There is a special mathematical constant, $e$e, called Euler's number which is of particular importance with regards to exponential functions. It is often referred to as the "natural base" and can be defined as $\lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)^n$limn→∞(1+1n)n. Other definitions and information about $e$e can be found here. It is an irrational number, a bit like $\pi$π , but it is approximately $2.718281828\dots$2.718281828…. To display $e$e on a calculator you often have to type $e^1$e1.

Practice questions

question 1

Use a calculator or other technology to approximate the following values correct to four decimal places:

$e^4$e4
$e^{-1}$e−1
$e^{\frac{1}{5}}$e15

question 2

The natural base $e$e (Euler’s number) is defined as $e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$e=limn→∞(1+1n)n

The table shows the value of $\left(1+\frac{1}{n}\right)^n$(1+1n)n using various values of $n$n.

$n$`n`	$\left(1+\frac{1}{n}\right)^n$(1+1`n`)`n`
$1$1	$2$2
$100$100	$1.01^{100}=2.704813$1.01100=2.704813 ...
$1000$1000	$1.001^{1000}=2.716923$1.0011000=2.716923 ...
$10000$10000	$1.0001^{10000}=2.718145$1.000110000=2.718145 ...
$100000$100000	$1.00001^{100000}=2.718268$1.00001100000=2.718268 ...

Evaluate $\left(1+\frac{1}{n}\right)^n$(1+1n)n for $n=1000000$n=1000000, correct to six decimal places.
Which of the following is the closest approximation of $e$e?
$2.718280821$2.718280821
A
$2.718281828$2.718281828
B
$2.718281820$2.718281820
C
$2.718281818$2.718281818
D

question 3

It is possible to compute $e^x$ex using the following formula.

$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\text{. . .}+\frac{x^n}{n!}+\text{. . .}$ex=1+x+x22!+x33!+x44!+. . .+xnn!+. . .

The more terms we use from the formula, the closer we get to the true value of $e^x$ex.

Use the first five terms of the formula to estimate the value of $e^{0.7}$e0.7.

Give your answer to six decimal places.
Use the $\editable{e^x}$ex key on your calculator to find the value of $e^{0.7}$e0.7.

Give your answer to six decimal places.
What is the difference between the two results?

Give your answer to six decimal places.

The graph of $y=e^x$`y`=`ex`

In our previous lesson we explored exponential functions of the form $y=a^x$y=ax, a very important member of this family of curves is $y=e^x$y=ex.

As a member of this family of curves we can see it has the same properties and sits between the functions $y=2^x$y=2x and $y=3^x$y=3x.

Graphs of $y=2^x$`y`=2`x`, $y=e^x$`y`=`ex` and $y=3^x$`y`=3`x`

Key features: Exponential growth: As the $x$`x`-values increase, the $y$`y`-values increase at an increasing rate. The $y$`y`-intercept is $\left(0,1\right)$(0,1), since when $x=0$`x`=0, $y=a^0=1$`y`=`a`0=1, for any positive value $a$`a`. $y=0$`y`=0 is a horizontal asymptote for each graph. As $x\ \rightarrow-\infty$`x` →−∞, $y\ \rightarrow0^+$`y` →0+ Domain: $x$`x` is any real number Range: $y>0$`y`>0

The three functions above can be viewed as horizontal dilations of each other. Using transformations we can in fact write any exponential function with a base of $e$e. Using a base of $e$e will become fundamental in exponential applications involving calculus. The function $f\left(x\right)=e^x$f(x)=ex has the remarkable property that $f'\left(x\right)=f\left(x\right)$f′(x)=f(x) and we will see using a base of $e$e will often simplify required processes and calculations.

Just as with functions of the form $y=a^x$y=ax we can perform transformations on the graph $y=e^x$y=ex.

Worked example

(a) For the function $y=e^x$y=ex, describe the transformations required to obtain the graph of $y=2\times e^x+3$y=2×ex+3.

Think: For the function $y=A\times e^{\left(bx-h\right)}+k$y=A×e(bx−h)+k, what impact does each parameter have? Which parameters have been altered?

Do: We have $A=2$A=2 and $k=3$k=3, hence, the function has been vertically dilated by a factor of $2$2 and a vertical translation by a $3$3 units upwards.

(b) Sketch the function.

Think: From the basic graph of $y=e^x$y=ex a vertical dilation by a factor of $2$2 will stretch each point away from the $x$x-axis by a factor of $2$2 (double the $y$y values). Hence, the point $\left(0,1\right)$(0,1) becomes $\left(0,2\right)$(0,2), $\left(1,e\right)$(1,e) becomes $\left(1,2e\right)$(1,2e) and so forth. Then we can shift the graph $3$3 units vertically, each point moves up $3$3 units (add $3$3 to the $y$y values) and the horizontal asymptote becomes $y=3$y=3.

Do:


Step 1. Dilate the graph by a factor of $2$2 from the $x$`x`-axis.	Step 2. Translate the graph upwards $3$3 units, this includes the horizontal asymptote.