1. 1. LIMIT & DIFFERENTIATION Course: BBA Subject: Business Mathematics Unit-3
2. 2. Definition of Derivative: Let f(x) be a function of the variable x. The value of f(x) at x = a is f(a). When x = a+h, the value of the function f is f(a+h) so (a + h) f (a) is the change in the value of the function f, when the value of x changes from a to a+h. Shows the relative changes in the value of the function f, when the value of a is increased by h. (a h) f(a)f h
3. 3. If the limit of (l) exists, when h 0, it is called the derivative of the function f(x) at x = a and it is denoted by f(a). Therefore, For any x in the domain of f, 0 ( ) ( ) '( ) lim h f a h f a f a h 0 ( ) ( ) '(x) lim h f x h f x f h
4. 4. f(x) is called the derivative of f(x) with respect to x. If y= f (x), then the derivative of f is also denoted by dy / dx or d / dx [f (x)].
5. 5. Formulae for Differentiation: If y = c, where c is a constant, dy / dx = 0. If y = xn, where n is rational no. dy / dx = n x n-1 If y = ex, then dy / dx = ex if y = ax, where a > 0, then dy/dx = ax logea If y = logex, then dy/dx = 1/x.
6. 6. Composite function A composite function is a function that is composed of two other functions. For the two functions f and g, the composite function or the composition of f and g, is defined by
7. 7. The function g(x) is substituted for x into the function f(x). For example, the function F(x)=(2x+6)4 could be considered as a composition of the functions, f(x)=x4 and g(x)=2x+6. However, it could also be written as a composition of f(x)=(2x)4 and g(x)=x+3. Often, a function can be written as a composition of several different combinations of functions.
8. 8. Implicit Differentiation A function whose value can only be computed indirectly fr om one or more of the independent variables. For example, in the equationx2 + y2 = 1, y is an implicit fu nction of x.
9. 9. Applications of derivative in economic theory Today's sophisticated international markets have helped foster the rapid growth in derivative instruments. In the hands of knowledgeable investors, derivatives can derive profit from: Changes in interest rates and equity markets around the world Currency exchange rate shifts Changes in global supply and demand for commodities such as agricultural products, precious and industrial metals, and energy products such as oil and natural gas
10. 10. The two most widely recognized benefits attributed to derivative instruments are price discovery and risk management. They Improve Market Efficiency for the Underlying Asset Derivatives Also Help Reduce Market Transaction Costs
11. 11. Market equilibrium Market equilibrium is a market state where the supply in the market is equal to the demand in the market. The equilibrium price is the price of a good or service when the supply of it is equal to the demand for it in the market. If a market is at equilibrium, the price will not change unless an external factor changes the supply or demand, which results in a disruption of the equilibrium.
12. 12. The operation of the market depends on the interaction between suppliers and demanders. Market equilibrium exists when quantity supplied is equal to quantity demanded. Note that there is just one price where this is true! The equilibrium price is the price that will generally prevail in a perfectly competitive market that is not subject to governmental intervention. When a market is in equilibrium, there is no tendency for the market price to change. In other words, the equilibrium price is stable under the existing market conditions.
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14. 14. Marginal Revenue Marginal revenue ( MR ) is the change in total revenue resulting from selling an extra unit of goods. MR = TR/Q, where TR = change in TR due to change in Q, Q = change in Q
15. 15. Price elasticity of supply Price elasticity of supply (PES) measures the responsiveness of quantity supplied to a change in price. It is necessary for a firm to know how quickly, and effectively, it can respond to changing market conditions, especially to price changes. The following equation can be used to calculate PES. While the coefficient for PES is positive in value, it may range from 0, perfectly inelastic, to infinite, perfectly elastic. 2.
16. 16. Income Elasticity Demand The Income Elasticity of Demand measures the rate of response of quantity demand due to a raise (or lowering) in a consumers income. The formula for the Income Elasticity of Demand (IEoD) is given by: IEoD = (% Change in Quantity Demanded)/(% Change in Income)
17. 17. How Do We Interpret the Income Elasticity of Demand? Income elasticity of demand is used to see how sensitive the demand for a good is to an income change. The higher the income elasticity, the more sensitive demand for a good is to income changes. A very high income elasticity suggests that when a consumer's income goes up, consumers will buy a great deal more of that good. A very low price elasticity implies just the opposite, that changes in a consumer's income has little influence on demand.
18. 18. Elasticity of Supply We can study the resulting changes in the supply of a commodity, when its price changes, using the elasticity of supply. Elasticity of supply is defined as Elasticity of supply = relative change in supply / relative change in price If the supply function is x = g (p), the elasticity of supply is given by Elasticity of supply = (p/x ). (dx/ dp)
19. 19. Suppose P0 is the initial price of a commodity and X0 is its supply. Suppose, when the price becomes p1 the supply becomes s1. In this case, the elasticity of supply is given by Elasticity of supply = 1 01 2 1 2 1 0 x xp p p p x x
20. 20. Marginal Revenue Marginal revenue is defined as follows: Marginal Revenue (M. R.) = Change in total revenue / Change in demand If the demand function is p = f(x), then the total revenue R is given by R = xp = x. f(x). The first derivative of R w.r.t. x is called the marginal revenue . Thus, M.R. = dR / dx.
21. 21. Relation between average revenue, Marginal revenue and Price elasticity of demand, There is a crucial relationship between the AR, MR and elasticity of demand, which is used extensively in the theory of pricing. The relationship is expressed in the form of formula,
22. 22. 3.
23. 23. But, AQ is marginal revenue and SQ is average revenue corresponding to point B at OQ level of output. Hence, equation (2.9) can be written as 4.
24. 24. The above relationship can be utilised to find out the marginal revenue corresponding to the average revenue at any given level of quantity sold, provided the price elasticity of demand is known. The relation between AR, MR and elasticity of demand (e) can now be written as 5.
25. 25. Maximum and Minimum Values DEFINITION: A function f has an absolute maximum (or global maximum) at c if f(c) f(x) for all x in D, where D is the domain of f. The number f(c) is called the maximum value of f on D. Similarly, f has an absolute minimum (or global minimum) at c if f(c) f(x) for all x in D and the number f(c) is called the minimum value of f on D. The maximum and minimum values of f are called the extreme values of f.
26. 26. DEFINITION: A function f has a local maximum (or relative maximum) at c if f(c) f(x) when x is near c. [This means that f(c) f(x) for all x in some open interval containing c.] Similarly, f has a local minimum (or relative minimum) at c if f(c) f(x) when x is near c.
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29. 29. Profit-maximizing The profit-maximizing level of output is a production level that achieves the greatest level of economic profit given existing market conditions and production cost. For a perfectly competitive firm, this entails adjusting the production level in response to the going market price.
30. 30. Three Views Profit-maximizing output can be identified in one of three ways--directly with economic profit, with a comparison of total revenue and total cost, and with a comparison of marginal revenue and marginal cost.
31. 31. (1) Profit: First, profit maximization can be illustrated with a direct evaluation of profit. If the profit curve is at its peak, then profit is maximized. In the top panel, the profit curve achieves its highest level at 7 pounds of zucchinis. At other output levels, profit is less 8.
32. 32. (2) Total Revenue and Total Cost: Second, profit maximization can be identified by a comparison of total revenue and total cost. The quantity of output that achieves the greatest difference of total revenue over total cost is profit maximization. In the middle panel, the vertical gap between the total revenue and total cost curves is the greatest at 7 pounds of zucchinis. For smaller or larger output levels, the gap is either less or the total cost curve lies above the total revenue curve. 9.
33. 33. (3) Marginal Revenue and Marginal Cost: Third, profit maximization can be identified by a comparison of marginal revenue and marginal cost. If marginal revenue is equal to marginal cost, then profit cannot be increased by changing the level of production. Increasing production adds more to cost than revenue, meaning profit declines. Decreasing production subtracts more from revenue than from cost, meaning profit also declines. In the bottom panel, the marginal revenue and marginal cost curves intersect at 7 pounds of zucchinis. At larger or smaller output levels, marginal cost exceeds marginal revenue or marginal revenue exceeds marginal cost. 10.
34. 34. Cost Minimization Strategies: Cost minimization aims to achieve the most cost-effective way of delivering goods and services to the required level of quality
35. 35. Two approaches to cost minimization (1) Possible sources of cost reductions (1): Eliminating waste & avoiding duplication ( lean production ) Simplifying processes and procedures Outsourcing non-core activities (e.g. transaction processing, payroll, call handling) Negotiating better pricing with suppliers
36. 36. (2) Possible sources of cost reductions (2) Pruning product ranges and customer accounts to eliminate unprofitable business Using the most effective methods of training and recruitment Introducing flexible working practices Aggressive control of overheads (e.g. banning first/business class travel)
37. 37. References: 1.https://lh3.ggpht.com/4jGqQc8XqZ8YrsVQ6LG0cPWEDDZnsqPM a_zElCl2jWnA1reUw1H53VvUTahTpDsFqylfdw=s110 2.https://lh6.ggpht.com/IwP2fWoHPdaGqS_7mTM9xMT2VibScixvst nvfvDkV2FDxbos_bcydDknlfylMJ04kM4dKw=s170. 3.https://lh6.ggpht.com/SHhJAGE4Fr4xXx7OpJPP2IOXx5yu7v7oY6 LvDZgq_anJKyRqPN4zwU03TQKKNUHwWg=s95. 4.https://lh6.ggpht.com/Lb7MNLpKXHaPudD_16mEoFlr1eVa8nPl7F QlciKafgjBDvrLCxYgK4uFkicKPCNfvM7agQ=s170 5.https://lh6.ggpht.com/SHhJAGE4Fr4xXx7OpJPP2IOXx5yu7v7oY6 LvDZgq_anJKyRqPN4z-wU03TQKKNUHwWg=s95 6.https://lh6.ggpht.com/KaYAXA_Wzq8B5S7tF0TCZzbgIkFd9SrkR Y9VGw0ld_0HPi7hqIpN_z8eCgAp-8eBxLncXg=s114
38. 38. 7.https://lh5.ggpht.com/EiVgOUFxNQ_WubV8Y1Rl84bkUMrd kmkjfutANj7Sd_cmeQTcWVQJpjy9iPboWXcauITw=s135 8. https://lh5.ggpht.com/eIXqTyak-aF0bimeguqNRGJN5UN71- 5qsCWuJ5Bj-J0tt6bO_IXhsb-o2_RSviygwVHQg=s135 9.https://lh3.ggpht.com/mvp5ltydSYbE2o46kzwIfMY99bgDtKZ Wvbcm_SvOAH9wMZSugUSYymSOtLelaQaCxJxLSI=s89 10.https://lh3.ggpht.com/zmgXy_Je603OaHZNvbYjQQUNJ7TP aWh3fBZChjle1vS1ERB-At3aCiinvnEmlia7AsfrRg=s89 11. Business Mathematics by A. G. Patel and G.C Patel Atul Prakashan.
39. 39. Web Sources: www.thefreedictionary.com/implicit+function www.yourarticlelibrary.com/.../price-elasticity/...average- revenue-margin. economics.about.com/cs/micfrohelp/a/supply_elast.htm http://en.wikipedia.org/wiki/Profit_maximization