4 Exponential decay

As we have noted, the behaviour of ${\text{e}}^x$ as $x\to \infty$ is called exponential growth. In a similar manner we characterise the behaviour of the function ${\text{e}}^{-x}$ as $x\to \infty$ as exponential decay . The graphs of ${\text{e}}^x$ and ${\text{e}}^{-x}$ are shown in Figure 3.

Figure 3 :

Exponential growth is very rapid and similarly exponential decay is also very rapid. In fact ${\text{e}}^{-x}$ tends to zero so quickly as $x\to \infty$ that, no matter how large the constant $K$ is,

$x^K{\text{e}}^{-x}\to 0\text{as}x\to \infty$

The next Task investigates this.

Task!

Choose $K=10$ in the expression $x^K{\text{e}}^{-x}$ and calculate $x^K{\text{e}}^{-x}$ using your calculator for $x=5,10,15,20,25,30,35$ .

Answer

The topics of exponential growth and decay are explored further in Section 6.5.

Exercises

1. Find, to 3 d.p., the values of
   1. $2^{-8}$
   2. ${\left(5.1\right)}^4$
   3. ${\left(2∕10\right)}^{-3}$
   4. ${\left(0.111\right)}^6$
   5. $2^{1∕2}$
   6. ${\pi }^{\pi }$
   7. ${\text{e}}^{\pi ∕4}$
   8. ${\left(1.71\right)}^{-1.71}$
2.  Plot $y=x^3$ and $y={\text{e}}^x$ for $0<x<7$ . For which integer values of $x$ is ${\text{e}}^x>x^3$ ?

Answer