The Chemistry Maths Book, Second Edition

3.6 The exponential function 81

The terms are decreasing rapidly in magnitude, and the first five are sufficient

(compare Example 1.10):

exp(−0.02) 1 ≈ 1 0.980 1198167

0 Exercises 33, 34

The graphs of e

x

and its reciprocal e

−x

are shown in Figure 3.18. The graphs of all the

exponential functions a

x

are very similar, with properties (for alla 1 > 10 )

a

0

1 = 1 1, a

x

1 → 1 ∞asx 1 → 1 ∞, a

x

1 → 1 0asx 1 → 1 −∞ (3.32)

It will be seen in Chapter 4 that the unique property of e

x

is that the slope of its graph

at any point is equal to the value of the function at that point:

(3.33)

The exponential function occurs in nearly all branches of applied mathematics,

including statistics, kinetics, electromagnetic theory, quantum mechanics, and statis-

tical mechanics.

0 Exercises 35

EXAMPLE 3.20Exponential growth and decay

Exponential growth arises when the rate of growth of a system at any time is

proportional to the size of the system at that time. Ifx(t)is the size at time tthen

(see Chapter 4) the rate of change of xis

dx

dt

=±kx t()

de

dx

e

x

=

e

x

e

−x

o

1

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Figure 3.18