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:

1. 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

2. 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.

Exponential Functions: Growth and Decay

This page introduces the concept of exponential functions and their applications in growth and decay scenarios. It provides a comprehensive overview of the fundamental principles and practical examples.

Definition: An exponential function is defined as f(x) = a·bˣ, where 'a' is the initial value and 'b' is the base.

The nature of exponential functions depends on the value of 'b':

• If b > 1, the function represents exponential growth.
• If 0 < b < 1, the function represents exponential decay.

Vocabulary:

• Growth factor: The value of 'b' when b > 1
• Growth rate: b - 1
• Decay factor: The value of 'b' when 0 < b < 1
• Decay rate: 1 - b

Example: In the function f(x) = 5·1.1ˣ, the growth rate is 1.1 - 1 = 0.1 or 10%.

The page includes several practical examples to illustrate the application of exponential functions:

1. Population Growth: A city with 75,000 people growing at 3% per year.

   Example: P(t) = 75,000 · 1.03ᵗ

2. Car Depreciation: A car depreciating at 10% per year.

   Example: A(t) = 19,000 · 0.9ᵗ

3. Radioactive Decay: A substance weighing 250 mg initially, decaying over time.

   Example: y = 250 · 0.933ᵗ

The page also covers methods for finding exponential functions given two points and solving problems involving future values.

Highlight: The document emphasizes the importance of understanding growth and decay factors in real-world applications of exponential functions.