4 Exponential decay

As we have noted, the behaviour of e x as x is called exponential growth. In a similar manner we characterise the behaviour of the function e x as x as exponential decay . The graphs of e x and e x are shown in Figure 3.

Figure 3 :

{ $y=e^x$ and $y=e^{-x}$}

Exponential growth is very rapid and similarly exponential decay is also very rapid. In fact e x tends to zero so quickly as x that, no matter how large the constant K is,

x K e x 0 as x

The next Task investigates this.

Task!

Choose K = 10 in the expression x K e x and calculate x K e x using your calculator for x = 5 , 10 , 15 , 20 , 25 , 30 , 35 .

x 5 10 15 20 25 30 35
x 10 e x 6.5 × 1 0 4 4.5 × 1 0 5 1.7 × 1 0 5 2.1 × 1 0 4 1324 55 1.7

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. ( 5.1 ) 4
    3. ( 2 10 ) 3
    4. ( 0.111 ) 6
    5. 2 1 2
    6. π π
    7. e π 4
    8. ( 1.71 ) 1.71
  2.  Plot y = x 3 and y = e x for 0 < x < 7 . For which integer values of x is e x > x 3 ?
    1. 0.004
    2. 676.520
    3. 125
    4. 0.0
    5. 1.414
    6. 36.462
    7. 2.193
    8. 0.400
  1. For integer values of x , e x > x 3 if x 5