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Differentials and Partial Derivatives Five Marks Questions

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
    10 x 5 = 50
  1. Let f , g : (a,b)→R be differentiable functions. Show that d(fg) = fdg + gdf

  2. Let g(x) = x2 + sin x. Calculate the differential dg.

  3. If the radius of a sphere, with radius 10 cm, has to decrease by 0 1. cm, approximately how much will its volume decrease?

  4. Let f (x, y) = 0 if xy ≠ 0 and f (x, y) =1 if xy = 0.
    (i) Calculate: \(\frac { \partial f }{ \partial x } (0,0),\frac { \partial f }{ \partial y } (0,0).\)
    (ii) Show that f is not continuous at (0,0)

  5. Let F(x, y) = x3 y + y2x + 7 for all (x, y)∈ R2. Calculate \(\frac { \partial F }{ \partial x } \)(-1,3) and \(\frac { \partial F }{ \partial y } \)(-2,1).

  6. Let w(x, y) = xy+\(\frac { { e }^{ y } }{ { y }^{ 2 }+1 } \) for all (x, y) ∈ R2. Calculate \(\frac { { \partial }^{ 2 }w }{ { \partial y\partial x } } \) and \(\frac { { \partial }^{ 2 }w }{ { \partial x\partial y } } \)

  7. Let (x, y) = e-2y cos(2x) for all (x, y) ∈ R2. Prove that u is a harmonic function in R2.

  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. Let g( x,y)= x3 - yx + sin(x+y), x(t) = e3t, y(t) = t2, t ∈ R. Find \(\frac { dg }{ dt } \) 

  10. Let g(x, y) = 2y + x2, x = 2r -s, y = r2+ 2s, r, s ∊ R. Find \(\frac { \partial g }{ \partial r } ,\frac { \partial g }{ \partial s } \)

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