Understanding Engineering Mathematics

behaviour of the exponential function by looking at the behaviour of the power function
for different values of the index, as shown in the review question.

Solution to review question 4.1.1

(i) If you have done RE 3.3.2A(iii) then you have already met this. The

values ofy= 2 nare:

n −4, − 3 − 2 −10123 4

2 n 161 18 14 12 124816

The corresponding graph (Figure 4.1) rises very steeply (note that the

scales on the axes differ).

(ii) The graph fory= 2 −nis also shown in Figure 4.1.

y

− 4 − 3 − 2 − 1 0 1 2 3 4 x

10

15

5

1

y = 2 x

y = 3 x

y = 2 −x

y = 3 −x

Figure 4.1Graphs of exponential functionse^2 x,e^3 x,e−^2 x,e−^3 x.

(iii) Drawing a smooth curve through the points fory= 2 ngives the curve

fory= 2 x. Similarly fory= 2 −x.y= 3 x( 3 −x)will be similar but

will increase (decrease) more rapidly (see Figure 4.1).

4.2.2 The general exponential functionax

➤

119 136➤

In Section 2.2.12 we covered the laws of indices, including such things asam×an=am+n.
Although it was not the intention, it was perhaps easy to get the impression that the indices