11x1 t08 07 rates of change
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Rates of Changee.g. A block of ice in the form of a cube has one edge 10 cm long. It is
melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
Rates of Change
x
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
Rates of Change
x ?dVdt
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
Rates of Change
x ?dVdt
110
dxdt
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
Rates of Change
x ?dVdt
110
dxdt
3V x
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
Rates of Change
x ?dVdt
110
dxdt
3V x
23dV xdx
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
Rates of Change
x ?dVdt
110
dxdt
3V x
23dV xdx
dV dV dxdt dx dt
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
Rates of Change
x ?dVdt
110
dxdt
3V x
23dV xdx
dV dV dxdt dx dt
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
2 1310
x
Rates of Change
x ?dVdt
110
dxdt
3V x
23dV xdx
dV dV dxdt dx dt
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
2 1310
x 23
10x
Rates of Change
x ?dVdt
110
dxdt
3V x
23dV xdx
dV dV dxdt dx dt
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
2 1310
x 23
10x
23 5when 5,
10dVxdt
Rates of Change
x ?dVdt
110
dxdt
3V x
23dV xdx
dV dV dxdt dx dt
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
2 1310
x 23
10x
23 5when 5,
10dVxdt
7.5
Rates of Change
x ?dVdt
110
dxdt
3V x
23dV xdx
dV dV dxdt dx dt
e.g. A block of ice in the form of a cube has one edge 10 cm long. It is melting so that its dimensions decrease at the rate of 1 mm/s.
At what rate is the volume decreasing when the edge is 5cm long?
2 1310
x 23
10x
23 5when 5,
10dVxdt
7.5
3volume is decreasing at 7.5 cm /s
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
x
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
x
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
x
x45
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
x
x45
x
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
x
x45
x 213
V r h
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
x
x45
x 213
V r h
213
x x
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
x
x45
x 213
V r h
213
x x
313
x
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
x
x45
x 213
V r h
213
x x
313
x3b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
x
x45
x 213
V r h
213
x x
313
x3b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?dxdt
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
x
x45
x 213
V r h
213
x x
313
x3b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?dxdt
15
dVdt
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
x
x45
x 213
V r h
213
x x
313
x3b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?dxdt
15
dVdt
313
V x
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
x
x45
x 213
V r h
213
x x
313
x3b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?dxdt
15
dVdt
313
V x2dV x
dx
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
x
x45
x 213
V r h
213
x x
313
x3b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?dxdt
15
dVdt
313
V x2dV x
dx
dx dx dVdt dV dt
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
x
x45
x 213
V r h
213
x x
313
x3b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?dxdt
15
dVdt
313
V x2dV x
dx
dx dx dVdt dV dt
21 1
5x
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
x
x45
x 213
V r h
213
x x
313
x3b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?dxdt
15
dVdt
313
V x2dV x
dx
dx dx dVdt dV dt
21 1
5x
21
5 x
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
x
x45
x 213
V r h
213
x x
313
x3b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?dxdt
15
dVdt
313
V x2dV x
dx
dx dx dVdt dV dt
21 1
5x
21
5 x
21when 4,
5 4dxxdt
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
x
x45
x 213
V r h
213
x x
313
x3b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?dxdt
15
dVdt
313
V x2dV x
dx
dx dx dVdt dV dt
21 1
5x
21
5 x
21when 4,
5 4dxxdt
180
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
x
x45
x 213
V r h
213
x x
313
x3b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?dxdt
15
dVdt
313
V x2dV x
dx
dx dx dVdt dV dt
21 1
5x
21
5 x
21when 4,
5 4dxxdt
180
depth is increasing 1at cm/min
80
(ii) A vessel is in the form of an inverted cone with a vertical angle of If the depth of the water in the vessel is x cm;a) find the volume of water.
90
x
x45
x 213
V r h
213
x x
313
x3b) If water is poured in at a rate of 0.2 cm /min, find the rate the depth
is increasing when the water depth is 4 cm.
?dxdt
15
dVdt
313
V x2dV x
dx
dx dx dVdt dV dt
21 1
5x
21
5 x
21when 4,
5 4dxxdt
180
depth is increasing 1at cm/min
80
Exercise 7H; 1a, 2a, 4,6, 7, 8, 10,
12, 14
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