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PROJECT WORK
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Grade: 11
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Project details
Subject: Business Mathematics
Topic: Application of the function in the real world
CONTENTS
Abstract
Acknowledgement
Introduction
Demand
Demand function
Equation of demand function
Supply
Supply function
Equation of supply
Profit function
Elasticity of demand
Price elasticity of demand
Price elasticity of supply
Income elasticity of demand
Budget and cost constraints
Budget
Cost
Equation of isocost
Supply and demand analysis ( quadratic)
Break – even function
Optimization
Exponential and Logarithmic Functions in Economic and Management
Unlimited growth function
Unlimited decay function
Limited growth function
Logistic growth function
Application/ recommendation/ practical significance
Conclusion
References
ABSTRACT
This project is made by me through different observations and evaluation mainly to generate more information about Linear, Quadratic, Exponential and Logarithmic functions and even to show the ways, how they relates or applied in real world. I have research and presented my findings about the application of the function in the real world by the mean of project. It is seen that Linear equation helps to figure out income over time, calculate mileage rates or predict profits. Similarly, Quadratic equation helps us in daily life in calculating areas, determining a product's profit or formulating the speed of an object. Moreover, Exponential and Logarithmic functions are widely used in modelling continuous growth and decay process such as bacterial growth, population growth, energy conservation, radioactive decay, salvage value, etc. Although the project covers various application but it is confined here. This project is supported by huge varieties of sources such as Wikipedia, Education world, spark notes, etc. Application of Linear, Quadratic, Exponential and Logarithmic function are from simple to complex. In simple word, one can describe Linear equation as the equation of straight line while Quadratic equation as the equation of parabola. I hope you all can imagine how different would be life in the absence of these functions.
ACKNOWLEDGEMENT
Firstly, I would like express my sincere thanks to my subject teacher, Mr. Amit Adhikari sir for providing me an opportunity to engage myself in such a useful task. This project has enhanced my knowledge about different application of the functions. I would also like to thank all those who have motivated me to do this project.
Lastly, I am thankful to all the websites that offered me to gather lots of information and facts related to the topic.
INTRODUCTION
The graph of a linear function is a straight line. The formula for a linear function is y = f(x) = a + bx. One independent variable and one dependent variable make up a linear function. The dependent variable is y, while x is the independent variable. The constant term, often known as the y-intercept, is denoted by the letter a. when x = 0, it is the value of the dependent variable. The independent variable’s coefficient is b. The rate of change of the dependent variable is given by the slope, which is also known as the slope.
Similarly, the graph of a Quadratic function is just like a parabola. The formula of a Quadratic function is y = f(x) = ax2 + bx + c, where a and b are independent variable’s coefficient and c is the constant. X is the independent variable whereas y is a dependent variable. The independent variable’s coefficient b and constant c can take any finite value, and a can take any finite value other than 0. Quadratic functions can be expressed in many different forms. The form written above is called standard form. Additionally, f(x) =a (x−x1) (x−x2) is known as factored form. The vertex form is f(x) =a (x−h) 2+k. The slope of a quadratic function, unlike the slope of a linear function, is constantly changing.
Moreover, In general, exponential functions are of the form f(x) = ax, where a is a positive constant. The function f(x) = 2x is called an exponential function because the variable, x, is the exponent. The inverse of an exponential function is called a logarithmic function. Therefore, the inverse of f(x) = ax is the logarithmic function with base a, such that y = loga x↔ ay = x. common and natural are two different types of logarithms.
BACKGROUND
Systems of linear equations arose in Europe with the introduction in 1637 by René Descartes of coordinates in geometry. In fact, in this new geometry, now called Cartesian geometry, lines and planes are represented by linear equations, and computing their intersections amounts to solving systems of linear equations. The first systematic methods for solving linear systems used determinants, first considered by Leibniz in 1693. In 1750, Gabriel Cramer used them for giving explicit solutions of linear systems, now called Cramer's rule.
Geometric methods were used to solve quadratic equations in Babylonia, Egypt, Greece, China, and India. The Egyptian Berlin Papyrus, dating back to the Middle Kingdom (2050 BC to 1650 BC), contains the solution to a two-term quadratic equation. Babylonian mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC used geometric methods of dissection to solve quadratic equations with positive roots. In 628 AD, Brahmagupta, an Indian mathematician, gave the first explicit (although still not completely general) solution of the quadratic equation ax2 + bx = c.
The possibility of defining logarithms as exponents was recognized by John Wallis in 1685 and by Johann Bernoulli in 1694. The invention of the common system of logarithms is due to the combined effort of Napier and Henry Briggs in 1624. In 1649, Alphonse Antonio de Sarasa, a former student of Grégoire de Saint-Vincent, related logarithms to the quadrature of the hyperbola, by pointing out that the area under the hyperbola.
MAKING A GRAPH OF LINEAR, QUADRATIC, EXPONENTIAL AND LOGARITHMIC FUNCTIONS
To graph a linear function, follow these steps:
Identify two points that satisfy the equation.
Plot them on the graph.
Draw a straight line connecting the points.
Example:
Y = 15 + 2x
When x = 1, y = 15 + 2*1 = 17
When x = 3, y = 15 + 2*3 = 21
So, points (x, y) are (1, 17), (3, 21).
To graph a quadratic function, follow these steps:
Identify four or more points (both +ve & -ve) that satisfy the equation.
Plot them on the graph.
Draw a curve connecting the points.
Example:
Y = x2
So, the points are (0, 0), (2, 4),
(-2, 4), (3, 9), (-3, 9).
To graph an exponential function, follow these steps:
Identify the satisfying value for exponential using table.
Plot the points.
Draw a curve passing through the points.
To graph a logarithmic function, follow these steps:
Identify the given function.
Plot the points just opposite of exponential function (i.e., y = x)
Draw a curve passing through the points.
Example:
Let,
y = log2x (logarithmic fun)
y = 2x (exponential fun)
For exponential,
When x = 0, then y = 1
In logarithmic values are just opposite (i.e., reflection though line y = x).
y = 2x
(0, 1) y = log2x
(1, 0)
y = x
DEMAND
Actually, demand is a desire supported by ability to pay and willingness to pay.
DEMAND FUNCTION
A demand function is a list of prices and the corresponding quantities that individuals are willing and able to buy at a fixed point of time. We may note at the outset that demand is a function (or schedule), not a specific quantity. It is formally defined as a schedule of the total quantities of a commodity or service that will be purchased at various prices at a particular point of time.
EQUATION OF DEMAND FUNCTION
Let us consider that the demand and price for copies are linked by a linear demand function of the form,
P = f (Q)
Where Q is the demand in dozens and P is the price in rupees.
We must compute the slope m in order to discover it.
m = (15-20)/ (10-5)
= -1
We can now write our demand function as with the slope, P = a – Q.
Substitute one of the ordered pairs into the equation to find a.
20 = a - 5
Or, 20 + 5 = a
∴ a = 25
P = 25 - Q is the equation of demand function.
SUPPLY
Supply is the total amount of a certain goods or services which are available to consumers.
SUPPLY FUNCTION
Supply function is a mathematical function or expression of the relationship between quantity demanded of a product or service, its price and other associated factors such as input costs, price of other goods, number of producers in the market, available technology, etc.
EQUATION OF SUPPLY FUNCTION
Let us consider that the quantity supplied and price for copies are linked by a linear supply function of the form,
Qs = f (P)
Where Qs is the Quantity supplied in dozens and P is the price in rupees.
We must compute the slope m in order to discover it.
m = (15-10)/ (10-5)
= 1
We can now write our supply function as with the slope, P = c + Q.
Substitute one of the ordered pairs into the equation to find c.
10 = c + 5
Or, 10 - 5 = c
∴ c = 5
P = 5 + Q is the equation of supply function.
PROFIT FUNCTION
A profit function is a mathematical equation that relates a company’s total profit to its output. It is maximal when the firm’s marginal revenue equal to its marginal cost and it equals total revenue minus total costs.
The difference between the revenue function R(x) and the total cost function C(x) is the profit function P(x).
As a result, P(x) = R(x) – C(x)
Profit = Revenue – Cost
P = R – C
For example: Find the profit function? If fixed expenses are Rs. 900, variable costs are Rs. 40 per item, and the selling price is Rs. 70 per unit.
Here, Let x be the total unit
Fixed cost = Rs. 900
Variable cost = Rs. 40x
Selling price = Rs. 70x
We know, to evaluate profit function, we need cost and revenue function.
So, Cost function C(x) = variable cost + fixed cost
= 40x + 900
Revenue function R(x) = 70x
Now,
Profit function P(x) = R(x) – C(x)
= 70x – (40x + 90)
= 30x – 900
Therefore, we obtain profit function as P = 30x – 900.
ELASTICITY OF DEMAND
According to the definition of demand elasticity, demand alters in response to price changes; this characteristic of demand is termed as Elasticity of demand (low or high). In the elasticity of demand, change sensitivity may be minor or non-existent.
To further comprehend the notion, let’s look at an example. Even a significant drop in demand for salt will not affect the decrease of its appreciable extension in demand. Similarly, a little decrease in the price of books will result in a significant change in demand. Demand elasticity can vary, but it is always either perfectly elastic or inelastic.
Mathematically,
Elasticity of demand = % change in quantity demanded
% change in determinants of demanded
TYPES OF ELASTICITY
Price elasticity of demand (Ed)
Price elasticity of supply (Es)
Income elasticity of demand (Ey)
PRICE ELASTICITY OF DEMAND
Price elasticity of demand is a measurement of the change in consumption of a product in relation to a change in its price. Expressed mathematically, it is:
ED = % change in quantity demanded
% change in price
TYPES OF PRICE ELASTICITY OF DEMAND
Elasticity demand
Inelastic demand
Unit elasticity
Perfectly inelastic
Perfectly elastic
PRICE ELASTICITY OF SUPPLY
The price elasticity of supply (PES) is the measure of the responsiveness in quantity supplied (QS) to a change in price for a specific good There are numerous factors that directly impact the elasticity of supply for a good including stock, time period, availability of substitutes, and spare capacity.
Mathematically,
Es = % change in quantity supplied
% change in price
TYPES OF PRICE ELASTICITY OF SUPPLY
Unitary elastic supply
Inelastic supply
Elasticity supply
Zero elastic supply
INCOME ELASTICITY OF DEMAND
Income elasticity of demand means the ratio of the percentage change in the quantity demanded to the percentage in income.
The demand for a product and consumer’s income are directly related to each other, unlike price-demand relationship.
Mathematically,
Ey = % change in quantity demanded
% change in income
Example:
Suppose the monthly income of an individual increases from Rs. 6,000 (Y) to Rs. 12,000 (Y1). Now, his demand for clothes increases from 30 units (Q) to 60 units (Q1).
Solution,
Ey = ∆Q/∆Y * Y/Q
Now,
∆Q = Q1 – Q = 60 – 30 = 30 units
∆Y = Y1 – Y
= 12000 – 6000
= Rs. 6000
We have,
Ey = 30/6000 * 6000/30
= 1 (equal to unity)
BUDGET AND COST CONSTRAINTS
BUDGET CONSTRAINTS
The budget constraints is the limit of the opportunity set all the different combination of consumption that a person can afford based on the cost of items and their income.
Equation of a Budget line
It is necessary to comprehend the aforementioned equation in order to fully comprehend the concept of a budget line. The following is a representation of the budget line equation:
M = Px Qx + Py Qy
Where,
Px denotes the price of product X.
The quantity of product X is Qx.
The cost of product Y is denoted by Py.
The quantity of product Y is Qy, and the consumer’s income is M.
Example:
Ram has 50 rupees to spend on an ice cream. He has a few alternatives for allocating his income to get the most out of his restricted pay.
The budget schedule offered can be outlined on a graph to acquire an appropriate budget line.
The budget implies that the combination of the two goods is within a consumer’s affordability range.
COST CONSTRAINTS
The cost constraints are a graphical representation of the possible combinations of two components, labour and capital, that a corporation or firm may buy or purchase with a specific amount of money.
Equation of isocost line
An isocost line is the producers what a budget line is to a consumer. While a budget line shows a consumer’s maximum income, an isocost line shows the maximum amount which a firm is willing to expend on production.
To calculate the isocost line for a firm, begin with the total cost equation, TC = (W x L) + (r x K) and solve for K. W= wages, L =labour, r = the rent (what you pay for the use of capital), and K = capital.
The interplay of a firm's isocost line and its isoquants determine the firm's production. The point at which the isocost line is tangent to the highest-possible isoquant is the point at which the firm maximizes its output keeping in view its cost constraints.
Example:
An isocost cost line can be drawn for any two factors of production if we know the total cost budget and prices per unit of each input. Assume you manage a large fast-food chain which has hundreds of outlets. You must decide whether to lease out self-serve kiosks (capital) or hire workers (labour). Your annual budget is $1 million, monthly lease cost of a kiosk is $500 and wage rate per hour is $15. There are two extreme cases of allocation of the budget between labour and capital: (a) spend the whole budget on kiosks and do not hire new workers or (b) hire new workers but no new unit of capital. In the first case, $1 million enables you to rent 2,000 kiosks ($1,000,000 divided by $500). In the second case, $1 million is sufficient to acquire 66,667 labour hours. If we plot capital on y-axis and labour on x-axis, there two cases give us two points on the graph: (0, 66667) and (2000, 0).
If C is total cost, r is the user cost of capital, w is the wage rate and K and L are the units of capital and labour respectively, we can write a general equation for isocost line as follows:
C = r *K + w*L
Converting this to the standard straight-line equation format, we get:
K = (C/r) – (w/r)*L
This shows that the slope of the isocost line is -w/r.
SUPPLY AND DEMAND ANALYSIS
Excess of demand when Qd > Qs, excess of supply when Qd < Qs and market equilibrium when Qd = Qs. In this section, we use quadratic functions to do the demand and supply analysis. We discuss this with the help of following example.
The demand and supply functions for a market are given by
Pd = 125 – Q2 and Ps = 0.2Q2 + 5 respectively. Find the equilibrium price and quantity.
Given,
Pd = 125 – Q2
Ps = 0.2Q2 + 5
For equilibrium condition, we have
Pd = Ps
Or, 125 – Q2 = 0.2Q2 + 5
Or, 1.2Q2 = 120
Or, Q2 = 100
∴ Q = 10
Substituting the value of Q in Pd = Ps , P = 125 – 102 = 25.
So, Equilibrium price = 25
Equilibrium quantity = 10
BREAK – EVEN ANALYSIS
In this section, we discuss break-even analysis with the help of quadratic functions. The concept is same as that of the linear function which was discussed earlier. The point at which total cost (TC) and total revenue (TR) are equal is called a break-even point.
We can say that at break-even the profit is zero.
OPTIMIZATION
A businessperson wants to minimize costs and maximize profits. In solving such practical problems, we have to convert the word problem into a mathematical optimization problem by setting up the function that is to be maximized or minimized
For the Quadratic function y = ax2 + bx +c (a ≠ 0), mainly there is two different cases.
If a > 0, then the parabola opens upwards and the vertex
[- b/2a, (4ac – b2)/4a] is the minimum point. That is minimum value of the function = (4ac – b2)/4a at x = - b/2a. In this case, maximum value of y doesn’t exist.
If a < 0, then the parabola opens downward and the vertex
[- b/2a, (4ac – b2)/4a] is the maximum point. That is, maximum value of the function = (4ac – b2)/4a at x = - b/2a. In this case, minimum value of y doesn’t exist.
Example:
The demand function of a good is given by P = 100 -2Q. Find total revenue function (TR) and sketch it. Find the maximum value of TR.
Here,
TR = P*Q
= 100Q – Q2
This is a Quadratic function
The main characteristics are:
Since the coefficient of Q2 is negative, the parabola opens downward.
When Q = 0, TR = 0
When TR = 0, 100Q -2Q2 =0
Or, Q (100 – 2Q) = 0
∴ Q = 0, 50
By symmetry, the parabola attains its maximum halfway between 0 and 50. Thus, the parabola attains its maximum at Q = 25
The corresponding revenue function
TR = 100(25) – 2(25)2
= 1250
Thus, the maximum value of TR is 1250 at Q = 25. This is shown in the following graph.
EXPONRNTIAL AND LOGARITHMIC FUNCTIONS IN ECONOMICS AND MANAGEMENT
Exponential and logarithmic functions are widely used in economics and management. These functions are used to calculate salvage value. Exponential functions are used to model population, composite investment, determine time of death, etc. and logarithm functions are used to calculate heavy calculation. They are also used to calculate complicated mode of time period.
Unlimited growth function
The function modelled by the equation f (t) = a ert, where a and r are constants, is called unlimited growth function. Investment and some models of population growth are the examples of unlimited growth function.
Unlimited decay function
The function modelled by the equation f (t) = a e-rt, where a and r are constants, is called unlimited decay function.
Limited growth function
The function modelled by the equation f (t) = M (1 – e-rt), where M and r are constants, is called limited growth function. Consumption functions, sales with advertising, etc. are some examples of it.
Logistic growth function
The function modelled by the equation f (t) = M/ (1 + a e-rMt), where M, a and r are constants, is called logistic growth function. Constrained population growths of epidemic, sales, etc… are some examples of logistic growth function.
Example:
The population of a nation is changing according to the equation
P = 5e-0.001t where t is in years, P is in millions. Calculate the number
of population at the end of 10 years.
Here,
P = 5e-0.001t
= 5e-0.001*10
= 5e-0.01
= 5/e0.01
= 4.950249169 million
= 4.950249169* 1000000
= 4950249.169
= 4950249
APPLICATION/ RECOMMENDATION/ PRACTICAL SIGNIFICANCE
Linear, Quadratic, Exponential and Logarithmic functions have a wide range of applications in real life. To use algebra to solve real-world problems, we generally prefer using the mathematical perspective that helps us to establish a kind connection between the unknowns (variables) and the knowledge available. While using mathematical perspective, following steps need to address:
Translate the problem statement into a mathematical statement and build it up as an algebraic expression in such a way that it accurately explains the situation.
Identify the unknowns in the problem and assign variables to these unknown quantities.
Read the problem several times and make a list of the data, phrases, and keywords. Organize the facts you’ve gathered in a logical order
Formulate an equation using the algebraic expression and the facts in the problem statement, and solve it using systematic equation-solving approaches.
Retrace your solution back to the problem statement and see if it meets the problem’s criteria.
So, there you have it!! Word problems can be easily solve using these methods and application of function.
CONCLUSION
Lastly, I would like to thank my subject teacher for this project. I have discovered the application of function has wide range coverage in real world. In fact these functions are being used from very earlier, since decades. Various infrastructures are built using such functions. Currently the uses of such functions evolving and emerging in the real world.
REFERENCES
Business Mathematics book (grade XI, Asmita publication), Wikipedia, investopedia, education world, spark notes, oasis business mathematics and other reliable sources I have used while creating this project
Some of the websites sources: