Higher Engineering Mathematics

32 NUMBER AND ALGEBRA

From the graph,whenx=2.2,y=3.87andwhen

y=1.6,x=−0.74.

Problem 17. Plot a graph ofy=^13 e−^2 xover

the rangex=− 1 .5tox= 1 .5. Determine from

the graph the value ofywhenx=− 1 .2 and the

value ofxwheny= 1 .4.

A table of values is drawn up as shown below.

x −1.5 −1.0−0.5 0 0.5 1.0 1.5

− 2 x 3210 − 1 − 2 − 3

e−^2 x 20.086 7.389 2.718 1.00 0.368 0.135 0.050

1

3

e−^2 x 6.70 2.46 0.91 0.33 0.12 0.05 0.02

A graph of^13 e−^2 xis shown in Fig. 4.5.

Figure 4.5

From the graph, whenx=−1.2,y=3.67 and

wheny=1.4,x=−0.72.

Problem 18. The decay of voltage,vvolts,

across a capacitor at timetseconds is given by

v=250 e

−t

(^3). Draw a graph showing the natural
decay curve over the first 6 seconds. From the
graph, find (a) the voltage after 3.4 s, and (b) the
time when the voltage is 150 V.
A table of values is drawn up as shown below.
t 01 2 3
e
−t
(^3) 1.00 0.7165 0.5134 0.3679
v=250 e
−t
(^3) 250.0 179.1 128.4 91.97
t 45 6
e
−t
(^3) 0.2636 0.1889 0.1353
v=250 e
−t
(^3) 65.90 47.22 33.83
The natural decay curve ofv=250 e
−t
(^3) is shown in
Fig. 4.6.
Figure 4.6
From the graph:
(a)when timet=3.4 s, voltagev=80 Vand
(b)when voltagev=150 V, timet=1.5 s.
Now try the following exercise.
Exercise 20 Further problems on exponen-
tial graphs

1. Plot a graph of y=3e^0.^2 xover the range
   x=−3tox=3. Hence determine the value
   ofywhenx= 1 .4 and the value ofxwhen
   y= 4 .5. [3.95, 2.05]


2. Plot a graph ofy=^12 e−^1.^5 xover a range
   x=− 1 .5tox= 1 .5 and hence determine the
   value ofywhenx=− 0 .8 and the value ofx
   wheny= 3 .5. [1.65,−1.30]