ELEMENTARY CALCULUS DIFFERENTIAL CALCULUS Introduction: The subject of differential calculus is based on the concept of limit and the discussion of limit depends upon the concept of function. A quantity which does not change is called a constant and a quantity which changes is called a variable. An association or correspondence from one variable to another is called a function. FUNCTION: A and B are two non empty sets. Let every element of A have a corresponding unique element in B. Such correspondence is called a function or mapping of set A into set B. Where ‘f’ indicates the rule which enables us to find the element of B which is associated with the elements of A. If y Є B is assigned to x Є A, we denote this as f(x) = y. Y is the image of X and X is the preimage of Y. If ‘x’ and ‘y’ are variables associated by ‘f’ then we write y = f(x) – x is called the independent variable and y is called the dependent variable. Types of functions: 1
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ELEMENTARY CALCULUS
DIFFERENTIAL CALCULUS
Introduction:
The subject of differential calculus is based on the concept of limit and the discussion of limit depends upon the concept of function.
A quantity which does not change is called a constant and a quantity which changes is called a variable.
An association or correspondence from one variable to another is called a function.
FUNCTION:
A and B are two non empty sets. Let every element of A have a corresponding unique element in B. Such correspondence is called a function or mapping of set A into set B. Where ‘f’ indicates the rule which enables us to find the element of B which is associated with the elements of A.
If y Є B is assigned to x Є A, we denote this as f(x) = y.Y is the image of X and X is the preimage of Y.
If ‘x’ and ‘y’ are variables associated by ‘f’ then we write y = f(x) – x is called the independent variable and y is called the dependent variable.
Types of functions:
1. Linear functions:
A function f(x) defined by f(x) = ax + b where a and b are real numbers is called a ‘linear function’.
E.g. 8x + 4, 3-4x are linear functions.
2. Quadratic functions:
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A function f(x) defined by f(x) = ax2 + bx + c, where a, b, c are real numbers is called a ‘quadratic function’.
E.g. 2X2 + 3X + 4, -X2 + 2X – 1 are quadratic functions.
3. Cubic functions :
A function f(x) defined by f(x) = ax3 + bx2 + cx + d where a, b. c. d are real numbers is called a cubic function.
4. Exponential function:
A function f(x) defined by f(x) = ax is called the exponential function.
E.g. e2x, ex/4, 2x, e-x/2 are all exponential functions.
5. Logarithmic functions:
A function f(x) defined by f(x) = loga x is called the logarithmic function.
Some functions in Economics:
1. Supply functions and demand functions:
If x denotes the quantity of commodity demanded and p is the price, x and p being variables then the demand functions can be written as follows :-
X = f(p) showing dependence of x on p or p = g(x), indicating the dependence of p on x.
2. Cost functions:
If C is the total cost of producing x quantities of a certain good, then we may write this in the implicit form:
g(c, x) = 0
3. Total Revenue functions:
Revenue is the amount of money derived from the sale of a product and depends upon the price of the product and the quantity of the product that is actually sold. Thus,
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R(x) = P.x ; p = R(x) X
Where R(x) is the toal revenue function and P is the price which is also average revenue function.
4. Profit functions:
The profit function of a firm is defined by P(x) = R(x) – C(x),Where R(x) is the revenue function and C(x) is the cost function.
DERIVATIVES;
Increment:
Increment of a variable is the difference of initial value from the final value. Let “x” change its value from 3 to 5. The the increment or change of ‘x’ = 5-3 = 2.
Notation:
‘x’ read as ‘delta’ x, is used to denote the change in ‘x’ or increment in ‘x’, and that of ‘y’ will be represented by ‘y’. It can be a positive change or a negative change.
‘’ or ‘∆’ just stands for the words ‘change in the variable or quantity’.
Consider a function y = x .....(1)Let y be the amount by which ‘y’ has increased when x is increased by x or that y + y is the new value of the function when ‘x’ is changed to ‘x + x’.
Therefore y + y = x + x .....(2)
gives y + y – y = x + x - x
Therefore, y = x
As x → 0, y → 0That is, as the change in ‘x’ tends to zero, the change in ‘y’ also tends to zero.
DIFFERENTIABLE FUNCTION:
Let y = f(x) be a continuous function of x. Let x be the increment (or change) given to the value of x. Then there will be a corresponding increment (or change) in the value of y. Let this be y.
Therefore, y + y = f(x + x) y = f(x)
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Consider the ratio of the change (increment)y = f(x + x) – f(x)x x
If the limit of the above ratio as x → 0 (but not equal to 0)exists, it is called the ‘derivative’ of ‘y’ with respect to ‘x’ or ‘the differential coefficient of ‘y’ with respect to ‘x’ and is denoted by dy .
dx
Thus, lim = y = dy = lim f(x + x) – f(x) x→ 0 x dx x → 0 x
The process of finding the derivative of a function is called the differentiation of that function.
Note: The derivative of a linear function y = 2x is 2. The derivative of a quadratic function y = x2 is 2x. The derivative of a constant is zero. Rules of Differentiation:
Sum Rule: If u = f(x) and v = g(x), then if y = u + v,
d (u + v) = du + dv dx dx dx
Product Rule : If y = u.v d (u.v) = u . dy + v . du dx dx dxQuotient Rule : If y = u/v, then
v. du - u. dy d . u = dy dx dx dx v v2
COST ANALYSIS:
Total Cost = Total Fixed Cost + Total Variable Cost
Average Fixed Cost + Total Cost divided by Output.
Average Variable Cost + Total Variable Cost divided by Output.
Average Total Cost = Total Variable Cost divided by Output.
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Marginal Cost is the cost of producing one extra unit of output.= MCn = TCn – TCn-1= Marginal Cost of producing the nth unit = Total cost of n units minus total cost of (n-1) units.
Linear Cost Function:
The linear cost function is TC = a + bQTC is the Total Cost, a = TFC, bQ = TVC
Quadratic Cost Function:
TC = a + bQ + cQ2
TC = Total Cost a = Total Fixed Cost bQ + cQ2 = Total Variable Cost
Average Cost = Total Cost divided by Quantity. = a/Q + b + cQ
Marginal Cost = b + 2cQ (2Q is the differential coeff. Of Q2)
Cubic Cost Function:TC = a + bQ – cQ2 + dQ3 (AC = a/Q + b – cQ + dQ2)
(MC = b – 2cQ + 3dQ2)
MATHEMATICS IN ECONOMICS
Meaning of some Basic Terms used in Mathematics:
A variable is something that can take on different values.
Endogenous variables - originating from within.Exogenous variables - originating from without.
A constant is a magnitude that does not change (opposite of a variable).(Givens) e.g. a in ax.
Endogenous variables ….a, b, c….OR. α, β, γExogenous variables ….a, b, c OR α, β, γ
Positive Integers - Whole numbers 1, 2, 3, 4, 5… Negative Integers -1, -2, -3, -4, -5 ….Together with number (0) which is neither – nor + , make up set of all integers.
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The values which fall between the integers are called fractions. e.g. ⅓, ⅔, ½, ⅝, ⅞ …. and -⅝, -⅜, -⅔, -½ ….make up set of all fractions.
Set of all integers and set of all fractions make up set of all ratio-nal numbers.
Irrational numbers are those which fall between rational numbers and integers. Thus Integers, Fractions, Irrational and rational numbers all put together form a set of “real numbers”.
There are also imaginary numbers such as square-root of negative numbers.
Function (f) – is the action of associating one thing with another.
In y = f(x),the functional notation “f” means a rule by which the set ‘x’ is converted or transformed into set ‘y’.The function converts x into y. f : x → y
In y = f(x), The domain of f = all permissible values x can take. all the y values into which x values are mapped is called the range of f.or set of all values which the ‘y’ variable will take is called the range of ‘f’.
Constant function:
Y = f(x) = 7 OR y = 7 or f(x) = 7.Regardless of value of x, value of Y remains static or the same.
Polynomial functions:
Polynomial means “multiterm”.Polynomial function of a single variable x has the general form (formula)
Y = a x + a x + a x + a x ….. + a x = a x 1+ a x + a x + a x ….. + a x = a + a x + a x + a x + ….. + a x
Rates of change in the equilibrium values of the variables:
Consider the rate of change of any variable ‘y’ in response to a change in another variable x, where the two variables are related to each other by the function
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Y = f(x)
Y represents the equilibrium value of an endogenous variable X will be some parameter.Presently we restrict to the simple case where there is only one parameter.
The Difference Quotient:
The symbol ∆ (the Greek Capital delta for “difference”) is used to denote “change” in value.
EXPONENTIAL LAWS OR BASIC LAWS OF INDICES
If ‘m’ and ‘n’ are positive integers and ‘a’ is a non-zero real number then,
Ist Law : am. an = am+n
2nd Law: a m = 1 if m > n = 1 if n > m an am-n an-m
3rd Law : (am)n = amn
4th Law : (ab)m = ambm where ‘b’ is a non-zero real number.
5th Law: (a)m = a m (b) bm
LOGARITHMS
Literally Log-arithm means “a rule to shorten arithmetic”. With its use, the harder operations of multiplications and divisions are replaced by simple operations of addition and substraction. We can work out the computations of powers and roots with the help of logarithms and save time, energy and labour.
Definition:If ‘a’ is a positive real number other than 1 such that ax = N, then ‘x’ is called the logarithm of N to the base ‘a’. We write x = loga N (a.0, a ≠ 0) and read as “x is logarithm of N to the base a”.
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“Logarithm of a number to a given base is the index of the power to which the base must be raised in order to get the given number.”
Remarks:1. Logarithm of any quantity to the same base is unity.(1). This is because any quantity raised to the power 1 is that quantity only.2. There are a few restrictions on the base. It should not be taken as ‘0’ or ‘1’ because zero raised to any power is meaningless and 1 raised to any power is only 3. The base cannot be a negative number, otherwise certain values will become imaginary.
I Law:Logarithm of aproduct of two numbers to any base (.0 and ≠ 1) is equal to the sum of the logarithms of the numbers to the same base.
Loga (mn) = loga m + loga n.
II Law:The logarithm of a quotient of two numbers to any base (non-zero positive real numbers) is the difference of logarithms to the base.
Hence, loga (m/n) = loga m – loga n.
III Law:Logarithm of the number raised to a power is equal to the index of the power multiplied by the logarithm of the number of the same base.i.e. Loga (mn) = n loga m
Change of base Law : If the logarithm of a number to any base is given, then the logarithm of the same number to any other base can be determined from the following law:
IV Law:The logarithm of ‘m’ to the base ‘a’ is the quotient of logarithms of ‘m’ to the base ‘b’ and the logarithm of ‘a’ to the base ‘b’.i.e. loga m = logb m logb a
Note: When no base is mentioned, it is understood to be 10. By the word ‘logarithm’ we generally mean common logarithm.
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Minimum Maths required for Managerial Economics – First Semester.
