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## Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.4

Question 1.

Find the partial derivatives of the following functions at the indicated points.

(i) f(x, y) = 3x^{2} – 2xy + y^{2} + 5x + 2, (2, -5)

(ii) g(x, y) = 3x^{2} + y^{2} + 5x + 2, (1, -2)

(iv) G (x, y) = e*^{x + 3y} log (x^{2} + y^{2}), (-1, 1)

Solution:

Question 2.

For each of the following functions find the f_{x}, f_{y} and show that f_{xy} = f_{yx}

Solution:

Question 3.

Solution:

Question 4.

Solution:

Question 5.

For each of the following functions find the g_{xy}, g_{xx}, g_{yy} and g_{yx}.

(i) g(x, y) = xe_{y} + 3x_{2}y

(ii) g(x, y) = log(5x + 3y)

(iii) g(x, y) = x_{2} + 3xy – 7y + cos(5x)

Solution:

Question 6.

Solution:

Question 7.

Solution:

Question 8.

Solution:

Question 9.

Solution:

Question 10.

A firm produces two types of calculators each week, x number of type A and j number of type B . The weekly revenue and cost functions (in rupees) are

R(x, y) = 80x + 90y + 0.04xy – 0.05.x^{2} – 0.05y^{2} and C(x, y) = 8x + 6y + 2000 respectively.

(i) Find the profit function P(x,y),

Solution:

(i) Profit = Revenue – Cost

= (80x + 90y + 0.04 xy – 0.05 x^{2} – 0.05y^{2}) – (8x + 6y + 2000)

= 80x + 90y + 0.04 xy – 0.05 x^{2} – 0.05y^{2} – 8x – 6y – 2000

P(x, y) = 72x + 84y + 0.04 xy – 0.05 x^{2} – 0.05y^{2} – 2000

### Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.4 Additional Problems

Question 1.

Solution:

Question 2.

If U = (x – y) (y – z) (z – x) then show that U_{x} + U_{y} + U_{z} = 0

Solution:

U_{x} = (y – z) {(x – y)(-1) + (z – x). 1}

= (y – z) [(z – x) – (x – y)]

Similarly U_{y} = (z – x) [(x – y) – (y – z)]

_{z} = (x – y)[(y – z) – (z – x]

U_{x} + U_{y} + U_{z} = (y – z) [(z – x) – (z – x)] + (x – y) [- (y – z) + (y – z)] + (z – x) [(x – y) – (x – y)]

∴ U_{x} + U_{y} + U_{z} = 0.

Hence proved.

Question 3.

Solution:

Question 4.

Solution: