answer: principal

Question 4:

The inverse operation of e^x is the _______ logarithm.

Correct Answer: natural

Question 5:

Before using the interest rate in the formula, it must be converted to a ____.

Correct Answer: decimal

Question 6:

In the continuous compounding formula, 't' represents the ____ in years.

Correct Answer: time

Question 7:

Taking the natural log of both sides of the equation allows you to isolate the variable in the ____.

Correct Answer: exponent

Question 8:

When solving for time to double, you divide both sides by the principal to get 2 = ______.

Correct Answer: e^(rt)

Question 9:

If A/P = 5, the investment has increased by a factor of ______.

Correct Answer: 5

Question 10:

Logarithms and exponential functions are _______ of one another.

Correct Answer: inverses

Educational Standards

A2.A.1.2:Use mathematical models to represent exponential relationships, such as compound interest, depreciation, and population growth. Solve these equations algebraically or graphically (including graphing calculator or other appropriate technology). A2.A.1.6:Solve common and natural logarithmic equations using the properties of logarithms. A2.F.1.4:Graph exponential and logarithmic functions. Identify the domain, range, asymptotes, and x- and y-intercepts using various methods and tools that may include calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically.

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