30 Higher Engineering Mathematics
x − 3. 0 − 2. 5 − 2. 0 − 1. 5 − 1. 0 − 0. 5 0
ex 0.05 0.08 0.14 0.22 0.37 0.611.00
e−x 20.09 12.18 7.39 4.48 2.72 1.651.00
x 0.5 1.0 1.5 2.0 2.5 3.0
ex 1.65 2.72 4.48 7.39 12.18 20.09
e−x 0.61 0.37 0.22 0.14 0.08 0.05
Figure 4.1 shows graphs ofy=exandy=e−x
y
20
16
y 5 e^2 x y 5 ex
12
8
4
23 22 2 10 1 2 3x
Figure 4.1
Problem 6. Plot a graph ofy=2e^0.^3 xover a
range ofx=−2tox=3. Hence determine the value
ofywhenx= 2 .2 and the value ofxwheny= 1 .6.
A table of values is drawn up as shown below.
x − 3 − 2 − 1 0 1 2 3
0. 3 x −0.9 −0.6 −0.3 0 0.3 0.6 0.9
e^0.^3 x 0.4070.5490.7411.0001.3501.8222.460
2e^0.^3 x 0.81 1.10 1.48 2.00 2.70 3.64 4.92
A graph ofy=2e^0.^3 xis shown plotted in Fig. 4.2.
From the graph,whenx=2.2,y=3.87and when
y=1.6,x=−0.74.
y
5 y^5 2e
0.3x
4
3
1.6
3.87
1
2 0.74 2.2
23 22 2 10 1 2 3x
2
Figure 4.2
Problem 7. 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 3 2 1 0 − 1 − 2 − 3
e−^2 x 20.0867.3892.7181.000.3680.1350.050
1
3
e−^2 x 6.70 2.46 0.910.33 0.12 0.05 0.02
A graph of^13 e−^2 xisshowninFig.4.3.
7
6
5
4
3
3.67
1.4
2
1
2 0.5 0.5
2 0.72
2 1.0
2 1.2
2 1.5 1.0 1.5
y
x
1
3 e
y 5 22 x
Figure 4.3
From the graph,whenx=−1.2,y=3.67andwhen
y= 1. 4 ,x=− 0 .72.
Problem 8. The decay of voltage,vvolts, across
a capacitor at timetseconds is given by
v=250e
−t
(^3). Draw a graph showing the natural