Exponential Functions and Compound Interest

Natural Exponential Function

The natural exponential function is a fundamental concept in calculus, widely used in modeling growth and decay processes. The function is defined as follows:

• Definition: The function is called the natural exponential function.

• Constant e: The number is defined by the limit and is approximately equal to 2.7.

• Application: The constant is the most convenient base to use in calculus, especially for continuous growth and decay models.

Applications of Exponential Functions

Exponential functions are commonly used to model real-world scenarios such as population growth, radioactive decay, and financial calculations involving compound interest.

• Revenue Modeling: If the price-demand function for a product is given by , where is the number of units produced and is the price per unit, the revenue for selling units is .

• Example: To find the revenue when 100 units are produced: Additional info: , so (rounded to the nearest dollar).

• Solving for Quantity: If the price per unit is set at , solve for : Additional info: , so units.

Compound Interest

Compound interest is a key application of exponential functions in finance. It describes how an initial principal grows over time when interest is added periodically.

• General Formula: If a principal is invested at an annual interest rate for years, and interest is compounded times per year, the future value is:

• Continuous Compounding: If interest is compounded continuously, the formula becomes:

• Example: What amount will be in an account after 5 years if $100 compounded:

  • Semiannually ():

  • Monthly ():

  • Daily ():

  • Continuously:

  Additional info: These formulas allow comparison of how different compounding frequencies affect the final amount.

Comparison of Compounding Methods

The following table summarizes the formulas for different compounding frequencies:

Key Terms

• Exponential Function: A function of the form , where and .

• Natural Exponential Function: The exponential function with base .

• Compound Interest: Interest calculated on both the initial principal and the accumulated interest from previous periods.

• Continuous Compounding: The process of calculating interest and adding it to the principal an infinite number of times per year.