Class Business Mathematics and Statistics English

ò Total cumulative sales after months. F( ) = f t dt ∫ dt t   t =- ,  = − ] = units (ii) Sales during the 5th month = dt t   t = −  = − ] = units Total sales due to the advertisement campaign. = −  ∞ ∞ ] 10000 = , untis. Example .

The price of a machine is , , if the rate of cost saving is represented by the function f ( t ) = , t . Find out the number of years required to recoup the cost of the function. Saving Cost S ( t ) = 20000 t dt t ò = 10000 t To recoup the total price, 10000 t = 640000 t = t = When t = years, one can recoup the price. .

. Revenue functions from Marginal revenue functions If R is the total revenue function when the output is x , then marginal revenue MR dR Integrating with respect to ‘ x ’ we get Revenue Function, R MR dx k = ( Where ‘ k ’ is the constant of integration which can be evaluated under given conditions, when x = , the total revenue R = , Demand Function, P ≠ . Example . For the marginal revenue function MR , find the revenue function and demand function.

Given MR = R = MR dx k k R = k Since R = when x k R = Demand function P P = x . Example . A firm has the marginal revenue function given by MR where x is the output and a, b, c are constants. Show that the demand function is given by x b p Given MR = a x − R = a x c dx R = a x c x k − R = − c x k When x ∴ = − − ( ) + k k = a R = − c x = − ab a x b x c x R = ax b x c x Demand function P = R P = b x P+c = b x b ( x+b ) = x = b P To find the Maximum Profit if Marginal Revenue and Marginal cost function are given: If ‘ P ’ denotes the profit function, then dP d dx R dR dC MR MC ) = Integrating both sides with respect to x gives , MR MC dx k Where k is the constant of integration.

However if we are given additional information, such as fixed cost or loss at zero level of output, we can determine the constant k . Once P is known, it can be maximum by using the concept of maxima and minima. Example . The marginal cost ′ ( ) C x and marginal revenue ′ ( ) = and R x .

The fixed cost is ` . Determine the maximum profit. Given C( x ) ∫ C x dx k k C( x ) k When quantity produced is zero, then the fixed cost is . i.e.

When x = , c = ⇒ k = Cost function is C(x) = ( ) The Revenue ′ R x ( ) = R ( x ) = ′ ∫ R x dx k k = x + k When no product is sold, revenue = i.e. When x Revenue R ( x ) = x ( ) Integral Calculus – II Profit P = Total Revenue – Total cost = dp = − x To get profit maximum, dp = ⇒ x = . d P = −< ∴ Profit is maximum when x = and Maximum Profit is P = ) − ( ) − – – Profit , . Example .

The marginal cost and marginal revenue with respect to commodity of a firm are given by and ′ ( ) = R x . Find the total Profit given that the total cost at zero output is zero. Given MC = + x C( x ) x dx k = k ( ) But given when x ⇒ k = ∴ C ( x ) = ( ) Given that MR = R ( x ) MR dx k k = x + k Revenue = , when x = ⇒ k = R ( x ) = x ( ) Total Profit functions P ( x ) = R ( x ) – C ( x ) P ( x ) = = Example . The marginal revenue function (in thousand of rupees ) of a commodity is Where x is the number of units sold.

Find the total revenue from the sale of units ( e − = . Given, Marginal revenue ′ ( ) = R x Total revenue from sale of units is R = + −   =  − = = . Total revenue = . × = ` , , Example .

The price of a machine is ` , , with an estimated life of years. The estimated salvage value is ₹ , . The machine can be rented at ₹ , per year. The present value of the rental payment is calculated at % interest rate.

Find out whether it is advisable to rent the machine. ( e − The present value of payment for t year = 72000 t dt t Present value of payment for12 years = 72000 dt t   72000 t = − ( ) −     72000 = −     = − ] = Cost of the machine = Hence it not advisable to rent the machine It is better to buy the machine. Inventory : Given the inventory on hand I ( x ) and the unit holding cost ( C ), the total inventory carrying cost is C I x dx T ò where T is the time period under consideration. Example .

A company receives a shipment of cars every days. From experience it is known that the inventory on hand is related to the number of days. Since the last shipment, I x ( ) = . .

Find the daily holding cost for maintaining inventory for days if the daily holding cost is ₹ . . Here I ( x ) = – . x C = ₹ .

T = Total inventory carrying cost = C I x dx x dx T - ò = = , Amount of an Annuity The amount of an annuity is the sum of all payments made plus all interest accumulated. Let an annuity consist of equal payments of Rs. p and let the interest rate of r percent annually be compounded continuously. Amount of annuity after N payments A = pe dt rt N ò Example .

Mr. Arul invests ₹ , in ABC Bank each year, which pays an interest of % per annum compounded continuously for years. How much amount will there be after years. e N 10000 .

, Annuity = 10000 dt t ò 10000 t  100000 100000 e . =100000 . ] = ₹ , Consumption of a Natural Resource Suppose that p ( t ) is the annual consumption of a natural resource in year t . If the consumption of the resource is growing exponentially at growth rate k , then the total consumption of the resource after T years is given by p e dt k e kt kT T Where p is the initial annual consumption at time t = .

Y T O P P ( ) = P e kt Fig. . year consumption Example . In year world gold production was metric tons and it was growing Integral Calculus – II exponentially at the rate of .

% per year. If the growth continues at this rate, how many tons of gold will be produced from to ? [e . = .

) Annual consumption at time t (In the year ) : metric ton. Total production of Gold from to = dt t ò  t 424500 e . = , . metric tons approximately.

. . The demand functions from elasticity of demand Elasticity of the function y ( ) at a point x is defined as the limiting case of ratio of the relative change in y to the relative change in x . ∴ h = E E lim ∆→ ∆ ∆ ⇒ h = x Elasticity of demand h d = − p dp − dp = dx d .

Integrating both sides w.r. to x ò Equation yields the demand function ‘ p ’ as a function of x . The revenue function can be found out by using integration. Example .

When the Elasticity function is x − Find the function when x = and y = . E E x − x − − . ò x − log y = log( ) log k = k ( x – ) when x = , y = ⇒ = k ( – ) k = = ( x – ) Example . The elasticity of demand with respect to price p for a commodity is h d .

Find demand function where price is ₹ and the demand is . h d = − p dp - dx dp ò dp log x = log( k ∴ x = k p When x = k ( ⇒ k = – Hence x = R = px Revenue = p( – p – p ) Exercise . . The cost of over haul of an engine is ₹ , The operating cost per hour is at the rate of x − where the engine has run x km.

Find out the total cost if the engine run for hours after overhaul. . Elasticity of a function Ey Ex is given by Ey Ex . Find the function when x .

The elasticity of demand with respect to price for a commodity is given by − where p is the price when demand is x . Find the demand function when price is and the demand is . Also find the revenue function. .

A company receives a shipment of scooters every days. From experience it is known that the inventory on hand is related to the number of days x . Since the shipment, I x ( ) = , the daily holding cost per scooter is ₹ . .

Determine the total cost for maintaining inventory for days. . An account fetches interest at the rate of % per annum compounded continuously An individual deposits ₹ , each year in his account. How much will be in the account after years.

( e . The marginal cost function of a product is given by dC where x is the output. Obtain the total and the average cost function of the firm under the assumption, that its fixed cost is ₹ . .

The marginal cost function is MC = and fixed cost is zero. Find out the total cost and average cost functions. . If the marginal cost function of x units of output is ax and if the cost of output is zero.

Find the total cost as a function of x . . Determine the cost of producing air conditioners if the marginal cost (is per unit) is ′ ( ) = x . .

The marginal revenue (in thousands of Rupees) functions for particular commodity is where x denotes the number of units sold. Determine the total revenue from the sale of units. (Given e − = approximately) . If the marginal revenue function for a commodity is MR .

Find the demand function. . Given the marginal revenue function x + − , show that the average revenue function is P . .

A firm’s marginal revenue function is MR Find the corresponding demand function. . The marginal cost of production of a firm is given by ′ C x , the marginal revenue is given by ′ R x and the fixed cost is ₹ . Find the profit function.

. If the marginal revenue function is ′ ( ) = R x . Find the revenue function and average revenue function. .

Find the revenue function and the demand function if the marginal revenue for x units is MR = x . Integral Calculus – II . The marginal cost function of a commodity is given by MC 14000 and the fixed cost is ₹ , . Find the total cost and average cost.

. If the marginal cost ( MC ) of a production of the company is directly proportional to the number of units ( x ) produced, then find the total cost function, when the fixed cost is ₹ , and the cost of producing units is ₹ , . . If MR = x , find total revenue function.

. If MR = x , find the demand function. . .

Consumer’s surplus: This theory was developed by the great economist Marshal. The demand function reveals the relationship between the quantities that the people would buy at a given price. It can be expressed as p Let us assume that the demand of the product x when the price is p . But there can be some consumer who is ready to pay q which is more than p for the same quantity x .

Any consumer who is ready to pay the price more than p gains from the fact that the price is only p . This gain is called the consumer’s surplus. It is represented in the following diagram Fig. .

P X CS P = f ( x ) o Y Quantity Price Mathematically the Consumer’s Surplus (CS) can be defined as CS = (Area under the demand curve from x = to x ) – (Area of the rectangle OAPB) CS = ∫ f x dx x p Example . The demand function of a commodity is y . Find the consumer’s surplus for y Given y and – x x CS ò (demand function) dx – (Price × quantity demanded)      − Hence the consumer’s surplus is = units. .

. Producer surplus A supply function g ( x ) represents the quantity that can be supplied at a price p . Let p be the market price for the corresponding supply x o . But there can be some producers who are willing to supply the commodity below the market price gain from the fact that the price is p .

This gain is called the producer’s surplus. It is represented in the following diagram. Y O Fig. .

PS P X Quantity Price Mathematically, producer’s surplus (PS) can be defined as, PS = (Area of the rectangle OAPB) − (Area below the supply function from to PS = x p – g x dx ò Example . Find the producer’s surplus defined by the supply curve g ( x ) = x + when x o = . g ( x ) x + and x = p ( ) + = PS x p – g x dx ∫ (  +   – ( + ) units Hence the producer’s surplus = units. Example .

The demand and supply function of a commodity are d = and s = . Find the consumer’s surplus and producer’s surplus at equilibrium price. The point of intersection of demand and supply curves is called equilibrium point. At equilibrium point q d = q s Given P d s ; We know that at equilibrium prices p d s x = 2x – = ) ( = = – or The value of x cannot be negative, x = When x ∴ p = ( ) ( ) CS f x dx x p =    − = −   − CS = units PS = x P g x dx − ∫ ( ) = ( = = units Hence at equilibrium price, Integral Calculus – II (i) the consumer’s surplus is units (ii) the producer’s surplus is units.

Exercise . . Calculate consumer’s surplus if the demand function p and . Calculate consumer’s surplus if the demand function p and .

The demand function p and supply function p . Calculate the equilibrium price and quantity demanded .Also calculate consumer’s surplus. . The demand function for a commodity is e x − .

Find the consumer’s surplus when p = . . . Calculate the producer’s surplus at x = for the supply function p .

If the supply function for a product is .Find the producer’s surplus when x = . . The demand function for a commodity is . Find the consumer’s surplus when the prevailing market price is ` .

. The demand and supply functions under perfect competition are p d = and p s = respectively. Find the producer’s surplus. .

Under perfect competition for a commodity the demand and supply laws are d s and respectively. Find the consumer’s and producer’s surplus. . The demand equation for a products is and the supply equation is .

Determine the consumer’s surplus and producer’s surplus, under market equilibrium. . Find the consumer’s surplus and producer’s surplus for the demand function d = and supply function s = . Exercise .

Choose the best answer form the given alternatives . Area bounded by the curve y between the limits and with x − axis is (a) sq.units (b) sq.units (c) sq.units (d) sq.units . Area bounded by the curve y − between the limits ≤ ≤∞ is (a) sq.units (b) sq.unit (c) sq.units (d) sq.units . Area bounded by the curve y = between the limits and is (a) log2 sq.units (b) log5 sq.units (c) log3 sq.units (d) log sq.units .

If the marginal revenue function of a firm is MR= e , then revenue is (a) − (b) (c) (d) e . If MR and MC denotes the marginal revenue and marginal cost functions, then the profit functions is (a) P MR MC dx k (b) P MR MC dx k (c) P MR MC dx k )( (d) P C dx k . The demand and supply functions are given by D x x and S x ( ) = ( ) = are under perfect competition, then the equilibrium price x is (a) (b) (c) (d) . The marginal revenue and marginal cost functions of a company are MR and MC = − where x is the product, then the profit function is (a) (b) (c) (d) k .

The given demand and supply function are given by D x x and S x ( ) = ( ) = if they are under perfect competition then the equilibrium demand is (a) (b) (c) (d) . If the marginal revenue MR , then the average revenue AR is (a) (b) (c) (d) . The profit of a function p ( x ) is maximum when (a) MC MR = (b) MC = (c) MR = (d) MC + MR = . For the demand function p ( x ), the elasticity of demand with respect to price is unity then (a) revenue is constant (b) cost function is constant (c) profit is constant (d) none of these .

The demand function for the marginal function MR is (a) − x (b) (c) (d) + x . When x and p the consumer’s surplus for the demand function d = is (a) units (b) units (c) units (d) units . When x and P the producer’s surplus for the supply function P s = is (a) units (b) units (c) units (d) units . Area bounded by y between the lines with y axis is (a) sq.units (b) sq.units (c) sq.units (d) sq.unit .

The producer’s surplus when the supply function for commodity is x and x is (a) (b) (c) (d) . The marginal cost function is MC = find AC given that TC = when the out put is zero is (a) (b) (c) (d) . The demand and supply function of a commodity are P x − and S x ( ) = then the equilibrium quantity x is (a) (b) (c) (d) . The demand and supply function of a commodity are D x ( ) = and S x ( ) = then the equilibrium price P is (a) (b) (c) (d) Integral Calculus – II .

If MR and MC denote the marginal revenue and marginal cost and MR MC then the maximum profit at x is equal to (a) (b) (c) (d) . If the marginal revenue of a firm is constant, then the demand function is (a) MR (b) MC (c) C x ( ) (d) AC . For demand function dp k dx k , if then isequal to = ∫ (a) h d (b) − h d (c) − h d (d) h d . Area bounded by y e x between the limits to is (a)( e − ) sq.units (b)( e + ) sq.units (c) sq.units (d) sq.units .

The area bounded by the parabola y bounded by its latus rectum is (a) sq units (b) sq units (c) sq units (d) sq units . Area bounded by y between the limits and is (a) 1sq.units (b) sq.units (c) sq.units (d) sq.units Miscellaneous problems . A manufacture’s marginal revenue function is given by MR= . Find the increase in the manufactures total revenue if the production is increased from to units.

. A company has determined that marginal cost function for product of particular commodity is given by MC . Where C is the cost of producing x units of the commodity. If the fixed cost is ₹ what is cost of producing units .

The marginal revenue function for a firm is given by MR = + . Show that the demand function is P . . For the marginal revenue function MR = x , Find the revenue function and demand function.