Page 375 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes 9. From mean value theorem: f(b) – f(a) = (b – a) f’ (x ); a < x < b if f(x) = 1/x , then x =
1 1 1
(a) ab
(b) a + b/2
(c) 2ab/q + b

(d) b – a/b + a
2
10. The minimum value of ax + by, where xy = r , is (r, ab >0)
(a) 2r ab
(b) 2ab r

(c) –2r ab
(d) None of these

14.8 Review Questions

1. The demand function of a particular commodity is given by y 15e x / 3 for 0 x 8 where
y is price per unit and x is the number of units demanded. Determine the price and
quantity for which revenue is maximum.
2
2. Total revenue from the sale of a good X is given by the equation R = 60Q – Q , for
0 Q 60 , where R is total revenue and Q is the quantity sold at price P. Calculate the
value of MR when the point price elasticity of demand is 2.

3. From the demand function Q = 600/P, show that total expenditure on the commodity
remains unchanged as price falls. Estimate the elasticity of demand at P = 4 and at
P = 2.
4. Following are the market demand and market supply equations for a product X:
Q d 10,000(12 2 ) and Q s 1,000(20 )
P
P
The government decides to collect a sales tax of 2.50 per unit sold.
(a) What effect does it have on the equilibrium price and quantity of commodity X?
(b) Find total amount of tax collected by the government.
(c) If the government wants to maximise total tax collections, find the rate of specific
tax it should impose.
5. Show that for a competitive market with linear demand and supply functions, the imposition
of a specific tax increases the demand price and decreases the supply price by less than the
tax rate.
6. Let x = a – bp and x = a + bp be the demand and supply functions respectively of a good in
d s
perfectly competitive market. A specific tax of t per unit is imposed. Find equilibrium
dp dp
price paid by consumer p and that received by seller p . Find dt and dt s and show
s
that equilibrium the price paid by consumer increases and that received by seller decreases
as tax rate is increased.
7. A cultural organisation is arranging a kathakali dance program in a city. It expects that 300
persons would attend the show if the entrance ticket is 8. It has also estimated that for a
unit decrease in entrance fee, 60 additional persons would attend the program. Express the
revenue of the organisation as a function of the entrance fee. What should be the entrance
fee so that the organisation gets maximum revenue?

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