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Exploring the Number e in Depth(vid45)

Feb 13, 2025

Lecture Notes: Exponential Functions and the Number e

Introduction to Number e

  • Definition:

    • e is an irrational number similar to Ï€ (pi), with an approximate value of 2.71.
    • Like Ï€, 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.