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4.2 The Natural Exponential Function

Focus on natural exponential and natural logarithm inverse.

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Lesson brief and rules

4.2 The Natural Exponential Function

$e^{x}$ has base $e$ ; $e^{0}=1$ and $e^{x}e^{y}=e^{x+y}$ .

Worked example

1

01 / Illustrate the limiting definition of

e

with the positive integers

2,10,50,250,2500

and a common-axis graph of

(1+\dfrac{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.

02 /

Positive integer	Approximation
$2$	$2.250000$
$10$	$2.593742$
$50$	$2.691588$
$250$	$2.712865$
$2500$	$2.717738$

03 /

e\approx2.718282

is the limiting value, not any one finite table entry. The natural exponential is

f(x)=e^{x}

04 / Reference plot. Curve

1

\left(1 + \dfrac{1}{x}\right)^{x}

. Curve

2

e

05 / The curve stays below the horizontal reference and approaches it. A finite table or graph illustrates the limiting definition but does not prove convergence.

06 /

P(1+\dfrac{r}{n})^{nt}=P\left[(1+\dfrac{1}{m})^{m}\right]^{rt}

after setting

m=\dfrac{n}{r}

for a positive rate. As frequency increases, the bracket approaches

e

, giving

A(t)=P e^{rt}

Worked example 2 · Define the natural base and derive continuous compounding

Illustrate the limiting definition of $e$ with the positive integers $3,15,75,375,3750$ and a common-axis graph of $(1+\dfrac{1}{x})^{x}$ 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.

Positive integer	Approximation
$3$	$2.370370$
$15$	$2.632879$
$75$	$2.700379$
$375$	$2.714666$
$3750$	$2.717919$

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+\dfrac{r}{n})^{nt}=P\left[(1+\dfrac{1}{m})^{m}\right]^{rt}

after setting

m=\dfrac{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.

15 questions with a worked answer for each.Start practice ↗

Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.

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4.2 The Natural Exponential Function

Ideas in this lesson

4.2 The Natural Exponential Function