Area bounded by the curve y = x (4 − x) between the limits 0 and 4 with x − axis is ________.

Area bounded by the curve y = e^−2x between the limits 0 ≤ x ≤ ∞ is ________.

The given demand and supply function are given by D(x) = 20 − 5x and S(x) = 4x + 8 if they are under perfect competition then the equilibrium demand is ________.

The profit of a function p(x) is maximum when ________.

When x₀ = 5 and p₀ = 3 the consumer’s surplus for the demand function p_d = 28 − x² is ________.

The cost of over haul of an engine is Rs. 10,000 The operating cost per hour is at the rate of 2x − 240 where the engine has run x km. Find out the total cost if the engine run for 300 hours after overhaul.

A company receives a shipment of 500 scooters every 30 days. From experience it is known that the inventory on hand is related to the number of days x. Since the shipment, I (x) = 500 − 0.03x², the daily holding cost per scooter is Rs. 0.3. Determine the total cost for maintaining inventory for 30 days.

If MR = 20 − 5x + 3x², find total revenue function.

The demand function of a commodity is y = 36 − x². Find the consumer’s surplus for y₀ = 11

Find the producer’s surplus defined by the supply curve g(x) = 4x + 8 when x_o= 5.

The demand and supply function of a commodity are p_d = 18− 2x − x² and p_s = 2x − 3 . Find the consumer’s surplus and producer’s surplus at equilibrium price.

The rate of new product is given by f (x) = 100 − 90 e^−x where x is the number of days the product is on the market. Find the total sale during the first four days. (e^–4 = 0.018)

In year 2000 world gold production was 2547 metric tons and it was growing exponentially at the rate of 0.6% per year. If the growth continues at this rate, how many tons of gold will be produced from 2000 to 2013? [e^0.078 = 1.0811)