#### Chapters

Chapter 2: Algebra

Chapter 3: Analytical Geometry

Chapter 4: Trigonometry

Chapter 5: Differential Calculus

Chapter 6: Applications of Differentiation

Chapter 7: Financial Mathematics

Chapter 8: Descriptive Statistics and Probability

Chapter 9: Correlation and Regression Analysis

Chapter 10: Operations Research

## Chapter 6: Applications of Differentiation

### 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]

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

- average cost
- average variable cost
- average fixed cost
- marginal cost and
- marginal average cost.

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

- cost when output is 4 units
- average cost when output is 10 units
- marginal cost when output is 3 units

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

- average cost
- marginal cost
- average revenue and
- marginal revenue.

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

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}

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}

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

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?

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

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

For the demand function p = 550 – 3x – 6x^{2} where x is quantity demand and p is unit price. Show that MR =

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

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

- profit function
- average profit at an output of 10 units
- marginal profit at an output of 10 units and
- marginal average profit at an output of 10 units.

Find the values of x, when the marginal function of y = x^{3} + 10x^{2} – 48x + 8 is twice the x.

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.

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.

Find the equilibrium price and equilibrium quantity for the following functions.

Demand: x = 100 – 2p and supply: x = 3p – 50.

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.

The cost function of a firm is C = x^{3} – 12x^{2} + 48x. Find the level of output (x > 0) at which average cost is minimum.

### 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]

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.

A television manufacturer finds that the total cost for the production and marketing of x number of television sets is C(x) = 300x^{2} + 4200x + 13500. If each product is sold for ₹ 8,400. show that the profit of the company is increasing.

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.

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.

Find the local minimum and local maximum of y = 2x^{3} – 3x^{2} – 36x + 10.

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.

### 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]

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:

- EOQ in units
- Minimum average cost
- EOQ in rupees
- EOQ in years of supply
- Number of orders per year

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

- EOQ
- Total optimum cost.

### 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]

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

If u = e^{xy}, then show that `(del^2"u")/(delx^2) + (del^2"u")/(del"y"^2)` = u(x^{2} + y^{2}).

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

Verify Euler’s theorem for the function u = x^{3} + y^{3} + 3xy^{2}.

Let u = x^{2}y^{3} cos`(x/y)`. By using Euler’s theorem show that `x*(del"u")/(delx) + y * (del"u")/(dely)`

### 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]

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

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

If the production function is z = 3x^{2} – 4xy + 3y^{2} where x is the labour and y is the capital, find the marginal productivities of x and y when x = 1, y = 2.

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]

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 p_{1} = p_{2} = 2.

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 p_{1} = 2, p_{2} = 1.

### 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]

#### Choose the Correct answer:

Average fixed cost of the cost function C(x) = 2x^{3} + 5x^{2} – 14x + 21 is:

`2/3`

`5/x`

`- 14/x`

`21/x`

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

20 – 6x

20 – 3x

20 + 6x

20 + 3x

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

x

^{2}+ 7x

^{2}- 7-x

^{2}+ 7-x

^{2}- 7

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

|η

_{d}| > 1|η

_{d}| = 1|η

_{d}| < 1|η

_{d}| = 0

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

0

1

`-1/"p"`

∞

Relationship among MR, AR and ηd is:

`eta_"d" = "AR"/("AR" - "MR")`

η

_{d}= AR – MRMR = AR = η

_{d}AR = `"MR"/eta_"d"`

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

`1/25`

`1/5 e^(5x)`

`1/125 e^(5x)`

25e

^{5x}

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

4

5

13

9

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

Profit P(x) is maximum when

MR = MC

MR = 0

MC = AC

TR = AC

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

1

`sqrt3/2`

`1/sqrt2`

`- 1/sqrt2`

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

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

8x + 4y + 4

4

2y + 32

0

If u = x^{3} + 3xy^{2} + y^{3} then `(del^2"u")/(del "y" del x)`is:

3

6y

6x

2

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

2x`e^(x^2)`

`e^(x^2)`

2`e^(x^2)`

0

Average cost is minimum when:

Marginal cost = marginal revenue

Average cost = marginal cost

Average cost = Marginal revenue

Average Revenue = Marginal cost

A company begins to earn profit at:

Maximum point

Breakeven point

Stationary point

Even point

The demand function is always

Increasing function

Decreasing function

Non-decreasing function

Undefined function

If q = 1000 + 8p_{1} – p_{2} then, `(del"q")/(del "p"_1)`is:

-1

8

1000

1000 - p

_{2}

If R = 5000 units/year, C_{1} = 20 paise, C_{3} = ₹ 20 then EOQ is:

5000

100

1000

200

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

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

- average cost function
- marginal cost function
- marginal average cost function.

**Find out the indicated elasticity for the following function:**

p = xe^{x}, x > 0; η_{s}

**Find out the indicated elasticity for the following function:**

p = `10 e^(- x/3)`, x > 0; η_{s}

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

For the demand function p x = 100 - 6x^{2}, find the marginal revenue and also show that MR = p`[1 - 1/eta_"d"]`

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.

For the cost function C = 2000 + 1800x - 75x^{2} + x^{3} find when the total cost (C) is increasing and when it is decreasing.

A certain manufacturing concern has total cost function C = 15 + 9x - 6x^{2} + x^{3}. Find x, when the total cost is minimum.

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.

Verify `(del^2 "u")/(del x del "y") = (del^2 "u")/(del "y" del x)` for u = x^{3} + 3x^{2} y^{2} + y^{3}.

If f(x, y) = 3x^{2} + 4y^{3} + 6xy - x^{2}y^{3} + 7, then show that f_{yy} (1,1) = 18.

## Chapter 6: Applications of Differentiation

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

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Concepts covered in Class 11th Business Mathematics and Statistics Answers Guide chapter 6 Applications of Differentiation are Applications of Differentiation in Business and Economics, Maxima and Minima, Applications of Maxima and Minima, Partial Derivatives, Applications of Partial Derivatives.

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