Understanding Engineering Mathematics

Solution to review question 4.1.3

A. We m a y d e fi n eeby the limit

e= lim

n→∞

(

1 +

1

n

)n

but it is more easily evaluated from the equivalent series

e= 1 + 1 +

1

2!

+

1

3!

+

1

4!

+···+

1

r!

+···(exwithx= 1 )

To 3 decimal placese= 2 .718. We cannot write down the exact

numerical value ofe, since it is an irrational number (6

➤

). Its

decimal never terminates or recurs.

B. (i) We only have to remember that the exponential function behaves

like any other power (18

➤

). We have

e^2 x=(ex)^2 =( 2 )^2 = 4

(ii) e−x=(ex)−^1 =( 2 )−^1 =^12

(iii) e^3 x=(ex)^3 =( 2 )^3 = 8

(iv) We only need (i) now:

e^4 x− 4 e^2 x=(e^2 x)^2 − 4 e^2 x=( 4 )^2 − 4 × 4 = 0

C. Sincee>1,y=exis an increasing function ofxande−xis a

decreasing function. Similarly,y=e^2 xis increasing,y=e−^3 xis

decreasing. The graphs are illustrated in Figure 4.3.

y

x

1

0

y = e−x

y = e−^3 x y = e 2 x

y = ex

Figure 4.3The exponential functionsex,e−x,e^2 x,e−^3 x.