Flashcards: Application of Derivatives | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


 
 
 
 
APPLICATION OF DERIVATIVES 
The notion of derivatives was introduced to solve problems of physics concerning motion of an object. 
The velocity of an object is a measure of the rate of change of distance with respect to time. Acceleration 
is a measure of the rate of change of velocity with respect to time. This rate of change is precisely given 
by the notion of derivative. The derivative is also used to find the equation of tangent to a curve at a 
specific point. 
  
Page 2


 
 
 
 
APPLICATION OF DERIVATIVES 
The notion of derivatives was introduced to solve problems of physics concerning motion of an object. 
The velocity of an object is a measure of the rate of change of distance with respect to time. Acceleration 
is a measure of the rate of change of velocity with respect to time. This rate of change is precisely given 
by the notion of derivative. The derivative is also used to find the equation of tangent to a curve at a 
specific point. 
  
 
 
 
 
TANGENT AND NORMAL 
Let ?? =?? (?? ) be the equation of a curve. At a point ?? , on the curve, the value of 
????
????
 (if defined) gives the 
slope the tangent (=tan??? ) to the curve at the point ?? . This is also said to give the slope of the curve at 
the point ?? 
  
Page 3


 
 
 
 
APPLICATION OF DERIVATIVES 
The notion of derivatives was introduced to solve problems of physics concerning motion of an object. 
The velocity of an object is a measure of the rate of change of distance with respect to time. Acceleration 
is a measure of the rate of change of velocity with respect to time. This rate of change is precisely given 
by the notion of derivative. The derivative is also used to find the equation of tangent to a curve at a 
specific point. 
  
 
 
 
 
TANGENT AND NORMAL 
Let ?? =?? (?? ) be the equation of a curve. At a point ?? , on the curve, the value of 
????
????
 (if defined) gives the 
slope the tangent (=tan??? ) to the curve at the point ?? . This is also said to give the slope of the curve at 
the point ?? 
  
 
 
 
 
 
TANGENT AND NORMAL 
(i) Equation of tangent at point ?? (?? 1
,?? 1
) to the curve ?? =?? (?? ) is given by the equation : 
?? -?? 1
=(
????
????
)
(?? 1
,?? 1
)
(?? -?? 1
) 
where (
????
????
)
(?? 1
,?? 1
)
 denotes value of 
????
????
 at the point (?? 1
,?? 1
) . 
  
Page 4


 
 
 
 
APPLICATION OF DERIVATIVES 
The notion of derivatives was introduced to solve problems of physics concerning motion of an object. 
The velocity of an object is a measure of the rate of change of distance with respect to time. Acceleration 
is a measure of the rate of change of velocity with respect to time. This rate of change is precisely given 
by the notion of derivative. The derivative is also used to find the equation of tangent to a curve at a 
specific point. 
  
 
 
 
 
TANGENT AND NORMAL 
Let ?? =?? (?? ) be the equation of a curve. At a point ?? , on the curve, the value of 
????
????
 (if defined) gives the 
slope the tangent (=tan??? ) to the curve at the point ?? . This is also said to give the slope of the curve at 
the point ?? 
  
 
 
 
 
 
TANGENT AND NORMAL 
(i) Equation of tangent at point ?? (?? 1
,?? 1
) to the curve ?? =?? (?? ) is given by the equation : 
?? -?? 1
=(
????
????
)
(?? 1
,?? 1
)
(?? -?? 1
) 
where (
????
????
)
(?? 1
,?? 1
)
 denotes value of 
????
????
 at the point (?? 1
,?? 1
) . 
  
 
 
 
TANGENT AND NORMAL 
(ii) Equation of normal at point ?? (?? 1
,?? 1
) (by definition, normal is a line ? to a tangent at ?? ) is given by 
?? -?? 1
=
(
 
-1
(
????
????
)
(?? 1
,?? 1
)
(?? -?? 1
)?( here, (
????
????
)
?? ?0)
 or (?? -?? 1
)+(
????
????
)
(?? 1
,?? 1
)
(?? -?? 1
)=0
 
  
Page 5


 
 
 
 
APPLICATION OF DERIVATIVES 
The notion of derivatives was introduced to solve problems of physics concerning motion of an object. 
The velocity of an object is a measure of the rate of change of distance with respect to time. Acceleration 
is a measure of the rate of change of velocity with respect to time. This rate of change is precisely given 
by the notion of derivative. The derivative is also used to find the equation of tangent to a curve at a 
specific point. 
  
 
 
 
 
TANGENT AND NORMAL 
Let ?? =?? (?? ) be the equation of a curve. At a point ?? , on the curve, the value of 
????
????
 (if defined) gives the 
slope the tangent (=tan??? ) to the curve at the point ?? . This is also said to give the slope of the curve at 
the point ?? 
  
 
 
 
 
 
TANGENT AND NORMAL 
(i) Equation of tangent at point ?? (?? 1
,?? 1
) to the curve ?? =?? (?? ) is given by the equation : 
?? -?? 1
=(
????
????
)
(?? 1
,?? 1
)
(?? -?? 1
) 
where (
????
????
)
(?? 1
,?? 1
)
 denotes value of 
????
????
 at the point (?? 1
,?? 1
) . 
  
 
 
 
TANGENT AND NORMAL 
(ii) Equation of normal at point ?? (?? 1
,?? 1
) (by definition, normal is a line ? to a tangent at ?? ) is given by 
?? -?? 1
=
(
 
-1
(
????
????
)
(?? 1
,?? 1
)
(?? -?? 1
)?( here, (
????
????
)
?? ?0)
 or (?? -?? 1
)+(
????
????
)
(?? 1
,?? 1
)
(?? -?? 1
)=0
 
  
 
 
 
TANGENT AND NORMAL 
(iii) Angle between two curves : Let ?? =?? (?? ) and ?? =?? (?? ) be the equation of two curves intersecting at 
point ?? (?? 1
,?? 1
) . Then the angle between curves ?? =?? (?? ) and ?? =?? (?? ) at their point of intersection ?? is 
defined as the angle ?? between the tangents ?? ?? and ?? ?? to the curves ?? =?? (?? ) and ?? =?? (?? ) respectively at 
their point of intersection.