This document discusses calculating rates of change using differentiation. It provides an example of using a formula to model the number of midges around a human over time. It calculates the number of midges initially and after 20 seconds, and the rate at which midges are arriving after 20 seconds by taking the derivative of the formula. The document explains that given any formula, the derivative can be used to find how quickly or slowly the quantity is changing. It provides another example calculating a person's remaining hairs over time using a formula, finding the number of hairs after set times, and the rate of hair loss by taking the derivative.

1. Block 2
Rates of Change

2. What is to be learned?
• How to use differentiation to calculate the
rate of change
• How to interpret this

3. Everything is Changing!
We can calculate how fast it is changing by:
Finding the derivative

4. Ex The amount of midges (M) around a
human in Kinlochmidgy after t seconds is
given by the formula:
M = t2
+ 5t + 20
a) How many midges are there when
Harry arrives?
b) i) How many midges after 20
seconds?
ii) How quickly are they arriving then?

5. Ex The amount of midges (M) around a
human in Kinlochmidgy after t seconds is
given by the formula:
M = t2
+ 5t + 20
a) How many midges are there when
Harry arrives?
b) i) How many midges after 20
seconds?
ii) How quickly are they arriving then?

6. Et The amount of midges (M) around a
human in Kinlochmidgy after t seconds is
given by the formula:
M = t2
+ 5t + 20
a) How many midges are there when
Harry arrives?
b) i) How many midges after 20
seconds?
ii) How quickly are they arriving then?

7. Et The amount of midges (M) around a
human in Kinlochmidgy after t seconds is
given by the formula:
M = t2
+ 5t + 20
a) How many midges are there when
Harry arrives?
b) i) How many midges after 20
seconds?
ii) How quickly are they arriving then?
Rate of change 
Find the derivative

8. M = t2
+ 5t + 20
a) How many midges are there when
Harry arrives?
t = 0
M = 02
+5(0) + 20
= 20 midges

9. M = t2
+ 5t + 20
b) i) How many midges after 20 seconds?
t = 20
M = 202
+5(20) + 20
= 520 midges

10. M = t2
+ 5t + 20
ii) How quickly are they arriving then?
Find Derivative:
dM
/dt = 2t + 5
After 20 secs (t = 20)
dM
/dt = 2(20) + 5
= 45 midges per sec

11. Rates of Change
Given formula, find how fast (or slowly)
things are changing by:
Finding the derivative

12. Ex Henry’s hairs remaining (h) over a
period of t months can be given by
formula:
h = 2000 - 50√t
a) How many hairs will he have after
i) 1 month
ii) 3 years
b) How quickly is he losing hairs at
these times?
Original formula
Rate of change
find derivative

13. h = 2000 - 50√t
a) i) t = 1
h = 2000 - 50 √(1)
= 1950 hairs
i) t = 36
h = 2000 - 50 √(36)
= 1700 hairs

14. h = 2000 - 50√t
b) Rearrange h = 2000 – 50t
dh
/dt = -25t
= -25
= -25
i) t = 1
½½
-½-½
t½½
√t
hairs per monthdhdh
//dtdt == -25-25
//√1√1= -25= -25

15. h = 2000 - 50√t
b) Rearrange h = 2000 – 50t
dh
/dt = -25t
= -25
= -25
i) t = 1
is losing 25 hairs per month
i) t = 36 dh
/dt = -25
/√36 = -4.2 hairs per month
rate of hair loss is slowing
½½
-½-½
t½½
√t
hairs per monthdhdh
//dtdt == -25-25
//√1√1== --2525

16. The rabbit population P in Bunnywood after t
weeks can be calculated using the formula
P = t2
+ 3t + 8
a) How many rabbits are there to start with?
b) How many rabbits are there after 4 weeks?
c) How quickly is the rabbit population growing
at that time?
a) t = 0, P = 02
+ 3(0) + 8
= 8 rabbits.
b) t = 4, P = 42
+ 3(4) + 8
= 36 rabbits.
c) dP
/dt = 2t + 3, t = 4,
ROC = 2(4) + 3
Key
Question