Definition of Calculus:
It is the branch of mathematics that deals with the finding and properties of derivatives and integrals of functions by methods originally.
Its a particular method or system of calculation or reasoning.
Calculus and it’s Applications in Business:
In business we come across many such variables where one variable is a function of the other. For example, the quantity demanded can be said to be a function of price. Supply and price or cost and quantity demanded are some other such variables. Calculus helps us in finding the rate at which one such quantity changes with respect to the other. Marginal analysis in Economics and Commerce is the most direct application of differential calculus. In this context, differential calculus also helps in solving problems of finding maximum profit or minimum cost etc., while integral calculus is used to find he cost function when the marginal cost is given and to find total revenue when marginal revenue is given.
To studying the application of calculus, let us first define some functions which are used in business and economics.
Cost Function:
The total cost C of producing and marketing x units of a product depends upon the number of units (x). So the function relating C and x is called Cost-function and is written as C = C (x).
The total cost of producing x units of the product consists of two parts
- Fixed Cost: The fixed cost consists of all types of costs which do not change with the level of production. For example, the rent of the premises, the insurance, taxes, etc.
- Variable Cost: The variable cost is the sum of all costs that are dependent on the level of production. For example, the cost of material, labor cost, cost of packaging, etc. i.e. C (x) = F + V (x)
Demand Function:
An equation that relates price per unit and quantity demanded at that price is called a demand function.
If ‘p’ is the price per unit of a certain product and x is the number of units demanded, then we can write the demand function as x=f(p)
or p = g (x) i.e., price (p) expressed as a function of x.
Revenue function:
If x is the number of units of certain product sold at a rate of Rs. ‘p’ per unit, then the amount derived from the sale of x units of a product is the total revenue. Thus, if R represents the total revenue from x units of the product at the rate of Rs. ‘p’ per unit then
R= p.x is the total revenue
Thus, the Revenue function R (x) = p.x. = x .p (x)
Profit Function:
The profit is calculated by subtracting the total cost from the total revenue obtained by selling x units of a product. Thus, if P (x) is the profit function, then P(x) = R(x) − C(x)
Break-Even Point:
Break even point is that value of x (number of units of the product sold) for which there is no profit or loss.
i.e. At Break-Even point P (x ) = 0
or R(x) = C(x)
Average Cost:
Let C=C(x) be the total cost of producing and selling x units of a product, then the average cost (AC) is defined as AC=C/X
Thus, the average cost represents per unit cost.
Marginal Cost:
Let C = C(x) be the total cost of producing x units of a product, then the marginal cost (MC), is defined to be the rale of change of C (x) with respect to x. Thus MC=dC/dx or d/dx (c(x))
Marginal cost is interpreted as the approximate cost of one additional unit of output.
Average Revenue:
If R is the revenue obtained by selling x units of the product at a price ‘p’ per unit, then the term average revenue means the revenue per unit, and is written as AR.
AR=R/X
Hence, average revenue is the same as price per unit.
Marginal Revenue:
The marginal revenue (MR) is defined as the rate of change of total revenue with respect to the quantity demanded.
