Applied Calculus Math 215 - University of Hawaii

28 CHAPTER 1. SOME BACKGROUND MATERIAL
Exponential Functions grow fast.
Example 1.21 (Exponential Growth). It is not so apparent from the
graph how fast the exponential function grows. You may remember the tale
of the ancient king who, as payment for a lost game of chess, was willing to
put 1 grain of wheat on the first square on the chess board, 2 on the second,
4 on the third, 8 on the forth, etc., doubling the number of grains with each
square. The chess board has 64 squares, and that commits him to 2 63 grains
on the 64th square for a total of
2 64 − 1=18,446, 744, 073, 709, 551, 615
grains. In mathematical notation, you say that he puts
f(n) =2 n−1
grains on the n-th square of the chess board. So, let us graph the function
f(x) =2 x for 0 ≤ x ≤ 63, see Figure 1.14. On the given scale in the graph,
even an already enormous number like 2 54 , cannot be distinguished from 0.
18
8. 10
18
6. 10
18
4. 10
18
2. 10
10 20 30 40 50 60
Figure 1.14: Graph of f(x) =2 x
It is difficult to imagine how large these numbers are. The amount of
grain which the king has to put on the chess board suffices to feed the current
world population (of about 6 billion people) for thousands of years. ♦