4.2 The Natural Exponential Function
Ideas in this lesson
- Define the natural base and derive continuous compounding
- Evaluate and transform natural exponentials
- Construct hyperbolic functions and compare catenaries
- Locate nonlinear exponential extrema
- Interpret continuous decay and saturation models
- Evaluate logistic models and carrying capacities
- Compare continuous and periodic investment models
- Read continuous-investment target times graphically
Define the natural base and derive continuous compounding · Worked example
Illustrate the limiting definition of \(e\) with the positive integers \(4,20,100,500,5000\) and a common-axis graph of \((1+1/x)^x\) and \(e\). Explain the evidence limitation and derive continuous compounding from the periodic formula. Show the requested work; written explanations and sketches require teacher review. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Answer: \(e\approx2.718282\) and \(A(t)=P e^{rt}\).
| Positive integer | Approximation |
|---|---|
| 4 | \(2.441406\) |
| 20 | \(2.653298\) |
| 100 | \(2.704814\) |
| 500 | \(2.715569\) |
| 5000 | \(2.718010\) |
\(e\approx2.718282\) is the limiting value, not any one finite table entry. The natural exponential is \(f(x)=e^x\).
Reference plot. Curve 1: \(\left(1 + \dfrac{1}{x}\right)^{x}\). Curve 2: \(e\).
The curve stays below the horizontal reference and approaches it. A finite table or graph illustrates the limiting definition but does not prove convergence.
\(P(1+r/n)^{nt}=P\left[(1+1/m)^m\right]^{rt}\) after setting \(m=n/r\) for a positive rate. As frequency increases, the bracket approaches \(e\), giving \(A(t)=P e^{rt}\).
Written-work review:
- Compute every listed limiting approximation and show the curve against the horizontal natural-base line.
- Distinguish a finite table or plot from proof of convergence.
- Show the substitution that transforms periodic compounding into the defining limit and derive the continuous model.