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

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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) = 3x² – 2xy + y² + 5x + 2, (2, -5)
(ii) g(x, y) = 3x² + y² + 5x + 2, (1, -2)

(iv) G (x, y) = e*^{x + 3y} log (x² + y²), (-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 + 3x₂y
(ii) g(x, y) = log(5x + 3y)
(iii) g(x, y) = x₂ + 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.x² – 0.05y² and C(x, y) = 8x + 6y + 2000 respectively.
(i) Find the profit function P(x,y),

Solution:
(i) P (x, y) = R (x, y) – C (x, y)
= (80x + 90y + 0.04xy – 0.05x² – 0.05y²) – (8x + 6y + 2000)
P (x, y) = 72x + 84y + 0.04xy – 0.05x² – 0.05y² – 2000