4.05 The natural exponent

We will now look at a special number, the mathematical constant, $e$e. $e$e is defined as $\lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)^n$limn→∞(1+1n​)n and is the irrational number $2.7128\dots$2.7128…. In a 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=2x, $y=e^x$y=ex and $y=3^x$y=3x
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=a0=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. 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 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.

 

Practice questions

Question 1

Question 2

Question 3