3. Statistics
4. Bivariate Linear Correlation
5. Bivariate Linear Regression
6. Time Series Analysis
7. Index Numbers
8. Discrete Probability Distribution
9. Normal Distribution
4. Functions and Their Applications
• Introduction
• Quantity of Commodity demanded / supplied in the market
• Its cost of Production
• The Taxes on its production
• Rate of interest in banks on car loans , housing loans , consumer loans
404/05/2018 Suraj - 7709685533
5. Depedent and Independent Variable
• X is independent Variable Demand
• Y is Depedent Variable price
• Y= F (X ) x= f(y)
• Y is Double of X
• Y= 2 X
504/05/2018 Suraj - 7709685533
6. MATHEMATICAL FUNCTIONS
1. Constant Function , y = k , for any value of X
2. Linear Function , y = 2𝑥
3. Power Function , y = 𝑥7
4. Step Function , Y = [X] , 0,1,2
5. Exponential Function , y= 3 𝑥
6. Logarithm Function , y = log 𝑥
604/05/2018 Suraj - 7709685533
7. 1. Constant Function
• The value of the function remains constant for different values of x
• Y=k for 𝑓 𝑥 = 𝐾
• Graph Straight line parallel to x axis
Catego…
704/05/2018 Suraj - 7709685533
8. Linear Function
• 𝑦 = 4𝑥 + 7 or 𝑓 𝑥 = 3𝑥 − 2
• Its Graph is a straight line
804/05/2018 Suraj - 7709685533
9. Power Function
• 𝑦 = 𝑥5 𝑜𝑟 𝑓 𝑥 = 𝑥7
• 𝑓 𝑥 = 𝑥 𝑛
• Where n is a positive Integer
• Graph is a curve
904/05/2018 Suraj - 7709685533
10. Step Function
• 𝑦 = 𝑥 = 0 , 𝑓𝑜𝑟 0 ≤ 𝑥 < 1
=1 , for 1≤ 𝑥 < 2
=2 , for 2≤ 𝑥 < 3
Graph of this function is like step
1004/05/2018 Suraj - 7709685533
11. Exponential Function
• 𝑦 = 𝑓 𝑥 = 𝑎 𝑥 𝑤ℎ𝑒𝑟𝑒 𝑎 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
• 𝑦 = 3 𝑥 or 𝑓 𝑥 =
1
2
𝑥
• If we take the base as e , it becomes y or 𝑓 𝑥 = 𝑒 𝑥
• e is an irrational Number 2.71828183
• The graph is a curve
1104/05/2018
Type equation here.
Suraj - 7709685533
16. Exercise
• Find values of the following functions at
• X=0,2,-3,1.6,-1
• 𝑓 𝑥 = 7𝑥 + 9
• 𝑓 𝑥 = 7
• 𝑓 𝑥 = 𝑥2
1604/05/2018 Suraj - 7709685533
17. ECONOMIC FUNCTIONS
1. Demand Function
2. Supply Function
3. Total Revenue Function
4. Average Revenue Function
5. Total Cost Function
6. Average Cost Function
7. Profit Function
1704/05/2018 Suraj - 7709685533
18. Economic Functions : 1. Demand Function
The function relates the price P of a commodity with its Demand D
Demand function is the demand D expressed as a function of its price
P i.e.
D = 𝑓 𝑃
Demand function 𝑝 = 𝑔 𝐷 price is a decreasing function of
demand 𝑝 = 12 − 5𝐷 , 𝑃 = 2 − 𝐷2 𝑜𝑟 𝑝 =
7
𝐷
, 𝑒𝑡𝑐
1804/05/2018 Suraj - 7709685533
20. Total Revenue Function
• The function represents the amount which can be obtained by selling
D Units ( market Demand ) at the price p . Thus the Total Revenue
function (R) can be expressed as
• R= p. D
2004/05/2018 Suraj - 7709685533
21. Average Revenue Function
• Average revenue function is nothing but a price P
• 𝐴𝑅 =
𝑅
𝐷
= 𝑝
• Total Revenue per Unit
2104/05/2018 Suraj - 7709685533
22. Total Cost Function
• Total cost of producing x Units of a commodity
• It is denoted by C
• C is a function of X
• Total cost usually consists of two parts 1. fixed cost 2. variable cost
• 𝑐 = 20 + 3𝑥 𝑜𝑟 𝑐 = 2𝑥2 + 7𝑥 + 50
2204/05/2018 Suraj - 7709685533
23. Average Cost Function
• The cost of Production per unit is called Average Cost (AC)
And is expressed as
𝐴𝐶 =
𝐶
𝑥
2304/05/2018 Suraj - 7709685533
24. 7. Profit Function
• It is the Difference Between Revenue and cost
• If R= Total Revenue
• And C= Total Cost
• 𝑝𝑟𝑜𝑓𝑖𝑡 = 𝑅 − 𝐶
• If R>C , there will be a profit
• If R<C , there will be a loss
• If R=C , there will be no profit and loss
2404/05/2018 Suraj - 7709685533
25. Illustration 4
• The demand for a commodity is given by 𝑝 = 7 − 𝐷
And the supply by 𝑝 = 3 + 3𝐷
At equilibrium point
Demand = Supply
Ans : the equilibrium price is 6 and quantity is 1
2504/05/2018 Suraj - 7709685533
26. Illustration 5
• If the total cost function is 𝐶 = 4 + 3𝑥 + 𝑥2
find the cost when
𝑥 𝑖𝑠 10 𝑢𝑛𝑖𝑡𝑠 , 𝑎𝑙𝑠𝑜 𝑓𝑖𝑛𝑑 𝑎𝑣𝑒𝑟𝑔𝑎𝑟𝑒 𝑐𝑜𝑠𝑡 𝑤ℎ𝑒𝑛 𝑥 𝑖𝑠 4 𝑢𝑛𝑖𝑡𝑠
• Solution : 𝐶 = 4 + 3𝑥 + 𝑥2
• 𝐴𝐶 =
𝐶
𝑥
=
4
𝑥
+ 3 + 𝑥
Now to find C when x=10
𝐶 = 4 + 3 𝑥 10 + 102
= 134
Now AC when x=4,
𝐴𝐶 =
4
4
+ 3 + 4 = 8
04/05/2018 26Suraj - 7709685533
27. Illustration 6
• If the Demand function is p=10-2D , find total revenue and average
revenue when D is 3 Units .
Solution :
𝑝 = 10 − 2𝐷
𝑅 = 𝑝. 𝐷 = 10 − 2𝐷 . 𝐷 = 10𝐷 − 2𝐷2
𝐴𝑅 =
𝑅
𝐷
=
10𝐷−2𝐷2
𝐷
=10-2D
Now R at D=3 , 𝑅 = 10𝑥3 − 2𝑥32
= 12
AR at D=3
𝐴𝑅 = 10 − 2𝑥3 = 4
04/05/2018 27Suraj - 7709685533
28. Illustration 7
• The total cost function is C=500+15x and the total revenue function is
R=700+5x . Find the point at which there will be no profit , no loss i.e.
break even point
Solution : For break even point , profit =0
R=C
700 + 5𝑥 = 500 + 15𝑥 𝑜𝑟 200 = 10𝑥
x= 20
So the break even point is X=20
04/05/2018 28Suraj - 7709685533
29. Illustration 8 :
• A manufacturer has put ₹ 30,000 as initial cost and a variable cost of
₹ 20 per unit for production of batteries . If each battery can be sold at
₹ 40 , find total cost , total revenue , profit functions , Also find the
number of batteries to be produced to achieve the break – even point
Of no profit , no loss
Solution :
04/05/2018 29Suraj - 7709685533
30. Solution :
• The total cost function C consists of two parts , the fixed cost and the
variable cost
𝐶 = 30000 + 20𝑥
The revenue function R is given by
R=40X
The profit function is given by
Profit = Revenue – cost
= 40𝑥 − 30000 − 20𝑥 = 20𝑥 − 30000
For break even point , profit =0
20𝑥 − 30000 = 0 𝑜𝑟 𝑥 = 1500
Hence , the required no. of batteries to be produced for break – even point is 1500
04/05/2018 30Suraj - 7709685533
31. Illustration 9
• A company manufactures notebooks . The weekly total cost function
is given by C=15X+3000
1) If each notebook is sold at ₹ 25, what is minimum quantity that
needs to be produced to ensure no loss
2) If the selling Price of a notebook is increased by 20 %, what would
be the minimum quantity that needs to be produced and sold to
ensure no loss
3) If it is known in advance that at least 400 notebooks can be sold per
week , find the selling price to ensure the company , no loss
04/05/2018 31Suraj - 7709685533
32. Solution
1) Here , total revenue R = 25 X
Total cost C = 15X+3000
for no loss ( break – even point ) R = C
25 X = 15 X + 3000 or 10 X = 3000
X= 300
The minimum quantity produced and sold of the notebooks to ensure
no loss is 300
04/05/2018 32Suraj - 7709685533
33. 2) Selling price is increased by 20 %
New selling price = 25+20%of25= ₹ 30
New R=30 X
For break – even point , new R= C
30𝑥 = 15𝑥 + 3000 𝑜𝑟 15𝑥=3000
𝑥 = 200
So 200 notebooks should be produced and sold
04/05/2018 33Suraj - 7709685533
34. 3) Let us assume the selling price of a notebook as p and x=400 ( given )
, so , R=400p
𝐶 = 15𝑥 + 3000 = 15𝑥400 + 3000 = 9000
For no loss, R=C
400p=9000 or p=22.5
Hence the 400 notebooks can be sold at ₹ 22.5 to have no loss
04/05/2018 34Suraj - 7709685533
