Advertisements
Advertisements
प्रश्न
Let z(x, y) = x tan–1(xy), x = t², y = s et, s, t ∈ R. Find `(delz)/(del"s")` and `(delz)/(del"t")` at s = t = 1
Advertisements
उत्तर
`(delz)/(del"z") = x 1/(1 + (xy)^2) xx y + tan^-1 (xy)`
`(xy)/(1 + x^2y^2) + tan^-1 (xy)`
`(delz)/(dely) = x 1/(1 + (xy)^2) xx x`
= `x^2/(1 + x^2y^2)`
`(delx)/(dely) = 0, (dely)/(del"s") = "e"^"t", (delx)/(del"t") = 2"t", (dely)/(del"t") = "s" "e"^"t"`
`(delz)/(del"s") = (delz)/(del"s") + (delz)/(dely) (dely)/(del"s")`
`(delz)/(del"s") = ([(xy)/( + x^2y^2)] + tan^-1 (xy)) xx 0 + x^2/(1 + x^2y^2) xx "e"^"t"`
= `"t"^4/(1 + "t"^2"s"^2"e"^(2"t")) "e"^"t" = ("e"^"t" "t"^4)/(1 + "t"^2"s"^2"e"^(2"t"))`
`(delz)/(del"t") = (delz)/(del"x") (delx)/(delt) + (delz)/(del"y") (dely)/(del"t")`
= `[(xy)/(1 + (xy)^2) + tan^-1(xy)] xx 2"t" + [x^2/(1 + (xy)^2)] xx "se"^"t"`
= `[("t"^2"se"^"t")/(1 + "t"^2"s"^2"e"^(2"t")) + tan^-1("t"^2"se"^"t")] xx 2"t" + ["t"^4/(1 + "t"^2"s"^2"e"^(2"t"))]"se"^"t"`
As s = t = 1,
`(delz)/(del"s") = "e"/(1 + "e"^2)`
`(delz)/(del"t") = (3"e")/(1 + "e"^2) + 2tan^-1("e")`
APPEARS IN
संबंधित प्रश्न
If u(x, y) = x2y + 3xy4, x = et and y = sin t, find `"du"/"dt"` and evaluate if at t = 0
Let u(x, y, z) = xy2z3 x = sin t, y = cos t, z = 1 + e2t, Find `"du"/"dt"`
Let U(x, y, z) = xyz, x = e–t, y = e–t cos t, z – sin t, t ∈ R, find `"dU"/"dt"`
Let w(x, y) = 6x3 – 3xy + 2y2, x = es, y = cos s, s ∈ R. Find `("d"w)/"ds"` and evaluate at s = 0
Let U(x, y) = ex sin y where x = st2, y = s2t, s, t ∈ R. Find `(del"U")/(del"s"), (del"u")/(del"t")` and evaluate them at s = t = 1
W(x, y, z) = xy + yz + zx, x = u – v, y = uv, z = u + v, u, v ∈ R. Find `(del"W")/(del"u"), (del"W")/(del"v")` and evaluate them at `(1/2, 1)`
In the following, determine whether the following function is homogeneous or not. If it is so, find the degree.
h(x, y) = `(6x^3y^2 - piy^5 + 9x^4y)/(2020x^2 + 2019y^2)`
In the following, determine whether the following function is homogeneous or not. If it is so, find the degree.
g(x, y, z) = `sqrt(3x^2+ 5y^2+z^2)/(4x + 7y)`
In the following, determine whether the following function is homogeneous or not. If it is so, find the degree.
U(x, y, z) = `xy + sin((y^2 - 2z^2)/(xy))`
If v(x, y) = `log((x^2 + y^2)/(x + y))`, prove that `x (del"v")/(delx) + y (del"u")/(dely) = 1`
Choose the correct alternative:
If w(x, y) = xy, x > 0, then `(del"w")/(delx)` is equal to
Choose the correct alternative:
If f(x, y) = exy, then `(del^2"f")/(delxdely)` is equal to
Choose the correct alternative:
f u(x, y) = x2 + 3xy + y – 2019, then `(delu)/(delx) "|"_(((4 , - 5)))` is equal to
Choose the correct alternative:
If w(x, y, z) = x2(y – z) + y2(z – x)+ z2(x – y) then `(del"w")/(delz) + (del"w")/(dely) + (del"w")/(delz)` is
Choose the correct alternative:
If f(x, y, z) = xy + yz + zx, then fx – fz is equal to
