PPT - Differentiation | Business Mathematics and Statistics - B Com PDF Download

 Page 1


  
Higher Maths         1            3    Differentiation 1
Page 2


  
Higher Maths         1            3    Differentiation 1
  
Higher Maths         1            3    Differentiation
The History of Differentiation
Differentiation is part of the science of Calculus, and was first 
developed in the 17
th
 century by two different Mathematicians.
Gottfried Leibniz
(1646-1716)
Germany
Sir Isaac Newton
(1642-1727)
England
2
Differentiation, or finding the 
instantaneous rate of change, is 
an essential part of:
  • Mathematics and Physics
  • Chemistry
  • Biology
  • Computer Science
  • Engineering
  • Navigation and Astronomy
Page 3


  
Higher Maths         1            3    Differentiation 1
  
Higher Maths         1            3    Differentiation
The History of Differentiation
Differentiation is part of the science of Calculus, and was first 
developed in the 17
th
 century by two different Mathematicians.
Gottfried Leibniz
(1646-1716)
Germany
Sir Isaac Newton
(1642-1727)
England
2
Differentiation, or finding the 
instantaneous rate of change, is 
an essential part of:
  • Mathematics and Physics
  • Chemistry
  • Biology
  • Computer Science
  • Engineering
  • Navigation and Astronomy
  
Higher Maths         1            3    Differentiation
Calculating Speed
2 4 6 8
4
8
2
6
10
0
0
Time (seconds)
Distance (m)
D
S T ×
÷ ÷
Example
Calculate the speed for each 
section of the journey opposite.
A
B
C
speed in A  =  
4
3
speed in B  =  
5
1
5 m/s 
 
=
speed in C  =  
2
5
0.4 m/s 
 
=
average speed  =  
9
1.22 m/s 
 
˜
11
˜
1.33 m/s 
 
Notice the following things:
• the speed at each instant is
   not the same as the average
• speed is the same as gradient
D
T
S  = 
 
y
x
= m =
3
Page 4


  
Higher Maths         1            3    Differentiation 1
  
Higher Maths         1            3    Differentiation
The History of Differentiation
Differentiation is part of the science of Calculus, and was first 
developed in the 17
th
 century by two different Mathematicians.
Gottfried Leibniz
(1646-1716)
Germany
Sir Isaac Newton
(1642-1727)
England
2
Differentiation, or finding the 
instantaneous rate of change, is 
an essential part of:
  • Mathematics and Physics
  • Chemistry
  • Biology
  • Computer Science
  • Engineering
  • Navigation and Astronomy
  
Higher Maths         1            3    Differentiation
Calculating Speed
2 4 6 8
4
8
2
6
10
0
0
Time (seconds)
Distance (m)
D
S T ×
÷ ÷
Example
Calculate the speed for each 
section of the journey opposite.
A
B
C
speed in A  =  
4
3
speed in B  =  
5
1
5 m/s 
 
=
speed in C  =  
2
5
0.4 m/s 
 
=
average speed  =  
9
1.22 m/s 
 
˜
11
˜
1.33 m/s 
 
Notice the following things:
• the speed at each instant is
   not the same as the average
• speed is the same as gradient
D
T
S  = 
 
y
x
= m =
3
  
Instantaneous Speed
Higher Maths         1            3    Differentiation
Time (seconds)
Distance (m)
Time (seconds)
Distance (m)
In reality speed does not often change instantly. The graph on the 
right is more realistic as it shows a gradually changing curve.
The journey has the same average speed, but the instantaneous 
speed is different at each point because the gradient of the curve is 
constantly changing. How can we find the instantaneous speed?
D
T
S  = 
 
y
x
=
m
=
4
Page 5


  
Higher Maths         1            3    Differentiation 1
  
Higher Maths         1            3    Differentiation
The History of Differentiation
Differentiation is part of the science of Calculus, and was first 
developed in the 17
th
 century by two different Mathematicians.
Gottfried Leibniz
(1646-1716)
Germany
Sir Isaac Newton
(1642-1727)
England
2
Differentiation, or finding the 
instantaneous rate of change, is 
an essential part of:
  • Mathematics and Physics
  • Chemistry
  • Biology
  • Computer Science
  • Engineering
  • Navigation and Astronomy
  
Higher Maths         1            3    Differentiation
Calculating Speed
2 4 6 8
4
8
2
6
10
0
0
Time (seconds)
Distance (m)
D
S T ×
÷ ÷
Example
Calculate the speed for each 
section of the journey opposite.
A
B
C
speed in A  =  
4
3
speed in B  =  
5
1
5 m/s 
 
=
speed in C  =  
2
5
0.4 m/s 
 
=
average speed  =  
9
1.22 m/s 
 
˜
11
˜
1.33 m/s 
 
Notice the following things:
• the speed at each instant is
   not the same as the average
• speed is the same as gradient
D
T
S  = 
 
y
x
= m =
3
  
Instantaneous Speed
Higher Maths         1            3    Differentiation
Time (seconds)
Distance (m)
Time (seconds)
Distance (m)
In reality speed does not often change instantly. The graph on the 
right is more realistic as it shows a gradually changing curve.
The journey has the same average speed, but the instantaneous 
speed is different at each point because the gradient of the curve is 
constantly changing. How can we find the instantaneous speed?
D
T
S  = 
 
y
x
=
m
=
4
  
Introduction to Differentiation
Higher Maths         1            3    Differentiation
Differentiate means
D
T
speed  =  
‘rate of change of
distance with respect to time’
S
T
acceleration  =  
‘find out how fast something is changing in comparison 
with something else at any one instant’.
gradient  =  
y
x
‘rate of change of
speed with respect to time’
‘rate of change of
   -coordinate with respect to    -coordinate’ y x
5