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## Tamilnadu Samacheer Kalvi 11th Business Maths Solutions Chapter 6 Applications of Differentiation Ex 6.5

### Samacheer Kalvi 11th Business Maths Applications of Differentiation Ex 6.5 Text Book Back Questions and Answers

Question 1.
Find the marginal productivities of capital (K) and labour (L) if P = 8L – 2K + 3K2 – 2L2 + 7KL when K = 3 and L = 1.
Solution:
P = 8L – 2K + 3K2 – 2L2 + 7KL
Marginal productivity of labour, $$\frac{\partial \mathrm{P}}{\partial \mathrm{L}}$$ = 8 – 0 + 0 – 2(2L) + 7K(1)
= 8 – 4L + 7K
Marginal productivity of labour when K = 3 and L = 1 is
$$\left(\frac{\partial P}{\partial L}\right)_{(3,1)}$$ = 8 – 4 + 21
= 29 – 4
= 25
Marginal productivity of capital, $$\frac{\partial \mathrm{P}}{\partial \mathrm{K}}$$ = 0 – 2(1) + 3(2K) – 0 + 7L(1)
= -2 + 6K + 7L
Marginal productivity of capital when K = 3 and L = 1 is
$$\left(\frac{\partial \mathrm{P}}{\partial \mathrm{K}}\right)_{(3,1)}$$
= -2 + 18 + 7
= -2 + 25
= 23 Question 2.
If the production of a firm is given by P = 4LK – L2 + K2, L > 0, K > 0, Prove that L $$\frac{\partial \mathbf{P}}{\partial \mathbf{L}}$$ + K $$\frac{\partial \mathbf{P}}{\partial \mathbf{K}}$$ = 2P.
Solution:
P = 4LK – L2 + K2
P(K, L) = 4LK – L2 + K2
P(tK, tL) = 4(tL) (tK) – t2L2 + t2K2
= t2(4LK – L2 + K2)
= t2P
∴ P is a homogeneous function in L and K of degree 2.
∴ By Euler’s theorem, L $$\frac{\partial \mathbf{P}}{\partial \mathbf{L}}$$ + K $$\frac{\partial \mathbf{P}}{\partial \mathbf{K}}$$ = 2P.

Question 3.
If the production function is z = 3x2 – 4xy + 3y2 where x is the labour and y is the capital, find the marginal productivities of x and y when x = 1, y = 2.
Solution:
Marginal productivity of labour, $$\frac{\partial z}{\partial x}$$ = 6x – 4y
Marginal productivity of labour when x = 1, y = 2 is
$$\left(\frac{\partial z}{\partial x}\right)_{(1,2)}$$ = 6(1) – 4(1)
= 6 – 4
= 2
Marginal productivity of capital, $$\frac{\partial z}{\partial y}$$ = 0 – 4x(1) + 3(2y)
= -4x + 6y
Marginal productivity of qapital when x = 1, y = 2 is
$$\left(\frac{\partial z}{\partial y}\right)_{(1,2)}$$ = -4(1) + 6(2)
= -4 + 12
= 8 Question 4.
For the production function P = 3(L)0.4 (K)0.6, find the marginal productivities of labour (L) and capital (K) when L = 10 and K = 6. [use: (0.6)0.6 = 0.736, (1.67)0.4 = 1.2267]
Solution:
Given that P = 3(L)0.4 (K)0.6…….(1)
Differentiating partially with respect to L we get,  When L = 10, k = 6, $$\frac{\partial P}{\partial L}=1.2\left(\frac{6}{10}\right)^{0.6}$$
= 1.2(0.6)0.6
= 1.2(0.736)
i.e., the marginal productivity of labour = 0.8832
Again differentiating partially with respect to ‘k’ we get,
Marginal productivity of labour when L = 10, K = 6 is Marginal productivity of capital when k = 10, k = 6
= $$1.8\left(\frac{10}{6}\right)^{0.4}$$
= 1.8(1.66666)0.4
= 1.8(1.67)0.4
= 1.8 × 1.2267
= 2.2081

Question 5.
The demand for a quantity A is q = 13 – 2p1 – $$3 p_{2}^{2}$$. Find the partial elasticities $$\frac{\mathbf{E} q}{\mathbf{E} p_{1}}$$ and $$\frac{\mathbf{E q}}{\mathbf{E} p_{2}}$$ when p1 = p2 = 2.
Solution: When p1 = p2 = 2,  When p1 = p2 = 2,  Question 6.
The demand for a commodity A is q = 80 – $$p_{1}^{2}$$ + 5p2 – p1p2. Find the partial elasticities $$\frac{\mathbf{E q}}{\mathbf{E} p_{\mathbf{1}}}$$ and $$\frac{\mathbf{E q}}{\mathbf{E} p_{\mathbf{2}}}$$ when p1 = 2, p2 = 1.
Solution: When p1 = 2, p2 = 1, When p1 = 2, p2 = 1, 