082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

Table2.2
Values ofa x fora = 0.5, 2 and 3.
2.4 Review of some common engineering functions and techniques 81
x 0.5 x 2 x 3 x
(0.5) x
y
3 x
2 x
−3 8 0.125 0.037
−2 4 0.25 0.111
−1 2 0.5 0.333
0 1 1 1
1 0.5 2 3
2 0.25 4 9
3 0.125 8 27
Figure2.26
Some typical exponential functions.
x
Themostwidelyusedexponentialfunction,commonlycalledtheexponentialfunction,
is
f(x)=e x
where e is an irrational constant (e = 2.718281828...) commonly called the
exponential constant.
Most scientific calculators have values of e x available. The function is tabulated
in Table 2.3. The graph is shown in Figure 2.27. This particular exponential function
so dominates engineering applications that whenever an engineer refers to the exponential
function it almost invariably means this one. We will see later why it is so
important.
Asxincreasespositively,e x increasesveryrapidly;thatis,asx → ∞,e x → ∞.This
situationisknownasexponentialgrowth.Asxincreasesnegatively,e x approacheszero;
that is, asx → −∞, e x → 0. Thusy = 0 is an asymptote. Note that the exponential
function isnever negative.
Figure2.28showsagraphofe −x .Asxincreasespositively,e −x decreasestozero;that
is,asx → ∞,e −x → 0.Thisisknownasexponentialdecay.Thefunctionistabulated
inTable 2.4.
Table2.3
Thevalues ofthe exponential
function f (x) = e x for
variousvalues ofx.
x
e x
−3 0.050
−2 0.135
−1 0.368
0 1
1 2.718
2 7.389
3 20.086
e x 1
Figure2.27
Graph ofy = e x showing exponential growth.
x