Question Bank (unit-5)

Question-1: - Calculate 

                           f(x , y) = 3x³-5y²+2xy-8x+4y-20

Sol. To calculate   treat the variable y as a constant, then differentiate f(x,y) with respect to x by using differentiation rules,

          = [3x³-5y²+2xy-8x+4y-20]

              = 3x³] - 5y²] + [2xy] -8x] +4y] - 20]

              = 9x² - 0 + 2y – 8 + 0 – 0

              = 9x² + 2y – 8

Similarly partial derivative of f(x,y) with respect to y is:

                      = [3x³-5y²+2xy-8x+4y-20]

                           = 3x³] - 5y²] + [2xy] -8x] +4y] - 20]

                           = 0 – 10y + 2x – 0 + 4 – 0

                           = 2x – 10y +4.

Question-2: Calculate    and   for the following function

                               f( x, y) = sin(y²x + 5x – 8)

Sol. To calculate   treat the variable y as a constant, then differentiate f(x,y) with respect to x by using differentiation rules,

                       [sin(y²x + 5x – 8)]

                           = cos(y²x + 5x – 8)(y²x + 5x – 8)

                           = (y² + 50) cos(y²x + 5x – 8)

Similarly partial derivative of f(x,y) with respect to y is,

                      [sin(y²x + 5x – 8)]

                           = cos(y²x + 5x – 8)(y²x + 5x – 8)

                           = 2xy cos(y²x + 5x – 8)

Question-3 : if , then show that-

Sol. Here we have,

u =     …………………..(1)

Now partially differentiate eq.(1) w.r to x and y , we get

                                               =

Or

                                                          ………………..(2)

And now,

                                                =

                                              ………………….(3)

Adding eq. (1) and (3) , we get

                                   =  0

Hence proved.

Question-4: Example-3: if w = x² + y – z + sint and  x + y = t, find

                   (a)    y,z

                     (b)   t, z  

Sol. With x, y, z independent, we have

                           t = x + y, w = x² + y - z + sin (x + y).

        Therefore,

                           y,z = 2x + cos(x+y)(x+y)

                                            =  2x + cos (x + y)

             With x, t, z independent, we have

                                                          Y = t-x,   w=  x² + (t-x) + sin t

                                           thus     t, z   =  2x – 1

Question-5: if φ( cx – az ,  cy – bz) = 0 then show that ap + bq = c

                          Where p =      q = 

Sol.  We have,

                              φ( cx – az ,  cy – bz) = 0   

                                φ( r , s) = 0

Where,

We know that,

Again we do,

By adding the two results, we get

Question-6: : If z is the function of x and y , and x =   , y = , then prove that,

Sol. Here , it is given that,  z is the function of x and y & x , y are the functions of u and v.

So that,

                    ……………….(1)

And,

                     ………………..(2)

Also there is,

      x =   and  y = ,

Now,

         ,       ,      ,    

From equation(1) , we get

  ……………….(3)

And from eq. (2) , we get

           …………..(4)

Subtracting eq. (4) from (3), we get

  = ) – ( 

              = x

Hence proved.

Question-7: If u = x²(y-x) + y²(x-y), then show that    -2 (x – y)²

Solution - here,  u = x²(y-x) + y²(x-y)

                                u = x²y - x³ + xy² - y³,

         now differentiate u partially with respect to x and y respectively,

                                         =  2xy – 3x² + y²        --------- (1)

                                       = x² + 2xy – 3y²          ---------- (2)

Now adding equation (1) and (2), we get

                                 = -2x² - 2y² + 4xy

                                               = -2 (x² + y² - 2xy)

                                               = -2 (x – y)²   

Question-8: If u = xy + sin(xy), show that    =  .

Solution –            u = xy + sin(xy)

                                 =   y+ ycos(xy)

                              =  x+ xcos(xy)

                             x (- sin(xy).(y)) + cos(xy)

                                     = 1 – xysin(xy) + cos(xy)     -------------- (1)

                               1 + cos(xy) + y(-sin(xy) x)

                                        = 1 – xy sin(xy) + cos(xy)      -----------------(2)

From equation (1) and (2),

                                               = 

Question-9: If u(x,y,z) = log( tan x + tan y + tan z) , then prove that ,

Sol. Here we have,

                    u(x,y,z) = log( tan x + tan y + tan z)  ………………..(1)

Diff. Eq.(1) w.r.t. x , partially , we get

                     ……………..(2)

Diff. Eq.(1) w.r.t. y , partially , we get

                    ………………(3)

Diff. Eq.(1) w.r.t. z , partially , we get

     ……………………(4)

Now multiply eq. 2 , 3 , 4 by sin 2x , sin 2y , sin 2z respectively and adding , in order to get the final result,

We get,

                                                    =

So that, 

                                                       hence proved.

Question-10: If u = u(), then prove that .

Sol. We are given that,

                                     u = u () = u( r , s)

Where r = and s =

Or we can write as-

           r =     and s =

Differentiate them partially with respect to x , y and z. , we get

As we know that-

           Then,

Adding all these results, we get

                                   .