DETERMINING AVERAGE RATE of CHANGE
average rate of change:
the change in the quantity given by the dependent
variable (y) divided by the corresponding change
in the quantity represented by the independent
variable (x) over an interval ( x1 ≤ x ≤ x2 )
change in y
change in x
y
=
x
f (x 2 )  f (x 1 )
=
x 2  x1
Average rate of change =
(NOTE: an average rate of change is expressed using the units of the two related quantities)
Example 
The given table represents the growth of a bacteria population. Determine
the 2h interval in which the bacteria population grew the fastest.
Time (h)
Number of Bacteria
Time
Interval
(h)
b
ARC
t
(bacteria/h)
0
850
0≤t≤2
2
1122
2≤t≤4
4
1481
4≤t≤6
6
1954
8
2577
6≤t≤8
10
3400
8 ≤ t ≤ 10
 _____________________________________________________________
secant line:
a line that passes through two
points on the graph of a relation
Graphically, the average rate of change for any
function, y = f(x), over the interval, x1 ≤ x ≤ x2,
is equivalent to the slope of the secant line
passing through two points, (x1,y1) and (x2,y2).
Average rate of change = msecant
y
=
x
y  y1
= 2
x 2  x1
Example 
Graphically determine the 2h interval in which the bacteria population grew
the fastest.
The greatest change occurs when
the secant line is the steepest.
Why is the average rate of change of the bacteria population positive on each interval?
What would it mean if the average rate of change were negative?
Example 
A pebble is tossed upwards from a cliff that is 120m above water. The height
of the rock above the water is modelled by the given equation, where h(t) is
the height in metres and t is time in seconds.
h(t) = –5t2 + 10t + 120
a)
i)
0≤t≤1
Calculate the ARC during each of the following intervals:
ii) 1 ≤ t ≤ 2
iii) 2 ≤ t ≤ 3
b)
What does the average rate of change represent in this situation?
c)
Is the pebble falling at a constant rate?
Example 
The water is drained from a hot tub. The tub holds 1600 L of water. It takes
2 h for the water to drain completely. The volume, V, in litres, of water
remaining in the tub at various times, t, in minutes, is shown in the table and
on the graph.
Time (min)
0
10
20
30
40
50
60
70
80
90
100
110
120
a)
Volume (L)
1600
1344
1111
900
711
544
400
278
178
100
44
10
0
Calculate the ARC during each of the following time intervals:
i)
30 ≤ t ≤ 90
ii)
60 ≤ t ≤ 90
ii)
90 ≤ t ≤ 110
iv)
110 ≤ t ≤ 120
b)
Is the rate of change in volume negative or positive? Explain.
c)
Does the hot tub drain at a constant rate? Explain.