Core 4 Differential Equations

1
Core 4 Differential Equations

• Exponential Growth and Decay
• 2 Cracking Questions
• More Logs

2
1. On Rabbit Island the rate of growth at a time
(t) is proportional to the size of the population
(P)
a) If the population was initially 200 and after
10 months the population has grown to
1000. Find A and b (2dp) to give the
particular solution of the equation b) What
would be the population after 2 years? c) How
long would it take the population to grow to
50000?
2. The radiation level (R) at Chernobyl is
modelled by the equation
t is the time (in months after the accident)
a) What was the radiation level immediately
after the accident? b) How long would it take
for the radiation to be half this amount? c) As
t ? ? what value does the radiation settle on?
This represents background radiation
3
Exponential Growth example question
On Rabbit Island the rate of growth at a time (t)
is proportional to the size of the population (P)

• If the population was initially 200 and after 10
  months the
• population has grown to 1000.
• Find A and b (2dp) to give the particular
  solution of the equation
• b) What would be the population after 2 years?
• c) How long would it take the population to grow
  to 50000?

4
Exponential Growth example question
On Rabbit Island the rate of growth at a time (t)
is proportional to the size of the population (P)
It can be modelled by the equation

• If the population was initially 200 and after 10
  months the
• population has grown to 1000.
• Find A and b (2dp) to give the particular
  solution of the equation
• When t0, P200 When t10, P1000
• b) What would be the population after 2 years?
• c) How long would it take the population to grow
  to 50000?

b 0.16
So, A 200
5
Exponential Growth example question
On Rabbit Island the rate of growth at a time (t)
is proportional to the size of the population (P)

• If the population was initially 200 and after 10
  months the
• population has grown to 1000.
• b) What would be the population after 2 years?
• c) How long would it take the population to grow
  to 50000?

It can be modelled by the equation
2 years t 24 months
t 34.5 months
t 2 years 10.5 months
6
Exponential Decay example question
7
Exponential Decay example question
The radiation level (R) at Chernobyl is modelled
by the equation
t is the time (in months after the accident)

• What was the radiation level immediately after
  the accident?
• b) How long would it take for the radiation to
  be half this amount?
• c) As t ? ? what value does the radiation
  settle on?
• This represents background radiation

t 7.44 months
8

The laws of logs

• Laws of logs
• ln a k k ln a
• Example
• ln 35 5 ln 3
• a ex
• ln a ln ex
• ln a x ln e x
• eln a a

• Laws of logs
• ln a ln b ln ab
• Example
• ln 2 ln 8 ln 16
• ln a - ln b ln (a/b)
• Example
• ln 42 ln 6 ln 7

9
Page 67 E1-E2
a eln a
ax ex ln a
5x ex ln 5
ax e(ln a)x
5x e1.61x
If y ax
then y e(ln a)x
The derivative y (ln a) e(ln a)x
The derivative of ax is (ln a) ax
10
Further Exponential Functions

• Now work Page 76-77
• Work through E1 to E4
• Homework Page 76
• Exercise D Q1, 2, 4