Chapter 9
Exponential Growth and Decay:
Differential Equations
9.1
A differential equation is an equation in which some function is related to its own derivative(s). For
each of the following functions, calculate the appropriate derivative, and show that the function
satisfies the indicated
differential equation
(a)
f
(
x
) = 2
e

3
x
,
f
prime
(
x
) =

3
f
(
x
)
(b)
f
(
t
) =
Ce
kt
,
f
prime
(
t
) =
kf
(
t
)
(c)
f
(
t
) = 1

e

t
,
f
prime
(
t
) = 1

f
(
t
)
9.2
Consider the function
y
=
f
(
t
) =
Ce
kt
where
C
and
k
are constants. For what value(s) of these
constants does this function satisfy the equation
(a)
dy
dt
=

5
y
,
(b)
dy
dt
= 3
y
.
[Remark: an equation which involves a function and its derivative is called a differential equa
tion.]
9.3
Find a function that satisfies each of the following
differential equations
. [Remark: all your answers
will be exponential functions, but they may have different dependent and independent variables.]
(a)
dy
dt
=

y
,
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Math 102 Problems
Chapter 9
(b)
dc
dx
=

0
.
1
c
and
c
(0) = 20,
(c)
dz
dt
= 3
z
and
z
(0) = 5.
9.4
If 70% of a radioactive substance remains after one year, find its halflife.
9.5
Carbon 14
Carbon 14 has a halflife of 5730 years. This means that after 5730 years, a sample of Carbon 14,
which is a radioactive isotope of carbon will have lost one half of its original radioactivity.
(a) Estimate how long it takes for the sample to fall to roughly 0.001 of its original level of
radioactivity.
(b) Each gram of
14
C
has an activity given here in units of 12 decays per minute. After some time,
the amount of radioactivity decreases.
For example, a sample 5730 years old has only one
half the original activity level, i.e. 6 decays per minute. If a 1 gm sample of material is found
to have 45 decays per hour, approximately how old is it? (Note:
14
C
is used in radiocarbon
dating, a process by which the age of materials containing carbon can be estimated. W. Libby
received the Nobel prize in chemistry in 1960 for developing this technique.)
9.6
Strontium90
Strontium90 is a radioactive isotope with a halflife of 29 years. If you begin with a sample of 800
units, how long will it take for the amount of radioactivity of the strontium sample to be reduced
to
(a) 400 units
(b) 200 units
(c) 1 unit
9.7
More radioactivity
The halflife of a radioactive material is 1620 years.
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 Winter '09
 Allard

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