When a quantity changes in proportion to itself (for instance, bacteria reproduction or radioactive decay), the growth or decay is exponential in nature. There are many many examples of this to be found.
Incoming and Outgoing Shipments in 3 STEPS Using Odoo 17
Lesson 16: Exponential Growth and Decay
1. Section 3.4
Exponential Growth and Decay
V63.0121.034, Calculus I
October 26, 2009
Announcements
Quiz 3 this week in recitation
. . . . . .
2. Outline
Recall
The equation y′ = ky
Modeling simple population growth
Modeling radioactive decay
Carbon-14 Dating
Newton’s Law of Cooling
Continuously Compounded Interest
. . . . . .
4. Outline
Recall
The equation y′ = ky
Modeling simple population growth
Modeling radioactive decay
Carbon-14 Dating
Newton’s Law of Cooling
Continuously Compounded Interest
. . . . . .
9. The equation y′ = ky
Example
Find a solution to y′ (t) = y(t).
Find the most general solution to y′ (t) = y(t).
. . . . . .
10. The equation y′ = ky
Example
Find a solution to y′ (t) = y(t).
Find the most general solution to y′ (t) = y(t).
Solution
A solution is y(t) = et .
. . . . . .
11. The equation y′ = ky
Example
Find a solution to y′ (t) = y(t).
Find the most general solution to y′ (t) = y(t).
Solution
A solution is y(t) = et .
The general solution is y = Cet , not y = et + C.
(check this)
. . . . . .
13. In general
Example
Find a solution to y′ = ky.
Find the general solution to y′ = ky.
. . . . . .
14. In general
Example
Find a solution to y′ = ky.
Find the general solution to y′ = ky.
Solution
y = ekt
y = Cekt
. . . . . .
15. In general
Example
Find a solution to y′ = ky.
Find the general solution to y′ = ky.
Solution
y = ekt
y = Cekt
Remark
What is C? Plug in t = 0:
y(0) = Cek·0 = C · 1 = C,
so y(0) = y0 , the initial value of y.
. . . . . .
16. Exponential Growth
It means the rate of change (derivative) is proportional to the
current value
Examples: Natural population growth, compounded interest,
social networks
again,
. . . . . .
17. Outline
Recall
The equation y′ = ky
Modeling simple population growth
Modeling radioactive decay
Carbon-14 Dating
Newton’s Law of Cooling
Continuously Compounded Interest
. . . . . .
18. Bacteria
Since you need bacteria
to make bacteria, the
amount of new bacteria
at any moment is
proportional to the total
amount of bacteria.
This means bacteria
populations grow
exponentially.
. . . . . .
19. Bacteria Example
Example
A colony of bacteria is grown under ideal conditions in a
laboratory. At the end of 3 hours there are 10,000 bacteria. At
the end of 5 hours there are 40,000. How many bacteria were
present initially?
. . . . . .
20. Bacteria Example
Example
A colony of bacteria is grown under ideal conditions in a
laboratory. At the end of 3 hours there are 10,000 bacteria. At
the end of 5 hours there are 40,000. How many bacteria were
present initially?
Solution
Since y′ = ky for bacteria, we have y = y0 ekt . We have
10, 000 = y0 ek·3 40, 000 = y0 ek·5
. . . . . .
21. Bacteria Example
Example
A colony of bacteria is grown under ideal conditions in a
laboratory. At the end of 3 hours there are 10,000 bacteria. At
the end of 5 hours there are 40,000. How many bacteria were
present initially?
Solution
Since y′ = ky for bacteria, we have y = y0 ekt . We have
10, 000 = y0 ek·3 40, 000 = y0 ek·5
Dividing the first into the second gives
4 = e2k =⇒ 2k = ln 4 =⇒ k = ln 2. Now we have
10, 000 = y0 eln 2·3 = y0 · 8
10, 000
So y0 = = 1250.
8
. . . . . .
22. Could you do that again please?
We have
10, 000 = y0 ek·3
40, 000 = y0 ek·5
Dividing the first into the second gives
40, 000 y e5k
= 0 3k
10, 000 y0 e
4 = e2k
ln 4 = ln(e2k ) = 2k
ln 4 ln 22 2 ln 2
k= = = = ln 2
2 2 2
. . . . . .
23. Outline
Recall
The equation y′ = ky
Modeling simple population growth
Modeling radioactive decay
Carbon-14 Dating
Newton’s Law of Cooling
Continuously Compounded Interest
. . . . . .
24. Modeling radioactive decay
Radioactive decay occurs because many large atoms
spontaneously give off particles.
. . . . . .
25. Modeling radioactive decay
Radioactive decay occurs because many large atoms
spontaneously give off particles.
This means that in a sample
of a bunch of atoms, we can
assume a certain percentage
of them will “go off” at any
point. (For instance, if all
atom of a certain radioactive
element have a 20% chance
of decaying at any point,
then we can expect in a
sample of 100 that 20 of
them will be decaying.)
. . . . . .
28. Thus the relative rate of decay is constant:
y′
=k
y
where k is negative. So
y′ = ky =⇒ y = y0 ekt
again!
It’s customary to express the relative rate of decay in the units of
half-life: the amount of time it takes a pure sample to decay to
one which is only half pure.
. . . . . .
31. Carbon-14 Dating
The ratio of carbon-14 to
carbon-12 in an organism
decays exponentially:
p(t) = p0 e−kt .
The half-life of carbon-14 is
about 5700 years. So the
equation for p(t) is
ln2
p(t) = p0 e− 5700 t
Another way to write this
would be
p(t) = p0 2−t/5700
. . . . . .
36. Outline
Recall
The equation y′ = ky
Modeling simple population growth
Modeling radioactive decay
Carbon-14 Dating
Newton’s Law of Cooling
Continuously Compounded Interest
. . . . . .
37. Newton’s Law of Cooling
Newton’s Law of
Cooling states that the
rate of cooling of an
object is proportional to
the temperature
difference between the
object and its
surroundings.
. . . . . .
38. Newton’s Law of Cooling
Newton’s Law of
Cooling states that the
rate of cooling of an
object is proportional to
the temperature
difference between the
object and its
surroundings.
This gives us a
differential equation of
the form
dT
= k (T − T s )
dt
(where k < 0 again).
. . . . . .
39. General Solution to NLC problems
To solve this, change the variable y(t) = T(t) − Ts . Then y′ = T′
and k(T − Ts ) = ky. The equation now looks like
dy
= ky
dt
. . . . . .
40. General Solution to NLC problems
To solve this, change the variable y(t) = T(t) − Ts . Then y′ = T′
and k(T − Ts ) = ky. The equation now looks like
dy
= ky
dt
which we can solve:
y = Cekt
T − Ts = Cekt
=⇒ T = Cekt + Ts
. . . . . .
41. General Solution to NLC problems
To solve this, change the variable y(t) = T(t) − Ts . Then y′ = T′
and k(T − Ts ) = ky. The equation now looks like
dy
= ky
dt
which we can solve:
y = Cekt
T − Ts = Cekt
=⇒ T = Cekt + Ts
Here C = y0 = T0 − Ts .
. . . . . .
42. Example
A hard-boiled egg at 98◦ C is put in a sink of 18◦ C water. After 5
minutes, the egg’s temperature is 38◦ C. Assuming the water has
not warmed appreciably, how much longer will it take the egg to
reach 20◦ C?
. . . . . .
43. Example
A hard-boiled egg at 98◦ C is put in a sink of 18◦ C water. After 5
minutes, the egg’s temperature is 38◦ C. Assuming the water has
not warmed appreciably, how much longer will it take the egg to
reach 20◦ C?
Solution
We know that the temperature function takes the form
T(t) = (T0 − Ts )ekt + Ts = 80ekt + 18
To find k, plug in t = 5:
38 = T(5) = 80e5k + 18
and solve for k.
. . . . . .
48. Solution
Let time 0 be midnight. We know T0 = 31, Ts = 21, and
T(1) = 29. We want to know the t for which T(t) = 37.
. . . . . .
49. Solution
Let time 0 be midnight. We know T0 = 31, Ts = 21, and
T(1) = 29. We want to know the t for which T(t) = 37.
To find k:
29 = 10ek·1 + 21 =⇒ k = ln 0.8
. . . . . .
50. Solution
Let time 0 be midnight. We know T0 = 31, Ts = 21, and
T(1) = 29. We want to know the t for which T(t) = 37.
To find k:
29 = 10ek·1 + 21 =⇒ k = ln 0.8
To find t:
37 = 10et·ln(0.8) + 21
1.6 = et·ln(0.8)
ln(1.6)
t= ≈ −2.10 hr
ln(0.8)
So the time of death was just before 10:00pm.
. . . . . .
51. Outline
Recall
The equation y′ = ky
Modeling simple population growth
Modeling radioactive decay
Carbon-14 Dating
Newton’s Law of Cooling
Continuously Compounded Interest
. . . . . .
52. Interest
If an account has an compound interest rate of r per year
compounded n times, then an initial deposit of A0 dollars
becomes ( r )nt
A0 1 +
n
after t years.
. . . . . .
53. Interest
If an account has an compound interest rate of r per year
compounded n times, then an initial deposit of A0 dollars
becomes ( r )nt
A0 1 +
n
after t years.
For different amounts of compounding, this will change. As
n → ∞, we get continously compounded interest
( r )nt
A(t) = lim A0 1 + = A0 ert .
n→∞ n
. . . . . .
54. Interest
If an account has an compound interest rate of r per year
compounded n times, then an initial deposit of A0 dollars
becomes ( r )nt
A0 1 +
n
after t years.
For different amounts of compounding, this will change. As
n → ∞, we get continously compounded interest
( r )nt
A(t) = lim A0 1 + = A0 ert .
n→∞ n
Thus dollars are like bacteria.
. . . . . .
57. I-banking interview tip of the day
ln 2
The fraction can
r
also be approximated as
either 70 or 72 divided
by the percentage rate
(as a number between 0
and 100, not a fraction
between 0 and 1.)
This is sometimes called
the rule of 70 or rule of
72.
72 has lots of factors so
it’s used more often.
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