Flashcards: Application of Derivatives

``` 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.

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## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

## FAQs on Flashcards: Application of Derivatives - Mathematics (Maths) for JEE Main & Advanced

 1. How can derivatives be applied in real-life scenarios?
Ans. Derivatives can be applied in various real-life scenarios such as calculating the rate of change, finding maximum or minimum values, optimizing functions, and predicting trends in business and economics.
 2. What is the importance of derivatives in calculus and mathematics?
Ans. Derivatives play a crucial role in calculus and mathematics by helping to analyze the behavior of functions, solve optimization problems, and understand the rate of change of a function at a specific point.
 3. Can derivatives be used to calculate velocity and acceleration?
Ans. Yes, derivatives can be used to calculate velocity by finding the derivative of the position function with respect to time, and acceleration can be calculated by finding the derivative of velocity with respect to time.
 4. How are derivatives used in finance and economics?
Ans. Derivatives are used in finance and economics for risk management, pricing financial instruments, hedging against market fluctuations, and understanding the behavior of complex financial systems.
 5. In what ways can derivatives be applied in science and engineering?
Ans. Derivatives are extensively used in science and engineering to analyze motion, optimize designs, predict outcomes in experiments, model physical systems, and understand the behavior of complex systems.

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

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