Tamil Nadu Board of Secondary EducationHSC Commerce Class 11th

# Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide chapter 6 - Applications of Differentiation [Latest edition]

#### Chapters ## Chapter 6: Applications of Differentiation

Exercise 6.1Exercise 6.2Exercise 6.3Exercise 6.4Exercise 6.5Exercise 6.6Miscellaneous Problems
Exercise 6.1 [Pages 138 - 139]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 6 Applications of Differentiation Exercise 6.1 [Pages 138 - 139]

Exercise 6.1 | Q 1 | Page 138

A firm produces x tonnes of output at a total cost of C(x) = 1/10x^3 - 4x^2 - 20x + 7 find the

1. average cost
2. average variable cost
3. average fixed cost
4. marginal cost and
5. marginal average cost.
Exercise 6.1 | Q 2 | Page 138

The total cost of x units of output of a firm is given by C = 2/3x + 35/2. Find the

1. cost when output is 4 units
2. average cost when output is 10 units
3. marginal cost when output is 3 units
Exercise 6.1 | Q 3 | Page 138

Revenue function ‘R’ and cost function ‘C’ are R = 14x – x2 and C = x(x2 – 2). Find the

1. average cost
2. marginal cost
3. average revenue and
4. marginal revenue.
Exercise 6.1 | Q 4 | Page 138

If the demand law is given by p = 10e^(- x/2) then find the elasticity of demand.

Exercise 6.1 | Q 5. (i) | Page 139

Find the elasticity of demand in terms of x for the following demand laws and also find the value of x where elasticity is equal to unity.

p = (a – bx)2

Exercise 6.1 | Q 5. (ii) | Page 139

Find the elasticity of demand in terms of x for the following demand laws and also find the value of x where elasticity is equal to unity.

p = a – bx2

Exercise 6.1 | Q 6 | Page 139

Find the elasticity of supply for the supply function x = 2p2 + 5 when p = 3.

Exercise 6.1 | Q 7 | Page 139

The demand curve of a commodity is given by p = (50 - x)/5, find the marginal revenue for any output x and also find marginal revenue at x = 0 and x = 25?

Exercise 6.1 | Q 8 | Page 139

The supply function of certain goods is given by x = asqrt("p" - "b") where p is unit price, a and b are constants with p > b. Find elasticity of supply at p = 2b.

Exercise 6.1 | Q 9 | Page 139

Show that MR = p[1 - 1/eta_"d"] for the demand function p = 400 – 2x – 3x2 where p is unit price and x is quantity demand.

Exercise 6.1 | Q 10 | Page 139

For the demand function p = 550 – 3x – 6x2 where x is quantity demand and p is unit price. Show that MR =

Exercise 6.1 | Q 11 | Page 139

For the demand function x = 25/"p"^4, 1 ≤ p ≤ 5, determine the elasticity of demand.

Exercise 6.1 | Q 12 | Page 139

The demand function of a commodity is p = 200 - x/100 and its cost is C = 40x + 120 where p is a unit price in rupees and x is the number of units produced and sold. Determine

1. profit function
2. average profit at an output of 10 units
3. marginal profit at an output of 10 units and
4. marginal average profit at an output of 10 units.
Exercise 6.1 | Q 13 | Page 139

Find the values of x, when the marginal function of y = x3 + 10x2 – 48x + 8 is twice the x.

Exercise 6.1 | Q 14 | Page 139

The total cost function y for x units is given by y = 3x((x+7)/(x+5)) + 5. Show that the marginal cost decreases continuously as the output increases.

Exercise 6.1 | Q 15 | Page 139

Find the price elasticity of demand for the demand function x = 10 – p where x is the demand p is the price. Examine whether the demand is elastic, inelastic, or unit elastic at p = 6.

Exercise 6.1 | Q 16 | Page 139

Find the equilibrium price and equilibrium quantity for the following functions.
Demand: x = 100 – 2p and supply: x = 3p – 50.

Exercise 6.1 | Q 17 | Page 139

The demand and cost functions of a firm are x = 6000 – 30p and C = 72000 + 60x respectively. Find the level of output and price at which the profit is maximum.

Exercise 6.1 | Q 18 | Page 139

The cost function of a firm is C = x3 – 12x2 + 48x. Find the level of output (x > 0) at which average cost is minimum.

Exercise 6.2 [Page 145]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 6 Applications of Differentiation Exercise 6.2 [Page 145]

Exercise 6.2 | Q 1 | Page 145

The average cost function associated with producing and marketing x units of an item is given by AC = 2x – 11 + 50/x. Find the range of values of the output x, for which AC is increasing.

Exercise 6.2 | Q 2 | Page 145

A television manufacturer finds that the total cost for the production and marketing of x number of television sets is C(x) = 300x2 + 4200x + 13500. If each product is sold for ₹ 8,400. show that the profit of the company is increasing.

Exercise 6.2 | Q 3 | Page 145

A monopolist has a demand curve x = 106 – 2p and average cost curve AC = 5 + x/50, where p is the price per unit output and x is the number of units of output. If the total revenue is R = px, determine the most profitable output and the maximum profit.

Exercise 6.2 | Q 4 | Page 145

A tour operator charges ₹ 136 per passenger with a discount of 40 paise for each passenger in excess of 100. The operator requires at least 100 passengers to operate the tour. Determine the number of passengers that will maximize the amount of money the tour operator receives.

Exercise 6.2 | Q 5 | Page 145

Find the local minimum and local maximum of y = 2x3 – 3x2 – 36x + 10.

Exercise 6.2 | Q 6 | Page 145

The total revenue function for a commodity is R = 15x + x^2/3 - 1/36 x^4. Show that at the highest point average revenue is equal to the marginal revenue.

Exercise 6.3 [Page 149]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 6 Applications of Differentiation Exercise 6.3 [Page 149]

Exercise 6.3 | Q 1 | Page 149

The following table gives the annual demand and unit price of 3 items.

 Items Annual Demand (units) Unit Price A 800 0.02 B 400 1.00 C 13,800 0.20

Ordering cost is ₹ 5 per order and holding cost is 10% of unit price. Determine the following:

1. EOQ in units
2. Minimum average cost
3. EOQ in rupees
4. EOQ in years of supply
5. Number of orders per year
Exercise 6.3 | Q 2 | Page 149

A dealer has to supply his customer with 400 units of a product per week. The dealer gets the product from the manufacturer at a cost of ₹ 50 per unit. The cost of ordering from the manufacturers in ₹ 75 per order. The cost of holding inventory is 7.5 % per year of the product cost. Find

1. EOQ
2. Total optimum cost.
Exercise 6.4 [Page 152]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 6 Applications of Differentiation Exercise 6.4 [Page 152]

Exercise 6.4 | Q 1 | Page 152

If z = (ax + b) (cy + d), then find (∂z)/(∂x) and (∂z)/(∂y).

Exercise 6.4 | Q 2 | Page 152

If u = exy, then show that (del^2"u")/(delx^2) + (del^2"u")/(del"y"^2) = u(x2 + y2).

Exercise 6.4 | Q 3 | Page 152

Let u = x cos y + y cos x. Verify (del^2"u")/(delxdely) = (del^"u")/(del"y"del"x")

Exercise 6.4 | Q 4 | Page 152

Verify Euler’s theorem for the function u = x3 + y3 + 3xy2.

Exercise 6.4 | Q 5 | Page 152

Let u = x2y3 cos(x/y). By using Euler’s theorem show that x*(del"u")/(delx) + y * (del"u")/(dely)

Exercise 6.5 [Page 154]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 6 Applications of Differentiation Exercise 6.5 [Page 154]

Exercise 6.5 | Q 1 | Page 154

Find the marginal productivities of capital (K) and labour (L) if P = 8L – 2K + 3K2 – 2L2 + 7KL when K = 3 and L = 1.

Exercise 6.5 | Q 2 | Page 154

If the production of a firm is given by P = 4LK – L2 + K2, L > 0, K > 0, Prove that L (del"P")/(del"L") + "K"(del"P")/(del"K") = 2P.

Exercise 6.5 | Q 3 | Page 154

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.

Exercise 6.5 | Q 4 | Page 154

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]

Exercise 6.5 | Q 5 | Page 154

The demand for a quantity A is q = 13 - 2"p"_1 - 3"p"_2^2. Find the partial elasticities "E"_"q"/("E"_("p"_1)) and "E"_"q"/("E"_("p"_2))  when p1 = p2 = 2.

Exercise 6.5 | Q 6 | Page 154

The demand for a quantity A is q = 80 - "p"_1^2 + 5"p"_2 - "p"_1"p"_2. Find the partial elasticities "E"_"q"/("E"_("p"_1)) and "E"_"q"/("E"_("p"_2))  when p1 = 2, p2 = 1.

Exercise 6.6 [Pages 154 - 156]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 6 Applications of Differentiation Exercise 6.6 [Pages 154 - 156]

Exercise 6.6 | Q 1 | Page 154

Average fixed cost of the cost function C(x) = 2x3 + 5x2 – 14x + 21 is:

• 2/3

• 5/x

• - 14/x

• 21/x

Exercise 6.6 | Q 2 | Page 154

Marginal revenue of the demand function p = 20 – 3x is:

• 20 – 6x

• 20 – 3x

• 20 + 6x

• 20 + 3x

Exercise 6.6 | Q 3 | Page 155

If demand and the cost function of a firm are p = 2 – x and C = -2x2 + 2x + 7 then its profit function is:

• x2 + 7

• x2 - 7

• -x2 + 7

• -x2 - 7

Exercise 6.6 | Q 4 | Page 155

If the demand function is said to be inelastic, then:

• d| > 1

• d| = 1

• d| < 1

• d| = 0

Exercise 6.6 | Q 5 | Page 155

The elasticity of demand for the demand function x = 1/"p" is:

• 0

• 1

• -1/"p"

Exercise 6.6 | Q 6 | Page 155

Relationship among MR, AR and ηd is:

• eta_"d" = "AR"/("AR" - "MR")

• ηd = AR – MR

• MR = AR = ηd

• AR = "MR"/eta_"d"

Exercise 6.6 | Q 7 | Page 155

For the cost function C = 1/25 e^(5x), the marginal cost is:

• 1/25

• 1/5 e^(5x)

• 1/125 e^(5x)

• 25e5x

Exercise 6.6 | Q 8 | Page 155

Instantaneous rate of change of y = 2x2 + 5x with respect to x at x = 2 is:

• 4

• 5

• 13

• 9

Exercise 6.6 | Q 9 | Page 155

If the average revenue of a certain firm is ₹ 50 and its elasticity of demand is 2, then their marginal revenue is:

• ₹ 50

• ₹ 25

• ₹ 100

• ₹ 75

Exercise 6.6 | Q 10 | Page 155

Profit P(x) is maximum when

• MR = MC

• MR = 0

• MC = AC

• TR = AC

Exercise 6.6 | Q 11 | Page 155

The maximum value of f(x) = sin x is:

• 1

• sqrt3/2

• 1/sqrt2

• - 1/sqrt2

Exercise 6.6 | Q 12 | Page 155

If f(x, y) is a homogeneous function of degree n, then x (del "f")/(del x) + "y" (del "f")/(del y) is equal to:

• (n – 1)f

• n(n – 1)f

• nf

• f

Exercise 6.6 | Q 13 | Page 155

If u = 4x2 + 4xy + y2 + 4x + 32y + 16, then (del^2"u")/(del"y" del"x") is equal to:

• 8x + 4y + 4

• 4

• 2y + 32

• 0

Exercise 6.6 | Q 14 | Page 155

If u = x3 + 3xy2 + y3 then (del^2"u")/(del "y" del x)is:

• 3

• 6y

• 6x

• 2

Exercise 6.6 | Q 15 | Page 155

If u = e^(x^2) then (del"u")/(delx) is equal to:

• 2xe^(x^2)

• e^(x^2)

• 2e^(x^2)

• 0

Exercise 6.6 | Q 16 | Page 155

Average cost is minimum when:

• Marginal cost = marginal revenue

• Average cost = marginal cost

• Average cost = Marginal revenue

• Average Revenue = Marginal cost

Exercise 6.6 | Q 17 | Page 155

A company begins to earn profit at:

• Maximum point

• Breakeven point

• Stationary point

• Even point

Exercise 6.6 | Q 18 | Page 155

The demand function is always

• Increasing function

• Decreasing function

• Non-decreasing function

• Undefined function

Exercise 6.6 | Q 19 | Page 155

If q = 1000 + 8p1 – p2 then, (del"q")/(del "p"_1)is:

• -1

• 8

• 1000

• 1000 - p2

Exercise 6.6 | Q 20 | Page 156

If R = 5000 units/year, C1 = 20 paise, C3 = ₹ 20 then EOQ is:

• 5000

• 100

• 1000

• 200

Miscellaneous Problems [Page 156]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 6 Applications of Differentiation Miscellaneous Problems [Page 156]

Miscellaneous Problems | Q 1 | Page 156

The total cost function for the production of x units of an item is given by C = 10 - 4x3 + 3x4 find the

1. average cost function
2. marginal cost function
3. marginal average cost function.
Miscellaneous Problems | Q 2. (i) | Page 156

Find out the indicated elasticity for the following function:

p = xex, x > 0; ηs

Miscellaneous Problems | Q 2. (ii) | Page 156

Find out the indicated elasticity for the following function:

p = 10 e^(- x/3), x > 0; ηs

Miscellaneous Problems | Q 3 | Page 156

Find the elasticity of supply when the supply function is given by x = 2p2 + 5 at p = 1.

Miscellaneous Problems | Q 4 | Page 156

For the demand function p x = 100 - 6x2, find the marginal revenue and also show that MR = p[1 - 1/eta_"d"]

Miscellaneous Problems | Q 5 | Page 156

The total cost function y for x units is given by y = 4x((x+2)/(x+1)) + 6. Prove that marginal cost [MC] decreases as x increases.

Miscellaneous Problems | Q 6 | Page 156

For the cost function C = 2000 + 1800x - 75x2 + x3 find when the total cost (C) is increasing and when it is decreasing.

Miscellaneous Problems | Q 7 | Page 156

A certain manufacturing concern has total cost function C = 15 + 9x - 6x2 + x3. Find x, when the total cost is minimum.

Miscellaneous Problems | Q 8 | Page 156

Let u = log (x^4 - y^4)/(x - y). Using Euler’s theorem show that x (del"u")/(del"x") + y(del"u")/(del"y") = 3.

Miscellaneous Problems | Q 9 | Page 156

Verify (del^2 "u")/(del x del "y") = (del^2 "u")/(del "y" del x) for u = x3 + 3x2 y2 + y3.

Miscellaneous Problems | Q 10 | Page 156

If f(x, y) = 3x2 + 4y3 + 6xy - x2y3 + 7, then show that fyy (1,1) = 18.

## Chapter 6: Applications of Differentiation

Exercise 6.1Exercise 6.2Exercise 6.3Exercise 6.4Exercise 6.5Exercise 6.6Miscellaneous Problems ## Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide chapter 6 - Applications of Differentiation

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