Advertisements
Advertisements
प्रश्न
Let z(x, y) = x2y + 3xy4, x, y ∈ R, Find the linear approximation for z at (2, –1)
उत्तर
z(x, y) = x2y + 3xy4 at (2, –1)
Here `(x_0, y_0)` = (2, –1)
`(delz)/(delx) = 2xy + 3y^4`
`(delz)/(dely) = x^2 + 12xy^3`
At (2, –1)
z = `(2)^2(- 1) + 3(2) (- 1)^4`
= `- 4 + 6`
= 2
`(delz)/(delx) = 2(2)(- 1) + 3(- 1)^4`
= `- 4 + 3`
= –1
`(delz)/(dely) = (2)^2 12(2)(- 1)^3`
= `4 - 24`
= – 20
Linear approximation is given by
L(x, y) = `z(x_0 + y_0) + ((delz)/(delx))_(((x_0, y_0))) (x - x_0) + ((delz)/(dely))_(((x_0, y_0))) (y - y_0)`
= `2 + (- 1)(x - 2) - 20(y + 1)`
= `2 - x + 2 - 20y - 20`
= ` x - 20y 16`
= `- (x + 20y + 16)`
APPEARS IN
संबंधित प्रश्न
If z = (ax + b) (cy + d), then find `(∂z)/(∂x)` and `(∂z)/(∂y)`.
Verify Euler’s theorem for the function u = x3 + y3 + 3xy2.
Let u = x2y3 cos`(x/y)`. By using Euler’s theorem show that `x*(del"u")/(delx) + y * (del"u")/(dely)`
If u = x3 + 3xy2 + y3 then `(del^2"u")/(del "y" del x)`is:
Find the partial dervatives of the following functions at indicated points.
f(x, y) = 3x2 – 2xy + y2 + 5x + 2, (2, – 5)
Find the partial dervatives of the following functions at indicated points.
g(x, y) = 3x2 + y2 + 5x + 2, (2, – 5)
For the following functions find the fx, and fy and show that fxy = fyx
f(x, y) = `(3x)/(y + sinx)`
For the following functions find the fx, and fy and show that fxy = fyx
f(x, y) = `cos(x^2 - 3xy)`
For the following functions find the gxy, gxx, gyy and gyx
g(x, y) = xey + 3x2y
For the following functions find the gxy, gxx, gyy and gyx
g(x, y) = log(5x + 3y)
For the following functions find the gxy, gxx, gyy and gyx
g(x, y) = x2 + 3xy – 7y + cos(5x)
If w(x, y) = xy + sin(xy), then Prove that `(del^2w)/(delydelx) = (del^2w)/(delxdely)`
If v(x, y, z) = x3 + y3 + z3 + 3xyz, Show that `(del^2"v")/(delydelz) = (del^2"v")/(delzdely)`
A from produces two types of calculates each week, x number of type A and y number of type B. The weekly revenue and cost functions = (in rupees) are R(x, y) = 80x + 90y + 0.04xy – 0.05x2 – 0.05y2 and C (x, y) = 8x + 6y + 2000 respectively. Find the profit function P(x, y)
Let V (x, y, z) = xy + yz + zx, x, y, z ∈ R. Find the differential dV
Choose the correct alternative:
If g(x, y) = 3x2 – 5y + 2y2, x(t) = et and y(t) = cos t then `"dg"/"dt"` is equal to
