12 x1 t04 06 displacement, velocity, acceleration (2010)
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Travel Graphs
Travel Graphs 5 8x t t
e.g. A ball is bounced and its distance from the ground is graphed.x
t
20
40
80
60
2 4 6 8
Travel Graphs 5 8x t t
e.g. A ball is bounced and its distance from the ground is graphed.x
t
20
40
80
60
2 4 6 8
Distance = total amount travelled
Travel Graphs 5 8x t t
e.g. A ball is bounced and its distance from the ground is graphed.x
t
20
40
80
60
2 4 6 8
Distance = total amount travelledDisplacement = how far from the starting point
Travel Graphs 5 8x t t
(i) Find the height of the ball after 1 second
e.g. A ball is bounced and its distance from the ground is graphed.x
t
20
40
80
60
2 4 6 8
Distance = total amount travelledDisplacement = how far from the starting point
Travel Graphs 5 8x t t
when 1, 5 1 8 1t x (i) Find the height of the ball after 1 second
e.g. A ball is bounced and its distance from the ground is graphed.x
t
20
40
80
60
2 4 6 8
Distance = total amount travelledDisplacement = how far from the starting point
35
Travel Graphs 5 8x t t
when 1, 5 1 8 1t x (i) Find the height of the ball after 1 second
After 1 second the ball is 35 metres above the ground
e.g. A ball is bounced and its distance from the ground is graphed.x
t
20
40
80
60
2 4 6 8
Distance = total amount travelledDisplacement = how far from the starting point
35
(ii) At what other time is the ball this same height above the ground?
(ii) At what other time is the ball this same height above the ground?when 35,x
(ii) At what other time is the ball this same height above the ground? 5 8 35t t
8 7t t
1 7 0t t
when 35,x
2
2
8 78 7 0t t
t t
1 or 7t t
(ii) At what other time is the ball this same height above the ground? 5 8 35t t
ball is 35 metres above ground again after 7 seconds
8 7t t
1 7 0t t
when 35,x
2
2
8 78 7 0t t
t t
1 or 7t t
(ii) At what other time is the ball this same height above the ground? 5 8 35t t
ball is 35 metres above ground again after 7 seconds
change in displacementchange in time
8 7t t
1 7 0t t
when 35,x
2
2
8 78 7 0t t
t t
1 or 7t t
Average velocity =
(ii) At what other time is the ball this same height above the ground? 5 8 35t t
ball is 35 metres above ground again after 7 seconds
change in displacementchange in time
2 1
2 1
x xt t
8 7t t
1 7 0t t
when 35,x
2
2
8 78 7 0t t
t t
1 or 7t t
Average velocity =
(iii) Find the average velocity during the 1st second
(iii) Find the average velocity during the 1st second2 1
2 1
average velocity
35 01 0
35
x xt t
(iii) Find the average velocity during the 1st second2 1
2 1
average velocity
35 01 0
35
x xt t
average velocity during the 1st second was 35m/s
(iii) Find the average velocity during the 1st second2 1
2 1
average velocity
35 01 0
35
x xt t
average velocity during the 1st second was 35m/s
(iv) Find the average velocity during the fifth second
(iii) Find the average velocity during the 1st second2 1
2 1
average velocity
35 01 0
35
x xt t
average velocity during the 1st second was 35m/s
when 4, 5 4 8 4 =80
t x
(iv) Find the average velocity during the fifth second
(iii) Find the average velocity during the 1st second2 1
2 1
average velocity
35 01 0
35
x xt t
average velocity during the 1st second was 35m/s
when 4, 5 4 8 4 =80
t x
(iv) Find the average velocity during the fifth second
when 5, 5 5 8 5 =75
t x
(iii) Find the average velocity during the 1st second2 1
2 1
average velocity
35 01 0
35
x xt t
average velocity during the 1st second was 35m/s
when 4, 5 4 8 4 =80
t x
(iv) Find the average velocity during the fifth second2 1
2 1
average velocity
75 805 45
x xt t
when 5, 5 5 8 5 =75
t x
(iii) Find the average velocity during the 1st second2 1
2 1
average velocity
35 01 0
35
x xt t
average velocity during the 1st second was 35m/s
when 4, 5 4 8 4 =80
t x
(iv) Find the average velocity during the fifth second2 1
2 1
average velocity
75 805 45
x xt t
average velocity during the 5th second was 5m/s
when 5, 5 5 8 5 =75
t x
(iv) Find the average velocity during its 8 seconds in the air
(iv) Find the average velocity during its 8 seconds in the air2 1
2 1
average velocity
0 08 00
x xt t
(iv) Find the average velocity during its 8 seconds in the air2 1
2 1
average velocity
0 08 00
x xt t
average velocity during the 8 seconds was 0m/s
distance travelledtime taken
Average speed =
(iv) Find the average velocity during its 8 seconds in the air2 1
2 1
average velocity
0 08 00
x xt t
average velocity during the 8 seconds was 0m/s
distance travelledtime taken
Average speed =
(iv) Find the average velocity during its 8 seconds in the air2 1
2 1
average velocity
0 08 00
x xt t
average velocity during the 8 seconds was 0m/s
(v) Find the average speed during its 8 seconds in the air
distance travelledtime taken
Average speed =
(iv) Find the average velocity during its 8 seconds in the air2 1
2 1
average velocity
0 08 00
x xt t
average velocity during the 8 seconds was 0m/s
(v) Find the average speed during its 8 seconds in the airdistance travelledaverage speed
time taken
distance travelledtime taken
Average speed =
(iv) Find the average velocity during its 8 seconds in the air2 1
2 1
average velocity
0 08 00
x xt t
average velocity during the 8 seconds was 0m/s
(v) Find the average speed during its 8 seconds in the airdistance travelledaverage speed
time taken
1608
20
distance travelledtime taken
Average speed =
(iv) Find the average velocity during its 8 seconds in the air2 1
2 1
average velocity
0 08 00
x xt t
average velocity during the 8 seconds was 0m/s
(v) Find the average speed during its 8 seconds in the airdistance travelledaverage speed
time taken
1608
20
average speed during the 8 seconds was 20m/s
Applications of Calculus To The Physical World
Applications of Calculus To The Physical World
Displacement (x)Distance from a point, with direction.
Applications of Calculus To The Physical World
Displacement (x)Distance from a point, with direction.
x
dtdxv ,,Velocity
The rate of change of displacement with respect to time i.e. speed with direction.
Applications of Calculus To The Physical World
Displacement (x)Distance from a point, with direction.
x
dtdxv ,,Velocity
The rate of change of displacement with respect to time i.e. speed with direction.
vxdt
xddtdva ,,,,on Accelerati 2
2
The rate of change of velocity with respect to time
Applications of Calculus To The Physical World
Displacement (x)Distance from a point, with direction.
x
dtdxv ,,Velocity
The rate of change of displacement with respect to time i.e. speed with direction.
vxdt
xddtdva ,,,,on Accelerati 2
2
The rate of change of velocity with respect to time
NOTE: “deceleration” or slowing down is when acceleration is in the opposite direction to velocity.
Displacement
Velocity
Acceleration
Displacement
Velocity
Acceleration
differentiate
Displacement
Velocity
Acceleration
differentiate
Displacement
Velocity
Acceleration
differentiate integrate
Displacement
Velocity
Acceleration
differentiate integrate
t
x
1 2 3 4
12
-2-1
t
x
1 2 3 4
12
-2-1
slope=instantaneous velocity
t
x
1 2 3 4
12
-2-1
slope=instantaneous velocity
t1 2 3 4
x
t
x
1 2 3 4
12
-2-1
slope=instantaneous velocity
t1 2 3 4
slope=instantaneous accelerationx
t
x
1 2 3 4
12
-2-1
slope=instantaneous velocity
t1 2 3 4
slope=instantaneous accelerationx
t1 2 3 4
x
t
x
1 2 3 4
12
-2-1
slope=instantaneous velocity
t1 2 3 4
slope=instantaneous accelerationx
t1 2 3 4
x
e.g. (i) distance traveled
t
x
1 2 3 4
12
-2-1
slope=instantaneous velocity
t1 2 3 4
slope=instantaneous accelerationx
t1 2 3 4
x
e.g. (i) distance traveled m7
t
x
1 2 3 4
12
-2-1
slope=instantaneous velocity
t1 2 3 4
slope=instantaneous accelerationx
t1 2 3 4
x
e.g. (i) distance traveled m7
(ii) total displacement
t
x
1 2 3 4
12
-2-1
slope=instantaneous velocity
t1 2 3 4
slope=instantaneous accelerationx
t1 2 3 4
x
e.g. (i) distance traveled m7
(ii) total displacement m1
t
x
1 2 3 4
12
-2-1
slope=instantaneous velocity
t1 2 3 4
slope=instantaneous accelerationx
t1 2 3 4
x
e.g. (i) distance traveled m7
(ii) total displacement m1
(iii) average speed
t
x
1 2 3 4
12
-2-1
slope=instantaneous velocity
t1 2 3 4
slope=instantaneous accelerationx
t1 2 3 4
x
e.g. (i) distance traveled m7
(ii) total displacement m1
(iii) average speed m/s47
t
x
1 2 3 4
12
-2-1
slope=instantaneous velocity
t1 2 3 4
slope=instantaneous accelerationx
t1 2 3 4
x
e.g. (i) distance traveled m7
(ii) total displacement m1
(iii) average speed m/s47
(iv) average velocity
t
x
1 2 3 4
12
-2-1
slope=instantaneous velocity
t1 2 3 4
slope=instantaneous accelerationx
t1 2 3 4
x
e.g. (i) distance traveled m7
(ii) total displacement m1
(iii) average speed m/s47
(iv) average velocity m/s41
23 21ttx
e.g. (i) The displacement x from the origin at time t seconds, of a particle traveling in a straight line is given by the formula
23 21ttx
e.g. (i) The displacement x from the origin at time t seconds, of a particle traveling in a straight line is given by the formula
a) Find the acceleration of the particle at time t.
23 21ttx
426423
212
23
tattv
ttx
e.g. (i) The displacement x from the origin at time t seconds, of a particle traveling in a straight line is given by the formula
a) Find the acceleration of the particle at time t.
23 21ttx
426423
212
23
tattv
ttx
e.g. (i) The displacement x from the origin at time t seconds, of a particle traveling in a straight line is given by the formula
a) Find the acceleration of the particle at time t.
b) Find the times when the particle is stationary.
23 21ttx
426423
212
23
tattv
ttx
e.g. (i) The displacement x from the origin at time t seconds, of a particle traveling in a straight line is given by the formula
a) Find the acceleration of the particle at time t.
b) Find the times when the particle is stationary.
Particle is stationary when v = 0
23 21ttx
426423
212
23
tattv
ttx
e.g. (i) The displacement x from the origin at time t seconds, of a particle traveling in a straight line is given by the formula
a) Find the acceleration of the particle at time t.
0423i.e. 2 tt
b) Find the times when the particle is stationary.
Particle is stationary when v = 0
23 21ttx
426423
212
23
tattv
ttx
e.g. (i) The displacement x from the origin at time t seconds, of a particle traveling in a straight line is given by the formula
a) Find the acceleration of the particle at time t.
0423i.e. 2 tt
b) Find the times when the particle is stationary.
Particle is stationary when v = 0
14or 00143
tttt
23 21ttx
426423
212
23
tattv
ttx
e.g. (i) The displacement x from the origin at time t seconds, of a particle traveling in a straight line is given by the formula
a) Find the acceleration of the particle at time t.
0423i.e. 2 tt
b) Find the times when the particle is stationary.
Particle is stationary when v = 0
14or 00143
tttt
Particle is stationary initially and again after 14 seconds
(ii) A particle is moving on the x axis. It started from rest at t = 0 from the point x = 7.If its acceleration at time t is 2 + 6t find the position of the particle when t = 3.
(ii) A particle is moving on the x axis. It started from rest at t = 0 from the point x = 7.If its acceleration at time t is 2 + 6t find the position of the particle when t = 3.
cttvta
232
62
(ii) A particle is moving on the x axis. It started from rest at t = 0 from the point x = 7.If its acceleration at time t is 2 + 6t find the position of the particle when t = 3.
cttvta
232
62
0,0when vt
(ii) A particle is moving on the x axis. It started from rest at t = 0 from the point x = 7.If its acceleration at time t is 2 + 6t find the position of the particle when t = 3.
cttvta
232
62
0,0when vt
0000 i.e.
cc
(ii) A particle is moving on the x axis. It started from rest at t = 0 from the point x = 7.If its acceleration at time t is 2 + 6t find the position of the particle when t = 3.
cttvta
232
62
0,0when vt
0000 i.e.
cc
232 ttv
(ii) A particle is moving on the x axis. It started from rest at t = 0 from the point x = 7.If its acceleration at time t is 2 + 6t find the position of the particle when t = 3.
cttvta
232
62
0,0when vt
0000 i.e.
cc
232 ttv cttx 32
(ii) A particle is moving on the x axis. It started from rest at t = 0 from the point x = 7.If its acceleration at time t is 2 + 6t find the position of the particle when t = 3.
cttvta
232
62
0,0when vt
0000 i.e.
cc
232 ttv cttx 32
7,0when xt
7007 i.e.
cc
(ii) A particle is moving on the x axis. It started from rest at t = 0 from the point x = 7.If its acceleration at time t is 2 + 6t find the position of the particle when t = 3.
cttvta
232
62
0,0when vt
0000 i.e.
cc
232 ttv cttx 32
7,0when xt
7007 i.e.
cc
732 ttx
(ii) A particle is moving on the x axis. It started from rest at t = 0 from the point x = 7.If its acceleration at time t is 2 + 6t find the position of the particle when t = 3.
cttvta
232
62
0,0when vt
0000 i.e.
cc
232 ttv cttx 32
7,0when xt
7007 i.e.
cc
732 ttx
43 733,3when 32
xt
(ii) A particle is moving on the x axis. It started from rest at t = 0 from the point x = 7.If its acceleration at time t is 2 + 6t find the position of the particle when t = 3.
cttvta
232
62
0,0when vt
0000 i.e.
cc
232 ttv cttx 32
7,0when xt
7007 i.e.
cc
732 ttx
43 733,3when 32
xt
after 3 seconds the particle is 43 units to the right of O.
e.g. 2001 HSC Question 7c)A particle moves in a straight line so that its displacement, in metres, is given by
where t is measured in seconds.
22
txt
e.g. 2001 HSC Question 7c)A particle moves in a straight line so that its displacement, in metres, is given by
where t is measured in seconds.
22
txt
(i) What is the displacement when t = 0?
e.g. 2001 HSC Question 7c)A particle moves in a straight line so that its displacement, in metres, is given by
where t is measured in seconds.
22
txt
0 2when 0,0 2
= 1
t x
(i) What is the displacement when t = 0?
e.g. 2001 HSC Question 7c)A particle moves in a straight line so that its displacement, in metres, is given by
where t is measured in seconds.
22
txt
0 2when 0,0 2
= 1
t x
(i) What is the displacement when t = 0?
the particle is 1 metre to the left of the origin
e.g. 2001 HSC Question 7c)A particle moves in a straight line so that its displacement, in metres, is given by
where t is measured in seconds.
22
txt
0 2when 0,0 2
= 1
t x
(i) What is the displacement when t = 0?
the particle is 1 metre to the left of the origin4(ii) Show that 1
2x
t
Hence find expressions for the velocity and the acceleration in terms of t.
e.g. 2001 HSC Question 7c)A particle moves in a straight line so that its displacement, in metres, is given by
where t is measured in seconds.
22
txt
0 2when 0,0 2
= 1
t x
(i) What is the displacement when t = 0?
the particle is 1 metre to the left of the origin4(ii) Show that 1
2x
t
Hence find expressions for the velocity and the acceleration in terms of t.
4 2 412 2
tt t
e.g. 2001 HSC Question 7c)A particle moves in a straight line so that its displacement, in metres, is given by
where t is measured in seconds.
22
txt
0 2when 0,0 2
= 1
t x
(i) What is the displacement when t = 0?
the particle is 1 metre to the left of the origin4(ii) Show that 1
2x
t
Hence find expressions for the velocity and the acceleration in terms of t.
4 2 412 2
tt t
22
tt
e.g. 2001 HSC Question 7c)A particle moves in a straight line so that its displacement, in metres, is given by
where t is measured in seconds.
22
txt
0 2when 0,0 2
= 1
t x
(i) What is the displacement when t = 0?
the particle is 1 metre to the left of the origin4(ii) Show that 1
2x
t
Hence find expressions for the velocity and the acceleration in terms of t.
4 2 412 2
tt t
22
tt
412
xt
e.g. 2001 HSC Question 7c)A particle moves in a straight line so that its displacement, in metres, is given by
where t is measured in seconds.
22
txt
0 2when 0,0 2
= 1
t x
(i) What is the displacement when t = 0?
the particle is 1 metre to the left of the origin4(ii) Show that 1
2x
t
Hence find expressions for the velocity and the acceleration in terms of t.
4 2 412 2
tt t
2
4 12
vt
22
tt
412
xt
e.g. 2001 HSC Question 7c)A particle moves in a straight line so that its displacement, in metres, is given by
where t is measured in seconds.
22
txt
0 2when 0,0 2
= 1
t x
(i) What is the displacement when t = 0?
the particle is 1 metre to the left of the origin4(ii) Show that 1
2x
t
Hence find expressions for the velocity and the acceleration in terms of t.
4 2 412 2
tt t
2
4 12
vt
242
vt
22
tt
412
xt
e.g. 2001 HSC Question 7c)A particle moves in a straight line so that its displacement, in metres, is given by
where t is measured in seconds.
22
txt
0 2when 0,0 2
= 1
t x
(i) What is the displacement when t = 0?
the particle is 1 metre to the left of the origin4(ii) Show that 1
2x
t
Hence find expressions for the velocity and the acceleration in terms of t.
4 2 412 2
tt t
2
4 12
vt
242
vt
1
4
4 2 2 12
ta
t
22
tt
412
xt
e.g. 2001 HSC Question 7c)A particle moves in a straight line so that its displacement, in metres, is given by
where t is measured in seconds.
22
txt
0 2when 0,0 2
= 1
t x
(i) What is the displacement when t = 0?
the particle is 1 metre to the left of the origin4(ii) Show that 1
2x
t
Hence find expressions for the velocity and the acceleration in terms of t.
4 2 412 2
tt t
2
4 12
vt
242
vt
1
4
4 2 2 12
ta
t
382
at
22
tt
412
xt
(iii) Is the particle ever at rest? Give reasons for your answer.
(iii) Is the particle ever at rest? Give reasons for your answer.
242
vt
0
(iii) Is the particle ever at rest? Give reasons for your answer.
242
vt
0
the particle is never at rest
(iii) Is the particle ever at rest? Give reasons for your answer.
242
vt
0
the particle is never at rest
(iv) What is the limiting velocity of the particle as t increases indefinitely?
(iii) Is the particle ever at rest? Give reasons for your answer.
242
vt
0
the particle is never at rest
(iv) What is the limiting velocity of the particle as t increases indefinitely?limt
v
(iii) Is the particle ever at rest? Give reasons for your answer.
242
vt
0
the particle is never at rest
(iv) What is the limiting velocity of the particle as t increases indefinitely?limt
v 2
4lim2t t
(iii) Is the particle ever at rest? Give reasons for your answer.
242
vt
0
the particle is never at rest
(iv) What is the limiting velocity of the particle as t increases indefinitely?limt
v 2
4lim2t t
0
(iii) Is the particle ever at rest? Give reasons for your answer.
242
vt
0
the particle is never at rest
(iv) What is the limiting velocity of the particle as t increases indefinitely?limt
v 2
4lim2t t
0
OR v
t
1 242
vt
(iii) Is the particle ever at rest? Give reasons for your answer.
242
vt
0
the particle is never at rest
(iv) What is the limiting velocity of the particle as t increases indefinitely?limt
v 2
4lim2t t
0
OR v
t
1
the limiting velocity of the particle is 0 m/s
242
vt
(ii) 2002 HSC Question 8b)A particle moves in a straight line. At time t seconds, its distance x metres from a fixed point O in the line is given by sin 2 3x t
(ii) 2002 HSC Question 8b)A particle moves in a straight line. At time t seconds, its distance x metres from a fixed point O in the line is given by sin 2 3x t
(i) Sketch the graph of x as a function of t for 0 2t
(ii) 2002 HSC Question 8b)A particle moves in a straight line. At time t seconds, its distance x metres from a fixed point O in the line is given by sin 2 3x t
(i) Sketch the graph of x as a function of t for 0 2t
amplitude 1 unit
(ii) 2002 HSC Question 8b)A particle moves in a straight line. At time t seconds, its distance x metres from a fixed point O in the line is given by sin 2 3x t
(i) Sketch the graph of x as a function of t for 0 2t
shift 3 units
amplitude 1 unit
(ii) 2002 HSC Question 8b)A particle moves in a straight line. At time t seconds, its distance x metres from a fixed point O in the line is given by sin 2 3x t
(i) Sketch the graph of x as a function of t for 0 2t 2period2
shift 3 units
amplitude 1 unit
(ii) 2002 HSC Question 8b)A particle moves in a straight line. At time t seconds, its distance x metres from a fixed point O in the line is given by sin 2 3x t
(i) Sketch the graph of x as a function of t for 0 2t 2period2
divisions4
shift 3 units
amplitude 1 unit
(ii) 2002 HSC Question 8b)A particle moves in a straight line. At time t seconds, its distance x metres from a fixed point O in the line is given by sin 2 3x t
(i) Sketch the graph of x as a function of t for 0 2t 2period2
divisions4
shift 3 units
amplitude 1 unit
1
234x
4
2 3
4 5
4 3
2 7
4 2 t
(ii) 2002 HSC Question 8b)A particle moves in a straight line. At time t seconds, its distance x metres from a fixed point O in the line is given by sin 2 3x t
(i) Sketch the graph of x as a function of t for 0 2t 2period2
divisions4
shift 3 units
amplitude 1 unit
1
234x
4
2 3
4 5
4 3
2 7
4 2 t
(ii) 2002 HSC Question 8b)A particle moves in a straight line. At time t seconds, its distance x metres from a fixed point O in the line is given by sin 2 3x t
(i) Sketch the graph of x as a function of t for 0 2t 2period2
divisions4
shift 3 units
amplitude 1 unit
1
234x
4
2 3
4 5
4 3
2 7
4 2 t
(ii) 2002 HSC Question 8b)A particle moves in a straight line. At time t seconds, its distance x metres from a fixed point O in the line is given by sin 2 3x t
(i) Sketch the graph of x as a function of t for 0 2t 2period2
divisions4
shift 3 units
amplitude 1 unit
1
234x
4
2 3
4 5
4 3
2 7
4 2 t
(ii) 2002 HSC Question 8b)A particle moves in a straight line. At time t seconds, its distance x metres from a fixed point O in the line is given by sin 2 3x t
(i) Sketch the graph of x as a function of t for 0 2t 2period2
divisions4
shift 3 units
amplitude 1 unit
1
234x
4
2 3
4 5
4 3
2 7
4 2 t
sin 2 3x t
(ii) Using your graph, or otherwise, find the times when the particle is at rest, and the position of the particle at those times.
(ii) Using your graph, or otherwise, find the times when the particle is at rest, and the position of the particle at those times.
Particle is at rest when velocity = 0
(ii) Using your graph, or otherwise, find the times when the particle is at rest, and the position of the particle at those times.
Particle is at rest when velocity = 0
0dxdt
i.e. the stationary points
(ii) Using your graph, or otherwise, find the times when the particle is at rest, and the position of the particle at those times.
Particle is at rest when velocity = 0
when seconds, 4 metres4
t x
3 seconds, 2 metres4
t x
5 seconds, 4 metres4
t x
7 seconds, 2 metres4
t x
0dxdt
i.e. the stationary points
(iii) Describe the motion completely.
(ii) Using your graph, or otherwise, find the times when the particle is at rest, and the position of the particle at those times.
Particle is at rest when velocity = 0
when seconds, 4 metres4
t x
3 seconds, 2 metres4
t x
5 seconds, 4 metres4
t x
7 seconds, 2 metres4
t x
0dxdt
i.e. the stationary points
(iii) Describe the motion completely.
(ii) Using your graph, or otherwise, find the times when the particle is at rest, and the position of the particle at those times.
Particle is at rest when velocity = 0
when seconds, 4 metres4
t x
3 seconds, 2 metres4
t x
5 seconds, 4 metres4
t x
7 seconds, 2 metres4
t x
0dxdt
i.e. the stationary points
The particle oscillates between x=2 and x=4 with a period ofseconds
Integrating Functions of Time
t1 2 3 4
x
t1 2 3 4
x
4
0
ntdisplacemein change dtx
Integrating Functions of Time
t1 2 3 4
x
4
0
ntdisplacemein change dtx
4
3
3
1
1
0
distancein change dtxdtxdtx
Integrating Functions of Time
t1 2 3 4
x
4
0
ntdisplacemein change dtx
4
3
3
1
1
0
distancein change dtxdtxdtx
t1 2 3 4
x
Integrating Functions of Time
t1 2 3 4
x
4
0
ntdisplacemein change dtx
4
3
3
1
1
0
distancein change dtxdtxdtx
t1 2 3 4
x
4
0
yin velocit change dtx
Integrating Functions of Time
t1 2 3 4
x
4
0
ntdisplacemein change dtx
4
3
3
1
1
0
distancein change dtxdtxdtx
t1 2 3 4
x
4
0
yin velocit change dtx
4
2
2
0
speedin change dtxdtx
Integrating Functions of Time
Derivative GraphsFunction 1st derivative 2nd derivative
displacement velocity acceleration
Derivative GraphsFunction 1st derivative 2nd derivative
displacement velocity acceleration
stationary point x intercept
Derivative GraphsFunction 1st derivative 2nd derivative
displacement velocity acceleration
stationary point x intercept
inflection point stationary point x intercept
Derivative GraphsFunction 1st derivative 2nd derivative
displacement velocity acceleration
stationary point x intercept
inflection point stationary point x intercept
increasing positive
Derivative GraphsFunction 1st derivative 2nd derivative
displacement velocity acceleration
stationary point x intercept
inflection point stationary point x intercept
increasing positive
decreasing negative
Derivative GraphsFunction 1st derivative 2nd derivative
displacement velocity acceleration
stationary point x intercept
inflection point stationary point x intercept
increasing positive
decreasing negative
concave up increasing positive
Derivative GraphsFunction 1st derivative 2nd derivative
displacement velocity acceleration
stationary point x intercept
inflection point stationary point x intercept
increasing positive
decreasing negative
concave up increasing positive
concave down decreasing negative
graph type integrate differentiate
graph type integrate differentiate
horizontal line oblique line x axis
graph type integrate differentiate
horizontal line oblique line x axis
oblique line parabola horizontal line
graph type integrate differentiate
horizontal line oblique line x axis
oblique line parabola horizontal line
parabola cubic oblique lineinflects at turning pt
graph type integrate differentiate
horizontal line oblique line x axis
oblique line parabola horizontal line
parabola cubic oblique lineinflects at turning pt
Remember:• integration = area
graph type integrate differentiate
horizontal line oblique line x axis
oblique line parabola horizontal line
parabola cubic oblique lineinflects at turning pt
Remember:• integration = area• on a velocity graph, total area = distance
total integral = displacement
graph type integrate differentiate
horizontal line oblique line x axis
oblique line parabola horizontal line
parabola cubic oblique lineinflects at turning pt
Remember:• integration = area• on a velocity graph, total area = distance
total integral = displacement• on an acceleration graph, total area = speed
total integral = velocity
(ii) 2003 HSC Question 7b)The velocity of a particle is given by for , where v is measured in metres per second and t is measured in seconds
2 4cosv t 0 2t
(ii) 2003 HSC Question 7b)The velocity of a particle is given by for , where v is measured in metres per second and t is measured in seconds
2 4cosv t
(i) At what times during this period is the particle at rest?
0 2t
(ii) 2003 HSC Question 7b)The velocity of a particle is given by for , where v is measured in metres per second and t is measured in seconds
2 4cosv t
(i) At what times during this period is the particle at rest? 0v
0 2t
(ii) 2003 HSC Question 7b)The velocity of a particle is given by for , where v is measured in metres per second and t is measured in seconds
2 4cosv t
(i) At what times during this period is the particle at rest? 0v
0 2t
2 4cos 0t 1cos2
t
(ii) 2003 HSC Question 7b)The velocity of a particle is given by for , where v is measured in metres per second and t is measured in seconds
2 4cosv t
(i) At what times during this period is the particle at rest? 0v
0 2t
2 4cos 0t 1cos2
t
Q1, 4
(ii) 2003 HSC Question 7b)The velocity of a particle is given by for , where v is measured in metres per second and t is measured in seconds
2 4cosv t
(i) At what times during this period is the particle at rest? 0v
0 2t
2 4cos 0t 1cos2
t
Q1, 4 1cos2
3
(ii) 2003 HSC Question 7b)The velocity of a particle is given by for , where v is measured in metres per second and t is measured in seconds
2 4cosv t
(i) At what times during this period is the particle at rest? 0v
0 2t
2 4cos 0t 1cos2
t
Q1, 4 1cos2
3
, 2t 5,
3 3t
(ii) 2003 HSC Question 7b)The velocity of a particle is given by for , where v is measured in metres per second and t is measured in seconds
2 4cosv t
(i) At what times during this period is the particle at rest? 0v
0 2t
2 4cos 0t 1cos2
t
Q1, 4 1cos2
3
, 2t 5,
3 3t
5 particle is at rest after seconds and again after seconds3 3
(ii) 2003 HSC Question 7b)The velocity of a particle is given by for , where v is measured in metres per second and t is measured in seconds
2 4cosv t
(i) At what times during this period is the particle at rest? 0v
0 2t
2 4cos 0t 1cos2
t
Q1, 4 1cos2
3
, 2t 5,
3 3t
5 particle is at rest after seconds and again after seconds3 3
(ii) What is the maximum velocity of the particle during this period?
(ii) 2003 HSC Question 7b)The velocity of a particle is given by for , where v is measured in metres per second and t is measured in seconds
2 4cosv t
(i) At what times during this period is the particle at rest? 0v
0 2t
2 4cos 0t 1cos2
t
Q1, 4 1cos2
3
, 2t 5,
3 3t
5 particle is at rest after seconds and again after seconds3 3
(ii) What is the maximum velocity of the particle during this period? 4 4cos 4t
(ii) 2003 HSC Question 7b)The velocity of a particle is given by for , where v is measured in metres per second and t is measured in seconds
2 4cosv t
(i) At what times during this period is the particle at rest? 0v
0 2t
2 4cos 0t 1cos2
t
Q1, 4 1cos2
3
, 2t 5,
3 3t
5 particle is at rest after seconds and again after seconds3 3
(ii) What is the maximum velocity of the particle during this period? 4 4cos 4t
2 2 4cos 6t
(ii) 2003 HSC Question 7b)The velocity of a particle is given by for , where v is measured in metres per second and t is measured in seconds
2 4cosv t
(i) At what times during this period is the particle at rest? 0v
0 2t
2 4cos 0t 1cos2
t
Q1, 4 1cos2
3
, 2t 5,
3 3t
5 particle is at rest after seconds and again after seconds3 3
(ii) What is the maximum velocity of the particle during this period?
2 2 4cos 6t
maximum velocity is 6 m/s
4 4cos 4t
(iii) Sketch the graph of v as a function of t for 0 2t
amplitude 4 units
(iii) Sketch the graph of v as a function of t for 0 2t
shift 2 unitsflip upside down
amplitude 4 units
(iii) Sketch the graph of v as a function of t for
2period1
2
shift 2 unitsflip upside down
amplitude 4 units
(iii) Sketch the graph of v as a function of t for 0 2t
2period1
2
2divisions4
2
shift 2 unitsflip upside down
amplitude 4 units
(iii) Sketch the graph of v as a function of t for 0 2t
2period1
2
2divisions4
2
shift 2 unitsflip upside down
amplitude 4 units
(iii) Sketch the graph of v as a function of t for 0 2t
6v
-2
-1
4
2 3
2 2 t
1
23
5
2period1
2
2divisions4
2
shift 2 unitsflip upside down
amplitude 4 units
(iii) Sketch the graph of v as a function of t for 0 2t
6v
-2
-1
4
2 3
2 2 t
1
23
5
2period1
2
2divisions4
2
shift 2 unitsflip upside down
amplitude 4 units
(iii) Sketch the graph of v as a function of t for 0 2t
6v
-2
-1
4
2 3
2 2 t
1
23
5
2period1
2
2divisions4
2
shift 2 unitsflip upside down
amplitude 4 units
(iii) Sketch the graph of v as a function of t for 0 2t
6v
-2
-1
4
2 3
2 2 t
1
23
5
2period1
2
2divisions4
2
shift 2 unitsflip upside down
amplitude 4 units
2 4cosv t
(iii) Sketch the graph of v as a function of t for 0 2t
6v
-2
-1
4
2 3
2 2 t
1
23
5
(iv) Calculate the total distance travelled by the particle between t = 0 and t =
(iv) Calculate the total distance travelled by the particle between t = 0 and t =
3
03
distance = 2 4cos 2 4cost dt t dt
(iv) Calculate the total distance travelled by the particle between t = 0 and t =
3
03
distance = 2 4cos 2 4cost dt t dt
0
3 3= 2 4sin 2 4sint t t t
(iv) Calculate the total distance travelled by the particle between t = 0 and t =
3
03
distance = 2 4cos 2 4cost dt t dt
0
3 3= 2 4sin 2 4sint t t t
2= 0 0 2 4sin 2 4sin3 3
(iv) Calculate the total distance travelled by the particle between t = 0 and t =
3
03
distance = 2 4cos 2 4cost dt t dt
0
3 3= 2 4sin 2 4sint t t t
2= 0 0 2 4sin 2 4sin3 3
2 4 3=2 23 2
(iv) Calculate the total distance travelled by the particle between t = 0 and t =
3
03
distance = 2 4cos 2 4cost dt t dt
2=4 3 metres3
0
3 3= 2 4sin 2 4sint t t t
2= 0 0 2 4sin 2 4sin3 3
2 4 3=2 23 2
(iii) 2004 HSC Question 9b)A particle moves along the x-axis. Initially it is at rest at the origin. The graph shows the acceleration, a, of the particle as a function of time t for 0 5t
(iii) 2004 HSC Question 9b)A particle moves along the x-axis. Initially it is at rest at the origin. The graph shows the acceleration, a, of the particle as a function of time t for 0 5t
(i) Write down the time at which the velocity of the particle is a maximum
(iii) 2004 HSC Question 9b)A particle moves along the x-axis. Initially it is at rest at the origin. The graph shows the acceleration, a, of the particle as a function of time t for 0 5t
(i) Write down the time at which the velocity of the particle is a maximum
v adt
(iii) 2004 HSC Question 9b)A particle moves along the x-axis. Initially it is at rest at the origin. The graph shows the acceleration, a, of the particle as a function of time t for 0 5t
(i) Write down the time at which the velocity of the particle is a maximum
v adt is a maximum when 2adt t
(iii) 2004 HSC Question 9b)A particle moves along the x-axis. Initially it is at rest at the origin. The graph shows the acceleration, a, of the particle as a function of time t for 0 5t
(i) Write down the time at which the velocity of the particle is a maximum
v adt is a maximum when 2adt t
OR is a maximum when 0dvvdt
(iii) 2004 HSC Question 9b)A particle moves along the x-axis. Initially it is at rest at the origin. The graph shows the acceleration, a, of the particle as a function of time t for 0 5t
(i) Write down the time at which the velocity of the particle is a maximum
velocity is a maximum when 2 secondst
v adt is a maximum when 2adt t
OR is a maximum when 0dvvdt
(ii) At what time during the interval is the particle furthest from the origin? Give reasons for your answer.
0 5t
(ii) At what time during the interval is the particle furthest from the origin? Give reasons for your answer.
0 5t
Question is asking, “when is displacement a maximum?”
(ii) At what time during the interval is the particle furthest from the origin? Give reasons for your answer.
is a maximum when 0dxxdt
0 5t
Question is asking, “when is displacement a maximum?”
(ii) At what time during the interval is the particle furthest from the origin? Give reasons for your answer.
is a maximum when 0dxxdt
0 5t
Question is asking, “when is displacement a maximum?”
But v adt
(ii) At what time during the interval is the particle furthest from the origin? Give reasons for your answer.
is a maximum when 0dxxdt
0 5t
Question is asking, “when is displacement a maximum?”
But v adt We must solve 0adt
(ii) At what time during the interval is the particle furthest from the origin? Give reasons for your answer.
is a maximum when 0dxxdt
0 5t
Question is asking, “when is displacement a maximum?”
But v adt We must solve 0adt
i.e. when is area above the axis = area below
(ii) At what time during the interval is the particle furthest from the origin? Give reasons for your answer.
is a maximum when 0dxxdt
0 5t
Question is asking, “when is displacement a maximum?”
But v adt We must solve 0adt
i.e. when is area above the axis = area belowBy symmetry this would be at t = 4
(ii) At what time during the interval is the particle furthest from the origin? Give reasons for your answer.
is a maximum when 0dxxdt
0 5t
Question is asking, “when is displacement a maximum?”
But v adt We must solve 0adt
i.e. when is area above the axis = area belowBy symmetry this would be at t = 4
particle is furthest from the origin at 4 secondst
(iv) 2007 HSC Question 10a)An object is moving on the x-axis. The graph shows the velocity, , of the object, as a function of t.The coordinates of the points shown on the graph are A(2,1), B(4,5), C(5,0) and D(6,–5). The velocity is constant for
dxdt
6t
(iv) 2007 HSC Question 10a)An object is moving on the x-axis. The graph shows the velocity, , of the object, as a function of t.The coordinates of the points shown on the graph are A(2,1), B(4,5), C(5,0) and D(6,–5). The velocity is constant for
dxdt
6t
(i) Using Simpson’s rule, estimate the distance travelled between t = 0 and t = 4
(iv) 2007 HSC Question 10a)An object is moving on the x-axis. The graph shows the velocity, , of the object, as a function of t.The coordinates of the points shown on the graph are A(2,1), B(4,5), C(5,0) and D(6,–5). The velocity is constant for
dxdt
6t
(i) Using Simpson’s rule, estimate the distance travelled between t = 0 and t = 4 0distance 4 2
3 odd even nh y y y y
(iv) 2007 HSC Question 10a)An object is moving on the x-axis. The graph shows the velocity, , of the object, as a function of t.The coordinates of the points shown on the graph are A(2,1), B(4,5), C(5,0) and D(6,–5). The velocity is constant for
dxdt
6t
(i) Using Simpson’s rule, estimate the distance travelled between t = 0 and t = 4
t 0 2 4v 0 1 5
0distance 4 23 odd even nh y y y y
(iv) 2007 HSC Question 10a)An object is moving on the x-axis. The graph shows the velocity, , of the object, as a function of t.The coordinates of the points shown on the graph are A(2,1), B(4,5), C(5,0) and D(6,–5). The velocity is constant for
dxdt
6t
(i) Using Simpson’s rule, estimate the distance travelled between t = 0 and t = 4
t 0 2 4v 0 1 5
1 1 0distance 4 2
3 odd even nh y y y y
(iv) 2007 HSC Question 10a)An object is moving on the x-axis. The graph shows the velocity, , of the object, as a function of t.The coordinates of the points shown on the graph are A(2,1), B(4,5), C(5,0) and D(6,–5). The velocity is constant for
dxdt
6t
(i) Using Simpson’s rule, estimate the distance travelled between t = 0 and t = 4
t 0 2 4v 0 1 5
1 14 0distance 4 2
3 odd even nh y y y y
(iv) 2007 HSC Question 10a)An object is moving on the x-axis. The graph shows the velocity, , of the object, as a function of t.The coordinates of the points shown on the graph are A(2,1), B(4,5), C(5,0) and D(6,–5). The velocity is constant for
dxdt
6t
(i) Using Simpson’s rule, estimate the distance travelled between t = 0 and t = 4
t 0 2 4v 0 1 5
1 14 2 0 4 1 536 metres
0distance 4 23 odd even nh y y y y
(ii) The object is initially at the origin. During which time(s) is the displacement decreasing?
(ii) The object is initially at the origin. During which time(s) is the displacement decreasing?
is decreasing when 0dxxdt
(ii) The object is initially at the origin. During which time(s) is the displacement decreasing?
is decreasing when 0dxxdt
displacement is decreasing when 5 secondst
(ii) The object is initially at the origin. During which time(s) is the displacement decreasing?
is decreasing when 0dxxdt
displacement is decreasing when 5 secondst
(iii) Estimate the time at which the object returns to the origin. Justify your answer.
(ii) The object is initially at the origin. During which time(s) is the displacement decreasing?
is decreasing when 0dxxdt
displacement is decreasing when 5 secondst
(iii) Estimate the time at which the object returns to the origin. Justify your answer.Question is asking, “when is displacement = 0?”
(ii) The object is initially at the origin. During which time(s) is the displacement decreasing?
is decreasing when 0dxxdt
displacement is decreasing when 5 secondst
(iii) Estimate the time at which the object returns to the origin. Justify your answer.Question is asking, “when is displacement = 0?”
But x vdt
(ii) The object is initially at the origin. During which time(s) is the displacement decreasing?
is decreasing when 0dxxdt
displacement is decreasing when 5 secondst
(iii) Estimate the time at which the object returns to the origin. Justify your answer.Question is asking, “when is displacement = 0?”
But x vdt We must solve 0vdt
(ii) The object is initially at the origin. During which time(s) is the displacement decreasing?
is decreasing when 0dxxdt
displacement is decreasing when 5 secondst
(iii) Estimate the time at which the object returns to the origin. Justify your answer.Question is asking, “when is displacement = 0?”
But x vdt We must solve 0vdt
i.e. when is area above the axis = area below
(ii) The object is initially at the origin. During which time(s) is the displacement decreasing?
is decreasing when 0dxxdt
displacement is decreasing when 5 secondst
(iii) Estimate the time at which the object returns to the origin. Justify your answer.Question is asking, “when is displacement = 0?”
But x vdt We must solve 0vdt
i.e. when is area above the axis = area belowBy symmetry, area from t = 4 to 5 equals areafrom t = 5 to 6
(ii) The object is initially at the origin. During which time(s) is the displacement decreasing?
is decreasing when 0dxxdt
displacement is decreasing when 5 secondst
(iii) Estimate the time at which the object returns to the origin. Justify your answer.Question is asking, “when is displacement = 0?”
But x vdt We must solve 0vdt
i.e. when is area above the axis = area belowBy symmetry, area from t = 4 to 5 equals areafrom t = 5 to 6In part (i) we estimated area from t = 0 to 4 to be 6,
(ii) The object is initially at the origin. During which time(s) is the displacement decreasing?
is decreasing when 0dxxdt
displacement is decreasing when 5 secondst
(iii) Estimate the time at which the object returns to the origin. Justify your answer.Question is asking, “when is displacement = 0?”
But x vdt We must solve 0vdt
i.e. when is area above the axis = area belowBy symmetry, area from t = 4 to 5 equals areafrom t = 5 to 6In part (i) we estimated area from t = 0 to 4 to be 6,
4 6A
4A
(ii) The object is initially at the origin. During which time(s) is the displacement decreasing?
is decreasing when 0dxxdt
displacement is decreasing when 5 secondst
(iii) Estimate the time at which the object returns to the origin. Justify your answer.Question is asking, “when is displacement = 0?”
But x vdt We must solve 0vdt
i.e. when is area above the axis = area belowBy symmetry, area from t = 4 to 5 equals areafrom t = 5 to 6In part (i) we estimated area from t = 0 to 4 to be 6,
4 6A
4A
a
5
5 6a
(ii) The object is initially at the origin. During which time(s) is the displacement decreasing?
is decreasing when 0dxxdt
displacement is decreasing when 5 secondst
(iii) Estimate the time at which the object returns to the origin. Justify your answer.Question is asking, “when is displacement = 0?”
But x vdt We must solve 0vdt
i.e. when is area above the axis = area belowBy symmetry, area from t = 4 to 5 equals areafrom t = 5 to 6In part (i) we estimated area from t = 0 to 4 to be 6,
4 6A
4A
a
5
5 6a 1.2a
(ii) The object is initially at the origin. During which time(s) is the displacement decreasing?
is decreasing when 0dxxdt
displacement is decreasing when 5 secondst
(iii) Estimate the time at which the object returns to the origin. Justify your answer.Question is asking, “when is displacement = 0?”
But x vdt We must solve 0vdt
i.e. when is area above the axis = area belowBy symmetry, area from t = 4 to 5 equals areafrom t = 5 to 6In part (i) we estimated area from t = 0 to 4 to be 6,
4 6A
4A
a
5
5 6a 1.2a particle returns to the origin when 7.2 secondst
(iv) Sketch the displacement, x, as a function of time.
(iv) Sketch the displacement, x, as a function of time.
x
t2 4 6 8
6
8.5
(iv) Sketch the displacement, x, as a function of time.
x
t2 4 6 8
6
8.5
object is initially at the origin
(iv) Sketch the displacement, x, as a function of time.
x
t2 4 6 8
6
8.5
object is initially at the originwhen t = 4, x = 6
(iv) Sketch the displacement, x, as a function of time.
x
t2 4 6 8
6
8.5
object is initially at the originwhen t = 4, x = 6
by symmetry of areas t = 6, x = 6
(iv) Sketch the displacement, x, as a function of time.
x
t2 4 6 8
6
8.5
object is initially at the originwhen t = 4, x = 6
by symmetry of areas t = 6, x = 6
Area of triangle = 2.5when 5, 8.5t x
(iv) Sketch the displacement, x, as a function of time.
x
t2 4 6 8
6
8.5
object is initially at the originwhen t = 4, x = 6
by symmetry of areas t = 6, x = 6
Area of triangle = 2.5when 5, 8.5t x
returns to x = 0 when t = 7.2
7.2
(iv) Sketch the displacement, x, as a function of time.
x
t2 4 6 8
6
8.5
object is initially at the originwhen t = 4, x = 6
by symmetry of areas t = 6, x = 6
Area of triangle = 2.5when 5, 8.5t x
returns to x = 0 when t = 7.2
7.2
v is steeper between t = 2 and 4 than between t = 0 and 2
particle covers more distance between 2 and 4t
(iv) Sketch the displacement, x, as a function of time.
x
t2 4 6 8
6
8.5
object is initially at the originwhen t = 4, x = 6
by symmetry of areas t = 6, x = 6
Area of triangle = 2.5when 5, 8.5t x
returns to x = 0 when t = 7.2
7.2
v is steeper between t = 2 and 4 than between t = 0 and 2
particle covers more distance between 2 and 4t
when t > 6, v is constantwhen 6, is a straight linet x
(iv) Sketch the displacement, x, as a function of time.
x
t2 4 6 8
6
8.5
object is initially at the originwhen t = 4, x = 6
by symmetry of areas t = 6, x = 6
Area of triangle = 2.5when 5, 8.5t x
returns to x = 0 when t = 7.2
7.2
v is steeper between t = 2 and 4 than between t = 0 and 2
particle covers more distance between 2 and 4t
when t > 6, v is constantwhen 6, is a straight linet x
(v) 2005 HSC Question 7b)
The graph shows the velocity, , of a particle as a function of time.Initially the particle is at the origin.
dxdt
(v) 2005 HSC Question 7b)
The graph shows the velocity, , of a particle as a function of time.Initially the particle is at the origin.
dxdt
(i) At what time is the displacement, x, from the origin a maximum?
(v) 2005 HSC Question 7b)
The graph shows the velocity, , of a particle as a function of time.Initially the particle is at the origin.
dxdt
(i) At what time is the displacement, x, from the origin a maximum?
Displacement is a maximum when area is most positive, also when velocity is zero
(v) 2005 HSC Question 7b)
The graph shows the velocity, , of a particle as a function of time.Initially the particle is at the origin.
dxdt
(i) At what time is the displacement, x, from the origin a maximum?
Displacement is a maximum when area is most positive, also when velocity is zero
i.e. when t = 2
(ii) At what time does the particle return to the origin? Justify your answer
(ii) At what time does the particle return to the origin? Justify your answer
Question is asking, “when is displacement = 0?”
(ii) At what time does the particle return to the origin? Justify your answer
Question is asking, “when is displacement = 0?”
i.e. when is area above the axis = area below?
(ii) At what time does the particle return to the origin? Justify your answer
Question is asking, “when is displacement = 0?”
i.e. when is area above the axis = area below?
2 a
a 2
w
(ii) At what time does the particle return to the origin? Justify your answer
Question is asking, “when is displacement = 0?”
i.e. when is area above the axis = area below?
2 a
a 2
w
2w = 2w = 1
(ii) At what time does the particle return to the origin? Justify your answer
Question is asking, “when is displacement = 0?”
i.e. when is area above the axis = area below?
2 a
a 2
w
2w = 2w = 1
Returns to the origin after 4 seconds
2
2(iii) Draw a sketch of the acceleration, , as afunction of
time for 0 6
d xdt
t
2
2(iii) Draw a sketch of the acceleration, , as afunction of
time for 0 6
d xdt
t
t1 2 3 5 6
2
2d xdt
2
2(iii) Draw a sketch of the acceleration, , as afunction of
time for 0 6
d xdt
t
t1 2 3 5 6
2
2d xdt
you get the xaxis
differentiate a horizontal line
2
2(iii) Draw a sketch of the acceleration, , as afunction of
time for 0 6
d xdt
t
t1 2 3 5 6
2
2d xdt
you get the xaxis
differentiate a horizontal line
from 1 to 3 we have a cubic, inflects at 2, and is decreasing
differentiate, you get a parabola, stationary at 2, it is below the x axis
2
2(iii) Draw a sketch of the acceleration, , as afunction of
time for 0 6
d xdt
t
t1 2 3 5 6
2
2d xdt
you get the xaxis
differentiate a horizontal line
from 1 to 3 we have a cubic, inflects at 2, and is decreasing
differentiate, you get a parabola, stationary at 2, it is below the x axis
from 5 to 6 is a cubic, inflects at 6 and is increasing (using symmetry)
differentiate, you get a parabola stationary at 6, it is above the x axis
2
2(iii) Draw a sketch of the acceleration, , as afunction of
time for 0 6
d xdt
t
t1 2 3 5 6
2
2d xdt
you get the xaxis
differentiate a horizontal line
from 1 to 3 we have a cubic, inflects at 2, and is decreasing
differentiate, you get a parabola, stationary at 2, it is below the x axis
from 5 to 6 is a cubic, inflects at 6 and is increasing (using symmetry)
differentiate, you get a parabola stationary at 6, it is above the x axis
Exercise 3A; 4, 7, 8
Exercise 3B; 2, 4, 6, 8, 10, 12
Exercise 3C; 1 ace etc, 2 ace etc, 4a, 7ab(i), 8, 9a, 10, 13, 16, 18
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