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12th Standard English Medium Maths Subject Differentials and Partial Derivatives Book Back 3 Mark Questions with Solution Part - II

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 30

    3 Marks

    10 x 3 = 30
  1. Let f (x, y) = 0 if xy ≠ 0 and f (x, y) = 1 if xy = 0.
    Calculate: \(\frac { \partial f }{ \partial x } (0,0),\frac { \partial f }{ \partial y } (0,0).\)

  2. Evaluate \(\begin{matrix} lim \\ (x,y)\rightarrow (0,0) \end{matrix}cos=\left( \frac { { e }^{ x }siny }{ y } \right) \), if the limit exists.

  3. For each of the following functions find the fx, fy and show that fxy = fyx
    f(x, y) = cos (x2 - 3xy)

  4. For each of the following functions find the gxy, gxx, gyy and gyx.
    g(x, y) = xey + 3x2y

  5. For each of the following functions find the gxy, gxx, gyy and gyx.
    g(x, y) = log (5x + 3y)

  6. Let z(x, y) = x2y + 3xy4, x, y ∈ R. Find the linear approximation for z at (2, -1).

  7. Let W(x, y, z) = x2 - xy + 3 sin z, x, y, z ∈ R. Find the linear approximation for U at (2, -1, 0).

  8. Verify the above theorem for F(x, y) = x2 - 2y2 + 2xy and x(t) = cos t, y(t) = sin t, t ∈ [0, 2\(\pi\)]

  9. If w (x, y, z) = x2 + y2 + y2, x = et, y = esin t, z = et cos t, find \(\frac{dw}{dt}\)

  10. If u(x, y) = \(\frac { { x }^{ 2 }+{ y }^{ 2 } }{ \sqrt { x+y } } \), prove that \(x\frac { \partial u }{ \partial x } +y\frac { \partial u }{ \partial y } =\frac { 3 }{ 2 } u\)

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