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Module 4 · L3 of 15 ~35 min +95 XP available

The Number $e$ and Natural Exponentials

You've met $\pi$ — the ratio that circles chose. Now meet $e \approx 2.71828$, the number calculus itself chose. It's the unique base where the exponential function is its own derivative. By the end of this lesson you'll know why every growth and decay model in science defaults to it.

Today's hook — You invest $1 at 100% interest. Compound it yearly: you get $2. Monthly: $2.61. Daily: $2.714. Every second: $2.71828... It never exceeds $e$. Why does compounding more often hit a ceiling — and why is that ceiling so mathematically important?

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Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question

Recall — your gut answer first

+5 XP warm-up

You invest $\$1$ at $100\%$ interest. As you compound more often — yearly, monthly, daily, every second — the final amount climbs. Without using a formula: does it grow without limit, or does it approach a ceiling? Pick a guess: under 2, between 2–3, or above 3?

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The two moves

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Everything in this lesson spins out of two core skills. Lock these in and the rest is just applying them to new contexts.

Recognise $e$ and its properties

Identify $e \approx 2.718$ and remember that $\dfrac{d}{dx}(e^x) = e^x$.

Apply to continuous compounding

Use $A = Pe^{rt}$ for continuously compounded problems. Identify $P$, $r$, $t$ before substituting.

Use the $e^x$ button

$e$ is irrational — always use the $e^x$ or exp function on your calculator, not the approximation 2.718.

What you'll master

Know

Key facts

The definition $e = \lim_{n\to\infty}(1+\frac{1}{n})^n$
That $e \approx 2.71828$ is irrational
The continuous compounding formula $A = Pe^{rt}$

Understand

Concepts

Why $e^x$ is its own derivative
How $y = e^x$ and $y = e^{-x}$ relate as reflections
The difference between $e^x$ (exponential) and $x^e$ (power function)

Can do

Skills

Evaluate $e^x$ for any value using a calculator
Sketch $y = e^x$, $y = e^{-x}$, and transformations
Solve continuous compounding problems

Key terms

Euler's number ($e$)The irrational constant $e \approx 2.71828\ldots$, named after Leonhard Euler.

Natural exponentialThe exponential function with base $e$: $y = e^x$.

Continuous compoundingInterest compounded infinitely often, modelled by $A = Pe^{rt}$.

Self-derivative property$\frac{d}{dx}(e^x) = e^x$; $e^x$ is the unique function (up to constant) equal to its own derivative.

Horizontal asymptote$y = e^x$ approaches $y = 0$ as $x \to -\infty$ but never touches it.

$y$-interceptFor $y = e^x$, the $y$-intercept is $(0, 1)$ since $e^0 = 1$.

What is $e$?

core concept

The number $e$ is defined as the limit of $(1 + \frac{1}{n})^n$ as $n$ grows without bound. The table below shows how this converges:

As $n$ increases, $(1 + \frac{1}{n})^n$ approaches $e$ from below. It never exceeds $e$ — this is why continuously compounded interest hits a ceiling. The function $y = e^x$ is special because $\dfrac{d}{dx}(e^x) = e^x$ exactly — no scaling factor, no remainder. That's what makes it the natural base for calculus.

$$e = \lim_{n \to \infty} \left(1 + \dfrac{1}{n}\right)^n \approx 2.71828$$

$$\frac{d}{dx}(e^x) = e^x \qquad \int e^x \, dx = e^x + C$$

The graph of $y = e^x$ is an increasing exponential with asymptote $y = 0$, $y$-intercept $(0,1)$, and it passes through $(1, e) \approx (1, 2.718)$. Its mirror image $y = e^{-x}$ is a decreasing exponential, reflected in the $y$-axis.

Why $e$ matters beyond finance. In physics, radioactive decay follows $N = N_0 e^{-\lambda t}$. In biology, population growth uses $P = P_0 e^{kt}$. In every case, the reason $e$ appears is that it is the only base where the rate of change equals the current quantity — a statement that would be more complicated with any other base.

$e = \lim_{n \to \infty}(1 + \frac{1}{n})^n \approx 2.71828$ — irrational like $\pi$; $\frac{d}{dx}(e^x) = e^x$ — the self-derivative property

Pause — copy the definition of $e$ ($\approx 2.71828$, irrational) and the self-derivative property $\frac{d}{dx}(e^x) = e^x$ — the reason $e$ is the natural base into your book.

Did you get this? True or false: $e^x$ is the only function whose derivative equals itself (up to a constant multiple).

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · EVALUATING $e^x$

Evaluate $e^2$ and $e^{-1}$ to 3 decimal places.

$e^2 \approx 7.389$

Use the $e^x$ function on your calculator.

PROBLEM 2 · SKETCHING $y = e^x$ AND $y = e^{-x}$

Sketch $y = e^x$ and $y = e^{-x}$ on the same axes, identifying key features.

$y = e^x$: asymptote $y = 0$, passes through $(0, 1)$, increasing

Standard natural exponential growth curve.

PROBLEM 3 · CONTINUOUS COMPOUNDING

$\$10{,}000$ is invested at $5\%$ p.a. compounded continuously. Find the value after 10 years.

$A = Pe^{rt}$ where $P = 10000$, $r = 0.05$, $t = 10$

Identify the continuous compounding formula and all parameters.

Quick check: $\$5000$ is invested at $4\%$ p.a. compounded continuously for 5 years. Which expression gives the correct final amount?

Common errors · the 3 traps that cost marks

Trap 01

Using 2.718 instead of the $e^x$ button

$e$ is irrational and $2.718$ is only an approximation. For accurate answers, always use the $e^x$ or $\text{exp}(x)$ function on your calculator, especially in HSC exams.

Trap 02

Confusing $e^x$ with $x^e$

$e^x$ has the variable in the exponent (exponential function). $x^e$ has the variable in the base (power function). These are completely different functions with different derivatives and graphs.

Trap 03

Forgetting the chain rule when differentiating $e^{f(x)}$

$\frac{d}{dx}(e^x) = e^x$ only when the exponent is exactly $x$. For $e^{2x}$, use the chain rule: $\frac{d}{dx}(e^{2x}) = 2e^{2x}$. The derivative of $e^{f(x)}$ is $f'(x) \cdot e^{f(x)}$.

Complete the sentence: The continuous compounding formula is $A = P e^{[\,]}$, where $r$ is the annual rate as a decimal and $t$ is time in years.

Work mode · how are you completing this lesson?

Quick-fire practice · 5 problems

Evaluate $e^3$ and $e^{-2}$ to 2 decimal places.

State the exact value of $\dfrac{d}{dx}(e^x)$.

Sketch $y = e^x + 1$ showing the asymptote and $y$-intercept.

$\$5000$ is invested at $4\%$ p.a. compounded continuously. Find the value after 5 years.

Simplify $\ln(e^5)$.

Odd one out: Which of the following does NOT have $y$-intercept $(0, 1)$?

Revisit your thinking

Earlier you were asked: does continuous compounding at $100\%$ grow without limit? It approaches $e \approx 2.718$ — never exceeding it. The function $e^x$ is the only exponential whose derivative equals itself, which is why it sits at the heart of growth, decay, and modelling across all of science.

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Multiple choice

+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.

Short answer

ApplyBand 42 marks

Q1. Evaluate $e^{1.5}$ and $e^{-0.5}$ to 3 decimal places. (2 marks)

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ApplyBand 43 marks

Q2. $\$20{,}000$ is invested at $6\%$ p.a. compounded continuously. How many years until the investment doubles? Give your answer to 1 decimal place. (3 marks)

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AnalyseBand 54 marks

Q3. $\$10{,}000$ is invested at $8\%$ p.a. Compare the value after 5 years for: (i) annual compounding, (ii) monthly compounding, (iii) continuous compounding. Give answers to the nearest dollar. (4 marks)

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Comprehensive answers (click to reveal)

Drill 1: $e^3 \approx 20.09$; $e^{-2} \approx 0.14$ · 2: $e^x$ · 3: asymptote $y=1$, $y$-int $(0,2)$ · 4: $5000e^{0.2} \approx \$6107$ · 5: $5$

Q1 (2 marks): $e^{1.5} \approx 4.482$ [1]. $e^{-0.5} \approx 0.607$ [1].

Q2 (3 marks): $20000e^{0.06t} = 40000$ [0.5]. $e^{0.06t} = 2$ [0.5]. $0.06t = \ln 2$, so $t = \frac{\ln 2}{0.06} \approx \frac{0.693}{0.06} \approx 11.6$ years [2].

Q3 (4 marks): (i) Annual: $10000(1.08)^5 \approx \$14{,}693$ [1]. (ii) Monthly: $10000(1+\frac{0.08}{12})^{60} \approx \$14{,}890$ [1.5]. (iii) Continuous: $10000e^{0.4} \approx \$14{,}918$ [1.5].

Boss battle · The Compound Calculator

earn bronze · silver · gold

Five timed questions. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.

Enter the arena

Science Jump · platform challenge

Climb platforms by answering natural exponential questions. Lighter alternative to the boss.

Mark lesson as complete

Tick when you've finished the practice and review.

← Lesson 2 · Exponential Functions and Graphs Lesson 4 · Exponential Modelling and Equations →

Module overview · Maths Advanced · Checkpoint 1