Basic Engineering Mathematics, Fifth Edition

Exponential functions 121

Table 16.1

x − 3. 0 − 2. 5 − 2. 0 − 1. 5 − 1. 0 − 0. 5 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0

ex 0. 05 0. 08 0. 14 0. 22 0. 37 0. 61 1.00 1. 65 2. 72 4. 48 7. 39 12. 18 20. 09

e−x 20. 09 12. 18 7. 39 4. 48 2. 72 1. 65 1.00 0. 61 0. 37 0. 22 0. 14 0. 08 0. 05

Figure 16.1 shows graphs ofy=exandy=e−x.

y

20

16

y  e y  ex

x

12

8

4

 3  2  1 0123 x

Figure 16.1

Problem 7. Plot a graph ofy= 2 e^0.^3 xover a

range ofx=−2tox=3. Then 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

2 e^0.^3 x 0.81 1.10 1.48 2.00 2.70 3.64 4.92

A graph ofy= 2 e^0.^3 xis shown plotted in Figure 16.2.
From the graph,whenx=2.2,y=3.87andwhen
y=1.6,x=−0.74

Problem 8. Plot a graph ofy=

1

3

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

1

3

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

y

5

y  2 e0.3x

4

3

2

1.6

3.87

1

0.74 2.2

 3  2  1 0123 x

Figure 16.2

A graph of

1

3

e−^2 xis shown in Figure 16.3.

From the graph,whenx=−1.2,y=3.67andwhen

y=1.4,x=−0.72

y

6

5

4

2

3

1

7

0.5 0.5 1.0 1.5 x

1.2 0.72

1.5 1.0

3.67

1.4

y  13 e^2 x

Figure 16.3

Problem 9. 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 6seconds. From the

graph, find (a) the voltage after 3.4sand(b)the

time when the voltage is 150V