The number e and exponential functions with base e

Three quick questions from earlier lessons.

Compute $\left(1 + \frac{1}{x}\right)^x$ for $x = 1$ , $10$ , $100$ and $1000$ with a calculator.

What do you notice as $x$ gets larger?

The values approach a number called $e \approx 2.718$ . Functions with base $e$ describe continuous growth.

Continuous growth at rate

r

(as a decimal):

A(t) = A_0 \, e^{rt}

In $f(t) = a \, e^{kt}$ , increasing $t$ by 1 multiplies $f$ by $e^{k}$ .

The area of a mold is $a(t) = 10e^{t}$ mm² after $t$ days. By what factor does it grow each day, and what is the area after 2 days?

Daily factor:

e \approx 2.718

Here k = 1, so each day multiplies the area by e¹.

a(2) = 10e^{2}

Substitute t = 2.

a(2) \approx 73.9

mm²

e² ≈ 7.389.

Your turn: 500 dollars grows continuously at 4% per year. Fill in the missing line.

r = 0.04

4% as a decimal.

A(t) = 500e^{0.04t}

Use A₀ e^(rt).

An account holds 400 dollars and grows continuously at 3% per year. Write the amount $A(t)$ after $t$ years.
An account holds 100 dollars and grows continuously at 8% per year. Write the amount $A(t)$ after $t$ years.
An account holds 300 dollars and grows continuously at 3% per year. Write the amount $A(t)$ after $t$ years.
An account holds 600 dollars and grows continuously at 6% per year. Write the amount $A(t)$ after $t$ years.
An account holds 500 dollars and grows continuously at 5% per year. Write the amount $A(t)$ after $t$ years.

The area of mold is $f(t) = 2e^{ 4t }$ square millimetres after $t$ days. By what factor is the area multiplied each day? Give the exact value.
The area of mold is $f(t) = 3e^{ 1t }$ square millimetres after $t$ days. By what factor is the area multiplied each day? Give the exact value.
The area of mold is $f(t) = 7e^{ 4t }$ square millimetres after $t$ days. By what factor is the area multiplied each day? Give the exact value.
The area of mold is $f(t) = 2e^{ 2t }$ square millimetres after $t$ days. By what factor is the area multiplied each day? Give the exact value.
The area of mold is $f(t) = 3e^{ 4t }$ square millimetres after $t$ days. By what factor is the area multiplied each day? Give the exact value.

Three questions to check you have got it.

The area of mold is $f(t) = 2e^{ 4t }$ square millimetres after $t$ days. By what factor is the area multiplied each day? Give the exact value.
The area of mold is $f(t) = 3e^{ 1t }$ square millimetres after $t$ days. By what factor is the area multiplied each day? Give the exact value.
The area of mold is $f(t) = 7e^{ 4t }$ square millimetres after $t$ days. By what factor is the area multiplied each day? Give the exact value.

The number $e$

Watch out: "continuously at $r$ " is not "by $1 + r$ once a year".

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