Unit – 4

Unit – 4

Unit – 4

Application of Partial Differentiation and Successive Differentiation

Question and answer

1. Find out the maxima and minima of the function

   Given     …(i)

         Partially differentiating (i) with respect to x we get

                   ….(ii)

         Partially differentiating (i) with respect to y we get

                    ….(iii)

          Now, form the equations

                    Using (ii) and (iii) we get

                                  using above two equations

                     Squaring both side we get

                      Or 

                       This show that

                         Also we get

                 Thus we get the pair of value as

                  Now, we calculate

      Putting above values in

      At point (0,0)  we get

      So, the point (0,0) is a saddle point.

      At point we get

        So the point is the minimum point where

          In case 

         So the point is the maximum point where

2.     Find the maximum and minimum point of the function

        Partially differentiating given equation with respect to and x and y then equate them to zero                 

       On solving above we get 

         Also

            Thus we get the pair of values (0,0), (,0) and (0,

        Now, we calculate

        At the point (0,0)

       So function has saddle point at (0,0).

       At the point (

         So the function has maxima at this point (.

        At the point (0,

       So the function has minima at this point (0,.

        At the point (

        So the function has an saddle point at (

3.     Find the maximum and minimum value of

Let

Partially differentiating given function with respect to x and y and equate it to zero

          ..(i)

           ..(ii)

      On solving (i) and (ii) we get

Thus pair of values are

Now, we calculate

At the point (0,0)

  So further investigation is required

On the x  axis y  = 0  , f(x,0)=0

On the line y=x,

At the point

So that the given function has maximum value at

Therefore, maximum value of given function

At the point

So that the given function has minimum value at

Therefore, minimum value of the given function

4.     Divide 24 into three parts such that the continued product of the first, square of second and cube of third may be maximum.                                    

Let first number be x, second be y and third be z.

According to the question

Let the given function be  f

And the relation

By Lagrange’s Method

        ….(i)

Partially differentiating (i) with respect to x,y and z and  equate  them  to zero

     ….(ii)

      ….(iii)

      ….(iv)

  From (ii),(iii)  and (iv) we get

On solving

Putting it in given relation we get

             Or 

            Or

Thus the first number is 4 second is 8 and third is 12

5.     If ,Find the value of x and y for which is maximum.

Given function is

And relation is

By Lagrange’s Method

      []    ..(i)

Partially differentiating (i) with respect to  x, y and z and equate them to zero

                 Or       …(ii)

                   Or       …(iii)

                   Or       …(iv)

On solving (ii),(iii) and (iv) we  get

Using the given relation  we get

So that

Thus the point for the maximum value of the given function is

6.     Find the nth derivative of  

Since  

Differentiating both side with  respect to  x

       [

      Again differentiating with respect to x

    Again  differentiating  with respect to x

   Similarly the  nth  derivative is

7.     Find the  nth  derivative of

    Let 

]

Differentiating  with  respect  to   x  we get

  Again  differentiating  with  respect to x  we get

Again  differentiating  with  respect to x  we get

Similarly Again  differentiating  with  respect to x  we get

8.     Find the nth  derivative 

  Let

   Differentiating with respect to x.

  Again differentiating with respect to x.

    Again differentiating with respect to x.

    Again differentiating with respect to x.

    Again differentiating with respect to x.

  Similarly the  nth derivative with respect to  x.

9.     Find the nth derivative of  

  Let

Also

   By Leibnitz’s theorem

       …(i)

Here 

Differentiating with  respect  to  x, we get

      Again  differentiating  with respect to x, we get

Similarly  the  nth derivative will be

From  (i) and (ii) we have,

10. Find the value of

Let     

         Differentiating both side with respect to a we get

        Using Leibnitz’s Rule

Now,  integrating both side

As given integral

   Thus  

11. If ,prove that y satisfied the differential equation

   Given

         Differentiating both side with respect to x

         By Leibnitz’s Rule

Again, differentiating with respect to x we get

Using Leibnitz’s Rule

Hence proved