Higher Engineering Mathematics, Sixth Edition

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