Lecture Notes: Exponential Functions and the Number e

Introduction to Number e

• Definition:

  • e is an irrational number similar to π (pi), with an approximate value of 2.71.
  • Like π, e has an infinite decimal expansion.

• Importance of e:

  • The function y = e^x is described as having a "natural base."
  • Significant in natural processes such as bacterial growth and decay of compounds.
  • Useful in calculus for understanding growth rates and decay processes.

Derivation of e

• Method to Derive e:
  • Use the formula ((1 + \frac{1}{n})^n) as n becomes very large.
  • Examples:
    • n = 1000: yields approximately 2.7169
    • n = 10,000: yields approximately 2.718
    • n = 1,000,000: approximates the true value of e
  • Exact value of e: 2.7182818...

Graphical Representation

• Graphs: e^x² and e^-x²
  • e^x²: Rapid exponential growth on both sides of the y-axis.
  • e^-x²: Maximum value occurs at x = 0. The graph shows rapid decay.
  • Explanation: y = \frac{1}{e^{x²}} results in 1 over exponentially increasing numbers, leading to rapid decrease.

Application Example

• Wine Consumption Model:

  • Model formula: 34e^{0.029x}
  • x = 50 corresponds to the year 1950.

• Calculations:

  • Year 2000 (x = 100): 617 million gallons consumed.
  • Year 2015 (x = 115): 954 million gallons consumed.

• Future Projections:

  • Determine when consumption exceeds 1100 million gallons.
  • First year exceeding: 2020, with 1.1 billion gallons.

Conclusion

• Reviewed the exponential function with base e and its applications, specifically focusing on its role in growth models.

• Highlighted significant increase in wine consumption over 20 years, illustrating the application of exponential functions in real-world data.

• Upcoming topics will be discussed in the next session.