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## Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.5

Question 1.

If w(x, y) = x^{3} – 3xy + 2y^{2}, x, y ∈ R, find the linear approximation for w at (1, -1) .

Solution:

w(x, y) = x^{3} – 3xy + 2y^{2} ; at (1, -1)

Linear approximation is given by

(1) ⇒ Let(x, y) = 6 + 6(x – 1) – 7(y + 1)

= 6x – 6 – 7y – 7

= = 6x – 7y – 7

Question 2.

Let z(x, y) = x^{2}y + 3xy^{4}, x, y ∈ R, Find the linear approximation for z at (2, -1).

Solution:

Linear approximation is given by

Question 3.

If v(x, y) = x^{2} – xy + \(\frac{1}{4}\)y^{2} + 7, x, y ∈ R, find the differential dv.

Solution:

The differential is dv = (2x -y) dx + (- x + \(\frac{y}{2}\))dy

Question 4.

Let W(x, y, z) = x^{2} – xy + 3sinz,x,y, z ∈ R. Find the linear approximation at (2, -1, 0).

Solution:

w (x, y, z) = x^{2} – xy + 3 sin z

Here(x_{0}, y_{0}, y_{0}) = (2, -1, 0)

Linear approximation is given by

Question 5.

Let V (x, y, z) = xy + yz + zx, x, y, z ∈ R. Find the differential dV.

Solution:

V(x, y, z) = xy + yz + zx

V_{x} = y + z

V_{y} = x + z

V_{z} = y + x

The differential is dV = (y + z) dx + (x + z) dy + (y + x) dz