Chapter 9 Differentiation Ex 9.2 Solutions

Demand, D = Rate of change of demand = When price P = 3, Rate of change of demand, ∴ When price is 3, Rate of change of demand is -3.

Given, P = 20 + D – D2 Rate of change of price = = (20 + D – D2) = 0 + 1 – 2D = 1 – 2D Rate of change of price at D = 3 is = 1 – 2(3) = -5 ∴ Price is changing at a rate of -5, when demand is 3.

Total cost function, C = 5x3 + 7x2 + 7 Average cost = When x = 4, Marginal cost = = 15(4)2 + 4(4) = 240 + 16 = 256 ∴ the average cost and marginal cost at x = 4 are and 256 respectively.

The total cost function, ∴ Marginal cost at n = 10 is 25.

Total cost of ‘t’ toy cars, C = 5(2t) + 17 ∴ at t = 3, the Marginal cost is 40 log 2 and the Average cost is 19.

Demand function, D = Now, Marginal demand =

Given, P = 175 + 9D + 25D2 Total revenue, R = P.D = (175 + 9D + 25D2)D = 175D + 9D2 + 25D3 Average revenue = P = 175 + 9D + 25D2 Marginal revenue = = (175D + 9D2 + 25D3) = 175 (D) + 9 (D2) + 25 (D3) = 175(1) + 9(2D) + 25(3D2) = 175 + 18D + 75D2 When D = 10, Total revenue = 175(10) + 9(10)2 + 25(10)3 = 1750 + 900 + 25000 = 27650 Average revenue = 175 + 9(10) + 25(10)2 = 175 + 90 + 2500 = 2765 Marginal revenue = 175 + 18(10) + 75(10)2 = 175 + 180 + 7500 = 7855 ∴ When Demand = 10, Total revenue = 27650, Average revenue = 2765, Marginal revenue = 7855.

Given, S = P2 + 9P – 2 ∴ The marginal supply is 23, at P = 7.

Given, cost C = x2 + 15x + 81 If marginal cost = average cost, then 2x + 15 = x + 15 + ∴ x = ∴ x2 = 81 ∴ x = 9 …..[∵ x > 0]