Balbharati Maharashtra State Board 11th Commerce Maths Solution Book Pdf Chapter 9 Differentiation Ex 9.2 Questions and Answers.

## Maharashtra State Board 11th Commerce Maths Solutions Chapter 9 Differentiation Ex 9.2

I. Differentiate the following functions w.r.t. x.

Question 1.

\(\frac{x}{x+1}\)

Solution:

Question 2.

\(\frac{x^{2}+1}{x}\)

Solution:

Question 3.

\(\frac{1}{e^{x}+1}\)

Solution:

Question 4.

\(\frac{e^{x}}{e^{x}+1}\)

Solution:

Question 5.

\(\frac{x}{\log x}\)

Solution:

Question 6.

\(\frac{2^{x}}{\log x}\)

Solution:

Question 7.

\(\frac{\left(2 e^{x}-1\right)}{\left(2 e^{x}+1\right)}\)

Solution:

Question 8.

\(\frac{(x+1)(x-1)}{\left(e^{x}+1\right)}\)

Solution:

II. Solve the following examples:

Question 1.

The demand D for a price P is given as D = \(\frac{27}{P}\), find the rate of change of demand when the price is 3.

Solution:

Demand, D = \(\frac{27}{P}\)

Rate of change of demand = \(\frac{dD}{dP}\)

When price P = 3,

Rate of change of demand,

\(\left(\frac{\mathrm{dD}}{\mathrm{dP}}\right)_{\mathrm{P}=3}=\frac{-27}{(3)^{2}}=-3\)

∴ When price is 3, Rate of change of demand is -3.

Question 2.

If for a commodity; the price-demand relation is given as D = \(\frac{P+5}{P-1}\). Find the marginal demand when the price is 2.

Solution:

Question 3.

The demand function of a commodity is given as P = 20 + D – D^{2}. Find the rate at which price is changing when demand is 3.

Solution:

Given, P = 20 + D – D^{2}

Rate of change of price = \(\frac{dP}{dD}\)

= \(\frac{d}{dD}\)(20 + D – D^{2})

= 0 + 1 – 2D

= 1 – 2D

Rate of change of price at D = 3 is

\(\left(\frac{\mathrm{dP}}{\mathrm{dD}}\right)_{\mathrm{D}=3}\) = 1 – 2(3) = -5

∴ Price is changing at a rate of -5, when demand is 3.

Question 4.

If the total cost function is given by; C = 5x^{3} + 7x^{2} + 7; find the average cost and the marginal cost when x = 4.

Solution:

Total cost function, C = 5x^{3} + 7x^{2} + 7

Average cost = \(\frac{C}{x}\)

When x = 4, Marginal cost = \(\left(\frac{\mathrm{dC}}{\mathrm{d} x}\right)_{x=4}\)

= 15(4)^{2} + 4(4)

= 240 + 16

= 256

∴ the average cost and marginal cost at x = 4 are \(\frac{359}{4}\) and 256 respectively.

Question 5.

The total cost function of producing n notebooks is given by

C = 1500 – 75n + 2n^{2} + \(\frac{n^{3}}{5}\)

Find the marginal cost at n = 10.

Solution:

The total cost function,

∴ Marginal cost at n = 10 is 25.

Question 6.

The total cost of ‘t’ toy cars is given by C = 5(2t) + 17. Find the marginal cost and average cost at t = 3.

Solution:

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.

Question 7.

If for a commodity; the demand function is given by, D = \(\sqrt{75-3 P}\). Find the marginal demand function when P = 5.

Solution:

Demand function, D = \(\sqrt{75-3 P}\)

Now, Marginal demand = \(\frac{dD}{dP}\)

Question 8.

The total cost of producing x units is given by C = 10e^{2x}, find its marginal cost and average cost when x = 2.

Solution:

Question 9.

The demand function is given as P = 175 + 9D + 25D^{2}. Find the revenue, average revenue, and marginal revenue when demand is 10.

Solution:

Given, P = 175 + 9D + 25D^{2}

Total revenue, R = P.D

= (175 + 9D + 25D^{2})D

= 175D + 9D^{2} + 25D^{3}

Average revenue = P = 175 + 9D + 25D^{2}

Marginal revenue = \(\frac{dR}{dD}\)

= \(\frac{d}{dD}\) (175D + 9D^{2} + 25D^{3})

= 175 \(\frac{d}{dD}\) (D) + 9 \(\frac{d}{dD}\) (D^{2}) + 25 \(\frac{d}{dD}\) (D^{3})

= 175(1) + 9(2D) + 25(3D^{2})

= 175 + 18D + 75D^{2}

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.

Question 10.

The supply S for a commodity at price P is given by S = P^{2} + 9P – 2. Find the marginal supply when the price is 7.

Solution:

Given, S = P^{2} + 9P – 2

∴ The marginal supply is 23, at P = 7.

Question 11.

The cost of producing x articles is given by C = x^{2} + 15x + 81. Find the average cost and marginal cost functions. Find marginal cost when x = 10. Find x for which the marginal cost equals the average cost.

Solution:

Given, cost C = x^{2} + 15x + 81

If marginal cost = average cost, then

2x + 15 = x + 15 + \(\frac{81}{x}\)

∴ x = \(\frac{81}{x}\)

∴ x^{2} = 81

∴ x = 9 …..[∵ x > 0]