A beautiful problem involving the concept of relation-mapping from IIT JAM 2014.
Let be a differentiable function such that
for all
and
. Then
equals
Relation/Mapping
Differentiation
Integration
Answer: .
Question 33 (IIT JAM 2014)
Real Analysis (Willy)
The above problem can be done in many ways we will try to solve this by the simplest method.
Now, as the function is given as
So first try to change this into
. Try this. It's very easy !!!
To change into
we can easily do
Now we have to find the value of so we have to change the second degree term, i.e.,
into some linear form. Can you cook this up ???
Let us assume
i.e.,
Now you know from previous knowledge that integration is also known as anti-derivative. So can be changed into
by integrating it with respect to
. Try to do this integration and we are half way done !!!
On integrating both side w.r.t we get :
, (where
is a integrating constant.)
Now we find the value to
We know
i.e.,
Can you find the answer now ?
Now simply, putting
we get
A beautiful problem involving the concept of relation-mapping from IIT JAM 2014.
Let be a differentiable function such that
for all
and
. Then
equals
Relation/Mapping
Differentiation
Integration
Answer: .
Question 33 (IIT JAM 2014)
Real Analysis (Willy)
The above problem can be done in many ways we will try to solve this by the simplest method.
Now, as the function is given as
So first try to change this into
. Try this. It's very easy !!!
To change into
we can easily do
Now we have to find the value of so we have to change the second degree term, i.e.,
into some linear form. Can you cook this up ???
Let us assume
i.e.,
Now you know from previous knowledge that integration is also known as anti-derivative. So can be changed into
by integrating it with respect to
. Try to do this integration and we are half way done !!!
On integrating both side w.r.t we get :
, (where
is a integrating constant.)
Now we find the value to
We know
i.e.,
Can you find the answer now ?
Now simply, putting
we get