#### Applications of Differentiation Model Question Paper

11th Standard

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Time : 02:00:00 Hrs
Total Marks : 60
6 x 1 = 6
1. Average fixed cost of the cost function C(x) = 2x3 +5x2 - 14x +21 is _______.

(a)

$\frac { 2 }{ 3 }$

(b)

$\frac { 5}{ x }$

(c)

$\frac { -14 }{ x }$

(d)

$\frac { 21 }{ x }$

2. If demand and the cost function of a firm are p= 2–x and c = 2x2 +2x +7 then its profit function is ________.

(a)

x2 + 7

(b)

x2 - 7

(c)

- x2 + 7

(d)

- x2 - 7

3. For the cost function C =$\frac { 1 }{ 25 } { e }^{ 25 }$, the marginal cost is _________.

(a)

$\frac { 1 }{ 25 }$

(b)

$\frac { 1 }{ 5 } { e }^{ 5x }$

(c)

$\frac { 1 }{ 125 } { e }^{ 5x }$

(d)

25e5x

4. If u = x3 + 3xy2 + y3 then  $\frac { \partial ^{ 2 }u }{ \partial y\partial x }$ is _______.

(a)

3

(b)

6y

(c)

6x

(d)

2

5. if q = 1000 + 8p1 - p2 then, $\frac { \partial q }{ \partial { p }_{ 1 } }$ is _______.

(a)

-1

(b)

8

(c)

1000

(d)

1000 - p2

6. If R = 5000 units / year, C1 = 20 paise , C3 = Rs 20 then EOQ is _______.

(a)

5000

(b)

100

(c)

1000

(d)

200

7. 8 x 2 = 16
8. Find the elasticity of supply for the supply function x = 2p2 - 5p + 1, p > 3.

9. The demand function for a commodity is $p={4\over x}$, where p is unit price. Find the instantaneous rate of change of demand with respect to price at p = 4. Also interpret your result.

10. Show that MR = $p\left[ 1-\frac { 1 }{ { n }_{ d } } \right]$ for the demand function p = 400 - 2x - 3x2 where p is unit price and x is quantity demand

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

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

13. let u= x2y3 cos $\left( \frac { x }{ y } \right)$ by using Euler’s theorem show that  $x.\frac { \partial u }{ \partial x } +y.\frac { \partial u }{ \partial y } =5u$

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

15. Verify $\frac { \partial ^{ 2 }u }{ \partial x\partial y } \frac { { \partial }^{ 2 }u}{ \partial y\partial x }$ for u = x3 + 3x2 y2 + y3

16. 6 x 3 = 18
17. For the demand function $x={20\over p+1}$, find the elasticity of demand with respect to price at a point p = 3. Examine whether the demand is elastic at p = 3.

18. Find the elasticity of demand in terms of x for the demand law $p={(a-bx)^{1\over 2}}.$ Also find the values of x when elasticity of demand is unity.

19. If the demand law is given by p = 10e$-\frac { x }{ 2 }$ then find the elasticity of demand.

20. Find the stationary value and the stationary points f(x) = x2 + 2x – 5.

21. Show that the function x3 + 3x2 + 3x + 7 is an increasing function for all real values of x.

22. Verify Euler’s theorem for the function $u=\frac{1}{\sqrt{x^2+y^2}}$

23. 4 x 5 = 20
24. Verify the relationship of elasticity of demand, average revenue and marginal revenue for the demand law p = 50 - 3x.

25. Find the stationary values and stationary points for the function f(x) = 2x3 + 9x2 + 12x + 1

26. The manufacturing cost of an item consists of Rs. 1,600 as over head material cost Rs.30 per item and the labour cost Rs .$\left( \frac { { x }^{ 2 } }{ 100 } \right)$ for x  items produced. Find how many items be produced to have the minimum average cost.

27. A company uses 48000 units/year of a raw material costing RS.2.5 per unit. Placing each order costs Rs. 45 and the carrying cost is 10.8 % per year of the average inventory. Find the EOQ, total number of orders  per year and time between each order. Also verify that at EOQ carrying cost is equal to ordering cost.