Exponential Functions: Advanced Concepts and Applications
This page continues the discussion on exponential functions, focusing on more advanced concepts and their practical applications. It introduces formulas for compound interest and continuous compounding.
Definition: Compound interest is the addition of interest to the principal sum of a loan or deposit, resulting in interest on interest.
The page presents two key formulas for calculating future value (FV) in compound interest scenarios:
-
For regular compound interest:
FV = P(1 + r/n)^(nt)
Where:
- FV = Future Value
- P = Present Value (initial principal)
- r = Interest rate (as a decimal)
- n = Number of times interest is compounded per year
- t = Number of years
-
For continuous compounding:
FV = Pe^(rt)
Where:
- e ≈ 2.71828 (Euler's number)
- r = Interest rate (as a decimal)
- t = Time in years
Highlight: The concept of continuous compounding represents the theoretical limit of compound interest, where interest is calculated and added to the principal continuously.
The page includes visual aids and graphs to illustrate the behavior of exponential functions and the differences between various compounding frequencies.
Example: While specific numerical examples are not provided on this page, it's implied that these formulas can be applied to various financial scenarios, such as investment growth or loan repayments.
The document emphasizes the importance of understanding these advanced concepts for more complex financial calculations and modeling of exponential growth in various fields, including economics, biology, and physics.
