Let's start application of calculus in commerce and economics. So, this particular topic is very easy. We will read definitions in this a little earlier, so that we can understand the things given here. Now, if you are a science student, then you may have a little difficulty in understanding things, because this is not taught in science. If you are a commerce and arts student, you will understand easily, because you are already well versed in all these things.
You must have studied all this in economics. Let's start. So, as I told you earlier that we have studied DY by DX. In economics and commerce we come across many such variables.
Objectives after studying this lesson you will be able to define. Now see what all we will study and what all are in our course. So, what all I am going to teach here are all in your course. So, first of all we will be talking about what is cost function?
What is variable cost? What is total cost? What is average cost?
Then what is marginal cost? First we will study about cost, then we will study about total revenue, marginal revenue and average revenue. So I will give you some examples about cost.
If we start any business, the money that we have to invest in that business is known as cost. So when we start any particular business, if we are selling any product or service, what we have to give before that is, you have to invest in the form of land, electricity bill, machinery purchase you have to hire skilled labour and unskilled labour, which comes in cost we break cost into two parts, one is variable cost and other is total cost we will tell you everything about total cost and variable cost then when you sell something, what do you get? a lot of money when you get a lot of money, what happens? a lot of dance when you get a lot of money basically the money you are getting is your revenue revenue means the money earned by the company by selling some of its products now in revenue also, average revenue, marginal revenue, total revenue you will know what is this slowly after that we will talk about When you make things, you put a lot of money and after that you sell a lot and earn a lot of money and when you earn a lot of money, it becomes revenue If you earn more money than what you have invested, will there be profit or loss? There will be profit And if you invest a lot of money and get a little money, what will happen?
There will be loss and the household will be sold So we will talk about the profit and loss system Profit and loss depends upon two things The first is the profit and loss system The cost function, the total cost that we have spent and the total revenue that we have generated. So revenue more, cost less, profit, revenue less, cost more, loss. You have read this in the chapter of profit loss in the form of CPSP. Then total revenue, cost revenue, marginal cost, what is marginal, let's talk about all these things.
Let's start. First of all, let's talk about cost function. What is cost function? So, as you can see here, the total cost C of producing and marketing X units of product depends upon the number of units.
So, if we are manufacturing something, then in manufacturing that thing, if we are saying that we have manufactured X number of units, then the cost we are applying to produce that X number of units is If we write it in the form of a function, in terms of x, then we call that function cost function. Clear? So what does cost function tell you? Cost function tells you that the total cost that you are spending in manufacturing a particular product, if you are manufacturing x number of products, then the cost function c in that is known as cx.
Now what does cx mean? Cx means nothing. CX means that this is a particular function which will be in terms of X. What is X? Number of units produced.
How many units you have produced. Okay. So the total cost of producing X units of the product consists of two parts.
Now to produce anything, two types of costs are required. What are the costs? One is fixed cost and one is variable cost. Now as we can see from the name, fixed cost. What is fixed cost?
Can you tell us, sir? tomorrow my maths practical is there please help me Maths practical is to go without any hesitation, be smart and go do good afternoon, good evening, sit down if they ask you, tell them, if not, don't tell we will get 20 out of 20, don't worry don't know vectors at all, no problem, we will teach you vectors too Yes, tell me, there are two things in cost function, one is fixed cost and one is variable cost. Fixed cost means the cost which is fixed, as the name suggests.
Means if you want to start your business, if you have manufactured any product, then the fixed cost does not depend on how many items you have produced. For example, let's say you have opened a company and you are selling in that company. What will you sell?
What are you manufacturing today? You are manufacturing slippers. For example, you have a company and you are manufacturing slippers.
If you are manufacturing slippers, then the cost will be fixed on the machine. The cost on the land is fixed. Means the cost you have to pay once, whether you produce one or a thousand slippers, you will have to buy the machine, you will have to buy the land. So that is known as fixed cost.
Fixed cost does not depend upon the number of items produced. So please write it quickly. Put the cost function heading.
Whatever I am writing, just write that much. Open it before the exam and see that it will be a great revision. There will be no need to do anything else.
So the cost function is made up of two things. One is fixed cost and one is variable cost. Fixed cost is applied once which is repeated. No, it does not recur. Fixed cost, you can write an example like machinery, the cost you have put on the machine, the cost you have put on the land, all this is fixed cost.
Now irrespective of whether you are producing one item or 100 items, fixed cost is the same. Variable cost, what is variable cost? So variable cost, as you can see from the name, is variable. So this cost function, this type, this part of the cost function, this is the variable cost.
Variable cost is a cost that re-occur, that keeps on increasing The number of units you produce, the same way your variable cost will keep on increasing So let's say, not ours, there is a very good brother among us His name is Kartik, he owns a factory of kattas in Lucknow, he makes desi kattas So we will take the example of desi kattas here So the variable cost will be, we will take iron from the desi kattas, which will make these desi kattas That will be the variable cost the labor that they will bring to make it will be the variable cost because if there is more labor, there will be more production, more cuts will be made if they will make it, they will sell it third, electricity now, in molding it, in bending it, whatever electricity cost you are giving the more electricity you use, the more your variable cost will increase so what does variable cost mean? a cost that re-occurs, that depends on the number of items produced understood till here? so cost function is made up of two types of things, fixed cost and variable cost I have told you about fixed cost and variable cost now comes demand function, listen carefully to what it is what is demand function? demand function is an equation that relates price per unit and the quantity demanded at that price this is called demand function, you know what demand means? how much is its need That is known as demand So anything like iPhone has a lot of demand Why does iPhone have a lot of demand?
It becomes a status symbol that I bought an iPhone, it's expensive I feel like I'm rich Or it has a good IOS It has good security And the camera is good, whatever, xyz reasons So that becomes demand So demand depends on On what? On price If something has a lot of demand Then what happens to the price? demand is very less, so what happens to the price? now they have orders of kattas, let's say they have produced 100 desi kattas 200 people came to buy 100 desi kattas now if they were selling 1 desi katta for 1000 rupees in the market now because they have 200 people, they can't give it to all they will give half of the katta, it won't work, what's the use? so what will they do?
they will increase their cost the amount they were selling So as a result, because demand was increasing, so they had to increase the supply. So what is the demand function? We write it like this.
P is the price per unit. This price per unit depends on a function which is in terms of x. That means which is in terms of the number of units. produced.
So the number of units produced depends on P or P depends on the number of units. You can move from LHS, you can move from left to right, you can move from right to left. If your price is low, then demand will be high. If price is high, then demand will be low.
This is how they are related to each other. So, this is a function which relates price to the number of units produced. That function is known as a demand function. That function is known as a demand function. After that, let's talk about revenue function.
As you know, revenue function, what does revenue mean? It is a function of the number of units produced. The money we earned by selling our X items So if X is the number of units of certain products sold at a price of say P per unit What is the price of one unit?
P So what will be the price of X units? P x X If we are selling a piece of Rs.1000, then how many pieces will we sell? 10,000?
10 x 1000? 20? 20 x 1000?
20,000? So this revenue function which we write in the form of Rx, that is equals to P into X. P is the price per unit.
X is the number of units sold. Okay. So this is your revenue function. So revenue function depends on price and number of units, the number of units we sell depends on that. So revenue function R is equals to P into X which is known as the total revenue.
This is our total revenue. Because how much we have? by selling our item then we have profit function what is a profit function? profit function means when our revenue will be more and our cost will be less profit is equal to revenue minus cost if we write profit function in the form of function what will be the profit function? profit function will be revenue function minus the cost function if you minus cost function from revenue function then then you will get profit function after this comes very important break even point so you will get a lot of question or even some questions on which we will talk about break even point break even point means break even point is a point where you have sold some things let's say he made some desi kattas let's say he made 100 kattas out of 100 kattas he sold 50 kattas let's say he made 100 kattas The cost was 500 rupees per piece.
So how much did it cost? 5000 rupees. What happened to you? Cost?
Now he sold 50 pieces and he was selling one piece for 1000 rupees. So if he was selling one piece for 1000 rupees, then he, sorry, if he was selling one piece for 1000 rupees, then how much did he sell? 50,000 and here also 50,000 will come. Okay?
So basically, the cost was, to make 100 pieces, according to 500 rupees, the total cost was 50,000. he sold 50 cuts in 1000 rupees, even then the revenue generated from here is 50,000. So, break-even point is that particular point at which the amount of cost you have invested, you got that amount of money back.
The amount of money you invested, you got that amount of money back. This means, break-even point is that point where the cost is equal to the revenue. Where the cost is equal to the revenue, that point is called break-even point. Now if cost is equal to revenue, then we have just studied profit function, so what is equal to profit function?
Revenue function minus cost function. So at break even point, if we talk about break even point, then what will happen at break even point? If these two will be equal to each other, then how much will be the profit function? It will not be zero, sir.
How much will it be, sir? It will not be zero. Okay, so this is our break even point.
So at break even point, the revenue function is equal to the cost function or we can say the profit function is equal to zero you can say anything from these so you are going to get MCQs in section C you will get MCQs of 5 marks of each number in that some basic things can be asked in MCQ format that which of these is true at break even point so you have to choose from that is this clear? is everyone clear till here? let's move ahead now let's come to the questions we will do these questions For a new product, a manufacturer spends Rs. 1 lakh on the infrastructure and the variable cost is estimated to be Rs. 150 per unit of the product. The sale price per unit was fixed at Rs. 200. Find the cost function, the revenue function, the profit function and the break-even point. You have to find all these things.
You will be clear with this particular question. Now what is said in this question? A new product was found in the market.
The fixed cost was? 1 lakh rupees. Fixed cost is how much? 1 lakh rupees. So 1 lakh rupees is the fixed cost.
Then it says the variable cost is rupees. Variable cost is rupees 150. 150 per unit. That means, the cost of producing one unit is 150. If we assume here, let the number of units produced be equal to x.
If the number of units produced is x, then the variable cost will be 150 into x. If it costs 150 rupees to make one item, then how much will it cost to make 10 items? 150 x 10 How much will it cost to make 20 items? 150 x 20 How much will it cost to make X item?
150 x 150 So this is the variable cost. Now if we see the cost function from here, then cost function is a combination of, is the sum of fixed cost and variable cost So what is the fixed cost here? 1 lakh rupees. So 1 lakh rupees is the fixed cost plus variable cost. What is the variable cost?
150x, so what is this? Cost function. So in our first part, we said we have to find the cost function. So here comes our cost function.
No problem till here. Cost function is cleared. After this, what is he saying?
After this, he is saying find the revenue function. Now if we talk about revenue function, so revenue function is equal to price per unit into the number of items produced. Now here in the question, it is given that the product, the sale price per unit was fixed at Rs. 200. So how much are we selling? One item of 200 rupees.
If we produce X items, then we will only sell X items. We will sell as much as we produce. So, the revenue function will be equal to 200 x X.
200 is the selling price of one item into the total number of items, that is X. Is it clear? This is the revenue function.
Now, what is he asking you in the third part? Tell me the profit function. So, the profit function will be equal to... Revenue function minus the cost function.
So, this will be 200x minus, write in bracket so that there is no mistake, 1 lakh plus 150x. So, revenue function minus the cost function. Now, if we minus this, then who will get it?
This will be 50x minus 1 lakh. So, 50x minus 1 lakh is your profit function. Who is this equal to? It is equal to the profit function.
Okay? After that, what is he saying to you? The break-even point. What will be the break-even point for this particular situation?
So when does the break-even point happen? When our profit function becomes equal to zero. Profit function is 50x minus 1 lakh.
From which we equate this? From zero. And from here, whose value to be taken out? The value of x. How much will be the value of x?
How much will be the value of x? So, 1 lakh divided by 50. 2, 0, 0, 0. 2,000, no, it's not 2,000, the value of x is 2000. So, how much revenue function has come here? How much break-even point has come?
When you sell 2000 units, then how much money you have invested, you will get that amount of money back only at 2000 so it is saying that what is the break even point here see what is the second example saying here so the second example is saying a company produced a product with rupees 18000 as fixed cost so what is 18000? fixed cost FC this is fixed cost the variable cost is estimated to be 30% of the total revenue when it is sold at a rate of rupees 20 per unit now see there are very good things here You will have to understand English, if your English is strong then you will understand this The variable cost is estimated to be 30% of the total revenue What does off mean? It means multiply So the total revenue of the variable cost is 30% Now if we want to calculate the variable cost then what do we have to calculate?
Revenue because if we calculate the revenue then only we can calculate 30% of it have we been given any information to calculate the revenue? yes sir when it is sold at a rate of Rs.20 per unit we are selling one for Rs.20 so if we are selling one for Rs.20 then what will be our total revenue? so if we assume that the items that are being produced are X number of items then what will be the total revenue? 20X now what will be the total revenue of 20X?
30% So 30% of 20x. So 30 upon 100 into 20x. Zero se zero gaya, zero se zero gaya, kitna gaya?
6x. Toh variable cost kitna gaya? 6x ke bharabar. Is this clear to each and every one of you?
Kya yaha tak sabko clear hai? So we have taken out the revenue function. How? We have multiplied the SP of one item by the number of total items sold. 20X is here.
We have taken out 30% of it. 30% of 20X is 6X. So we have found out 6X. Now what is it saying from here? It is saying, find the total revenue, total cost and profit function.
Will you take it out? Who will be equal to the total revenue? 20X which we have already taken out. So we have got the first part of total revenue.
What is it saying in the second part? In the second part, it is saying total cost. So to calculate the total cost, you have to calculate the total cost equal to the fixed cost plus the variable cost. What was the fixed cost? The fixed cost is 18000 rupees.
So 18000 is the fixed cost plus the variable cost is 6x. So this is the total cost. In the third part, what is being said? The profit function. So the profit function will be equal to the revenue function minus the cost function.
Revenue function was 20x minus 18000 plus 6x. So this will be 14x minus 18x. Let's come to the next question. It says, a manufacturing company finds that the daily cost of producing x items of a product is given by Now see what is this given? Can someone tell us?
Look at it quickly and tell us. What is this given? Cost function.
Very good. This is the cost function. In the first part it says, if each item is sold for Rs. 350, then what is this 350 given?
SP is given, selling price. What is this? SP is given. Find the minimum number that must be produced and sold daily to ensure no loss. How many items are made daily so that we don't have any loss?
Now, if there is no loss, it means there is no profit. Okay. Means, in the first part, what do we have to find? Break-even point. Break-even point.
Now, here they have not directly said break-even point. They have told the story. But what do you have to find?
Break-even point. Now, how will you find break-even point? You have been given cost function. You can take out revenue function from here. Revenue function will be equal to SP into number of items sold.
So SP is 350, number of items sold. If we assume that it is X, then how much will it be? 350X.
Now, when does our break-even point come? When revenue function is equal to cost function. Either you say like this or you can say like this also that revenue function minus cost function equals to 0. You can say like this also.
Profit function is 0. That's why there will be no loss and no profit. So if we equate these two together 350x is equal to 210x plus 7000 So by solving this equation we can get the value of x Say yes or no We can get the value of x from here Single equation, single variable We will solve it and we will get the answer So this was the first part Let's come to the second part In the second part it says If the selling price is increased by rupees 35 per piece What does buy mean? The selling price is increased by 35 rupees per piece.
Har piece ki jo selling price hai, wo 35 kar di jae. Ya har piece ki jo selling price hai, wo 35 rupees badha di jae. Batao. Very good. Har piece ki 35 rupees badha di jae.
Increases by. Increases by means, uski value 35 rupees badha di jae. Toh pehle kitne ki bik rahi thi? 350 ki.
Ab kitne ki bik kegi? 385 So if we take out new SP from here, how much will it be? 385 Okay, it is saying what would be the break even point?
Now tell me the break even point So we will do the same now Now the revenue function will change here What will be the revenue function? This will be 385X And if we want to take out the break even point Then we will equate it with the cost function Which is 210X plus 7000 Okay, so see here when we When we equated this in the previous one, then what was the value of X? The value of X was 50 and when we increased it by 35 rupees, then what was the value of X? The value of X was 40. Obviously, when you sell on more money, then the break-even point will be at selling less. If you sell less, then you have to sell more, then the break-even point will be there.
So, this thing is clear here. So we have cleared the first topic. Now let's move to the next topic. Because the other questions are the same.
Chintu questions are the same. You can do them easily. What do you need to understand here?
Cost function. Cost function is made up of two things. From whom?
Fixed cost and variable cost. What is revenue function? Price per unit into the number of units. What is profit function?
Revenue function minus cost function. When does break even point come? When revenue function is equal to cost function or profit function is equal to zero then the first exercise is over.
Is it easy? Now tell me how easy is Section C? People say that Section C is difficult.
No problem, it's easy. Everything is like that. Everything will end in a story. Then we have, the next topic that we have is average and marginal functions.
Now you can understand it easily. Now see, if you have given two quantities, x and y. Right? Two quantities, x and y.
We are writing y in the form of a function, which is a function in terms of x. So y is equal to fx. Right?
So average. What is average? What does average mean? Mean, what is mean? Mean doesn't mean anything.
Mean means average. Average means mean or central value. You had read a formula to find mean in your childhood. Sum of observations upon number of observations. She is not laughing.
How do you know sir? How do you know what I have been through? It looks like a face.
No problem. Let's see. Mean is average. If you want to find the average cost of anything, if you want to find the average cost, average cost, how can you find the average cost?
You divide the total cost with the number of items. Now, the number of items we are producing, selling, what are we taking from this? X.
So, total cost divided by the number of units produced or sold, that is known as average cost. Right? This is how average revenue is. So what will be the average revenue?
Total revenue that is P into X divided by the number of units sold. Now what will happen here? If X is deducted from X, then it will be equal to what? Price per unit, not profit.
Profit is different, this is price per unit. The shoe you sold is your average revenue. The shoe they sold is average revenue. Okay? Is it clear?
This is the average revenue. what will be the average profit? average profit will be total profit that is profit function divided by the number of units sold just like that, divide it by x and you will get the average if it comes to cost then divide it by x and if it comes to revenue then divide it by x and if it comes to profit then divide it by x clear?
so this is known as average, now we are telling you about average cost this is average, ok, this story is not easy to understand this is how it is understood in desi language, after this comes marginal does anyone know what is marginal? if we talk in mathematics language if your word marginal comes anywhere, what word will come? marginal, so you have to find d by dx of that function d by dx, if marginal comes then what to do?
D by DX, we have to differentiate it in respect to X. What you will get? Marginal. Now, how can we do it?
So, let's say you have given cost function. You have to find marginal cost function. So, as soon as you know marginal cost function, as soon as you hear the name marginal, then what you have to do?
You have to make cost function D by DX. That means, you have to differentiate it with respect to X. If you have marginal revenue, so, D by...
So, did you understand marginal? Did you understand? Tick. So, let's talk about marginal cost.
So, marginal cost is d by dx of cost function. Okay. Let's take one cost function.
Let's say our cost function is let's say 3x square plus 2x plus 3. And we have to find the marginal cost function. So, how can we get marginal cost function from cost function? By differentiating cost function. So, if we differentiate it, what will we get?
6x plus 2. So, what will we get when 6x plus 2 comes? marginal cost short form kya hota hai MC then you will start laughing, wrong thing bro MC means marginal cost Maheshwari classes also happens, move ahead so if MC comes to find you then you have to do cost function D by DX, you have to differentiate it with respect to X you will get marginal cost, now see the example you will understand a lot from the example here For example, if the cost function is 0.2x square plus 5, then the marginal cost is 0.4x. We have differentiated it. Therefore, the marginal cost when 5 units are produced is, if we have to calculate the marginal cost when 5 units are produced, then what will we put in place of x?
So what we did is, when we took out marginal cost at x is equal to 5, we multiplied 0.4 by 5. So how much did we get? 2. Now what is this 2 telling you? What is the meaning of this 2? The meaning of this 2 is, When the production is increased from 5 units to 6 units, then the cost of additional unit is approximately rupees 2. So when we increase the production from 5 units to 6 units, then the cost will be approximately 2. So this marginal cost gives you an approximate value, not an exact value.
Marginal function, marginal cost gives you the approximate value So if you calculate the marginal cost, let's say at x is equal to 10 So if our marginal cost function was 0.4, then what will be the marginal cost at x is equal to 10? 4, so this means that when we produce the 11th unit from 10 units, then what will be the additional cost? approximately 4 rupees not exactly 4 rupees approximately 4 rupees this is marginal cost question has come on this last year that's why I am giving emphasis on this ok now see the question question is the cost function of a firm is given by this, find the average cost find the marginal cost when x is equal to 4 So we have given cost function.
What is equal to it? 2x square plus x minus 5. What we have to find is average cost. To find average cost, what we have to do is divide this cost function by x.
When we divide it by x, what will we get? 2x plus 1 minus 5 by x. That's it. Let's come to the second part.
It says marginal cost. So marginal cost means, differentiate the cost function. If you differentiate this, what will you get?
4x plus 1. This is marginal cost. Then what is being said? Marginal cost when x is equal to 4. So now what to do?
Keep the value of x as 4. So how much will be the marginal cost when we keep the value of x as 4? 17 will come. Very good. When we are producing the 5th unit from the 4th unit, then the approximate cost which is being taken to produce the 5th unit from the 4th unit, how much is it taking? 17 rupees.
Okay? So this is all about marginal and average. Let's give one more example.
Ok, you will not get such type of questions. Show that, slope of this, slope of that, leave it. I have given such questions in sheet.
I have given in book also. You don't need to do such questions. You will not get questions like prove that.
Section B and C, this is scoring. They don't want to make you do PhD in this. Ok. So, leave such type of questions.
questions like prove that. You will not get such long questions. Ok. Now see this.
Now this is also a question like prove that. Yes, see this question. This is a good question.
See this question carefully. So this is an increasing and decreasing question. This is a good question.
You can get such questions in your paper. What is he saying? If the total cost function of a product is given by What is this? Total cost function.
Means it will have variable cost and fixed cost also. Prove the marginal cost falls continuously as the output increases. So as you increase the x, your marginal cost will decrease.
You have to tell this here. So as x increases, the marginal cost will decrease. Now let's understand how to solve this question. So first of all we have given cost function.
First of all what we get out of this cost function? Marginal function, marginal cost, MC. To get marginal cost, we have differentiated this cost function.
Now to differentiate, what rule will we have to apply here? Question rule. By applying question rule, we have differentiated this. So let's move ahead, let's come directly here.
So when we differentiate marginal cost, So, what is the difference? This is our marginal cost. Now, look here.
It is clear that when x increases, as you increase x, what will happen to the whole fraction? Will it be less or more? It will be less. As you increase the denominator, if it is numerator upon denominator, as you increase the denominator, what will happen to the total fraction? because if denominator is bigger then the whole fraction will be less so as the value of x increases, the whole value will decrease so what we can see here is that as we increase the production, the marginal cost will decrease so you have to write this line Show that, you have shown that when marginal cost is taken out, as x is increasing, marginal cost is decreasing.
It is clear. After this comes the average revenue, marginal revenue. What to do for the average revenue?
To divide the revenue function by x. What to do for the marginal revenue? To differentiate the revenue function.
The talk is over. Let's come to the questions directly. So this is the question 41.7 It says the total revenue received from the sale of x units of a product is given by this What is this?
Revenue function is given It says take out the average revenue So what will we have to do to take out the average revenue? We have to divide it by x So average revenue will be equal to 12 plus 2x plus 6 by x Every term is divided by x Right? marginal revenue, so to get marginal revenue, what will we have to do to get MR? we have to differentiate, so what will come? 4x plus 12 will come as marginal revenue or you write 12 plus 4x, then also the answer is the same then it says marginal revenue at x is equal to 50 so what does this mean?
instead of x, what will you put? 50 so MR at x is equal to 50 so 4 x 5 is 20, how much will it come? 212 so what does 212 mean?
Very good. When we sell items from 50 to 51st, we will produce. Here, because of the revenue, we will sell.
That's the difference. When we sell items from 50 to 51st, how much will we get? Approximately, Rs. 212. Now, see what he is saying after this.
In the fourth part, he is saying, Find the actual revenue from selling the 51st item. Marginal revenue means that when we sell 50 to 51st item, we will get approximately Rs. 212. Here in the fourth part, it is saying the actual revenue from selling the 51st item. How much money will be earned in actual, when selling the 51st item? So what we have to do is, we have to get revenue for 51 items minus revenue for 50 items.
So what does X tell in revenue function? Number of units sold So if we subtract R51, what does R51 mean? The total revenue we got, the total money we got 51 by selling desi kattay Will come What we have to subtract is that when we went from 50 to 51 Then how much revenue did we get? So for this what we will do We will subtract revenue function on which?
On 50 total 50 desi katte bhechne pe humko kitna paisa mila in dono ko kya karenge subtract karlenge sab subtract karlenge to humko jo value milaegi wo kya mil jayega exact value jo aapko 51st katte ko bhechne pe milaegi clear hai thik to yaha pe r51 minus r50 kiya jab dono ko aap subtract karoge aya se to subtract karke aapko kya mila hai dekho 214 ruphe kya mila hai 214 ruphe ye jo 214 hai this is the exact value. How much was the approximate value? 212. Now you might be wondering, when we can know the exact value, why should we take the approximate value? We take the marginal value because it is very easy to take the marginal value. If you differentiate the revenue function and keep the value, then taking the approximate value is a very fast concept, very fast process.
But taking this is a lengthy process. Now how much is the difference? It is only 2 rupees, but you can save a lot of calculations.
Okay? That's why we use marginal revenue. Our calculation gets reduced there.
We are talking about when our revenue function is very small. If we take this revenue function of Ambani ji, that how many mobile we sell, how many fruits we sell, how many clothes we sell, then it becomes a very big revenue function. In that case, it takes a lot of time to compute.
That's why the concept of marginal revenue comes. Okay? See what the next question is saying. Now this next question is also a good question. You can also get such questions.
What is this great question saying? It is a great question, it is given in your book too It says, the demand function of a product for a manufacturer is x is equal to ax plus b What is this? Demand function And what is this demand function?
Is it linear, quadratic, cubic? What kind of function is it? It is a linear function How did you know? Because x is equal to 1 So this is a linear function You had read the linear equation in your childhood y is equal to mx plus c This equation is a linear equation It is the same equation So this is a linear function, demand function. He knows that he can sell 1250 units when the price is rupees 5 per unit.
How many units can he sell? 1250 units. At how much?
At 5 rupees. And he can sell 1500 units at a price of 4 per unit. He can sell 1500 units. At how much?
At 4 rupees. Obviously, if he reduces the price, then units will be sold more. Find the total average and marginal revenue function. Also find the price per unit when the marginal revenue is 0. So we had y is equal to this.
x is equal to ax plus b. What do you keep in place of this x? Price function is equal to ax plus b. This is how you made a function. Now what will you keep in place of p?
Price per unit. And what will you keep in place of x? Number of units sold.
So from here you will get two equations, two variables, two unknowns. Which are the unknowns? A and B. So when you put the first unit, the first value, it was 5. 5 is selling 1250 units.
1250 A plus B. What is your first equation? Second equation, when it is selling at Rs. 4, it is selling a total of 1500 items plus B. What is your second equation? Now two equations, two unknowns.
Did you learn to solve in your childhood? Which chapter was it? Simultaneous Linear Equation It was the chapter of class 10 If you solve both of them simultaneously then what will you get?
You will get A and B When you solve both of these equations then what value will you get? A and B When you get the value of A and B then what will be the value? A will be minus 1 upon 250 and B will be 10 So from here, the demand function Px, who is equal to that? This is equal to AX plus B So A is equal to minus 1 upon 250X plus B How much is the value of B?
10 Now what is this? Demand function Demand function equates On one side is price and on the other side is number of units produced So demand function adds two things Adds what? Adds price and number of units produced So this is your demand function Now if this is your demand function, then in which terms is this demand function? In price terms. If you want to find total revenue, then total revenue is equal to price into number of units sold.
Right? How much is the price? This much. How will you multiply this?
With x. So what will you get? Total revenue. So how much will be the total revenue?
Minus 1 upon 250 x square plus 10x. What is this? Total revenue. After this, what is he saying? If you take out the average revenue, then the average revenue will be equal to this.
This is equal to this. The average revenue will be equal to this. Right? After that, what is he saying?
If you take out the marginal revenue, then what will you have to do to take out the marginal revenue? Revenue, total revenue function will have to be differentiated from x. This is the total revenue function.
If you differentiate it from x, then what will come? This will come as minus 2 upon 250x plus 10. You cancel it a little more. So, minus 1 upon 125x.
plus 10. What is this? Marginal Revenue. So as soon as your total revenue function comes, everything else is easy.
To get the average revenue, divide it by x. To get the marginal revenue, divide it by x. You will get the complete sheet. I will share the complete sheet with you.
Download the Maheshwari Classes application, whose link is given in the description below. Go to the free study material, I will give you the complete sheet there. Next question is, the demand function of a monopolist.
Monopolist. is given by p is equal to 1500 minus 2x minus x square Do you know what is monopolist? Monopoly means that you are selling what you are selling and no one else is selling Can you tell me the name of a company that does monopoly in the market? That company is Boeing They make aircraft Boeing company makes aircraft At one time it was monopolistic It was the only company that made aircraft for the whole world It made aircraft for the whole world It would feel like today we will sell an aircraft for 100 rupees So it will sell for 100 he will feel like selling the same aircraft in 10 crores tomorrow because there is no competition, that is known as monoplistic after that, another company has come in it, Airbus so nowadays there are two types of aircraft, either Boeing or Airbus if you travel in an aeroplane, you will know either Boeing or Airbus so now that monoplistic is over, that is known as monoplistic so the demand function of a monopolist is given by P is equal to 1500 minus 2x minus x square find the Revenue function, find the marginal revenue, find the MR1x is equal to 20. Now, this demand function that you have given, according to whom is it given? P here is price and x here is number of units produced.
Because the demand function relates to whom? It relates to price with number of units produced. So, if this is demand function, it means this is price and this is number of units produced. Now, it is telling you to find revenue function. So, to find revenue function, what will you have to do?
With whom will you have to multiply this? With x, you will have to multiply. What will you get? Revenue function.
So how much will be the revenue function? 1500x minus 2x square minus x cube. What is this?
Revenue function. After this, what is he saying? Take out marginal revenue.
We will take it out, sir. What will we have to do for marginal revenue? Revenue function will have to be differentiated from x So how much will it be?
1500 minus 4x minus 3x square This is the marginal revenue Now it is saying find the MR when x is equal to 20 Now what you have to do is Keep 20 here Just keep the x where it is coming from Keep 20 and what will you get? Marginal revenue So if you keep 20 here What will be the value? Calculate it and tell me. So, it will be 1500 minus 80 minus 2400, 1200. So, if we get Rs. 1200, 300, 300, then it will be Rs. 220. It will be Rs. 220. Now, what is this Rs. 220?
Rs. 220 means that when we sell the 21st item from 20 items, we have sold 20 items. After 20, when we sell the 21st item, Rudranj, you didn't understand. I am telling you again out of your love.
When we sell the 21st item from the 20th item, then approximately how much revenue will we get? Rs. 220. revenue will be given. We will gain the revenue of Rs.220 when we will be selling the 21st item. We will be getting approximately Rs.220 as the revenue.
So let's say in one question, it is given that the cost function is given, let's say 2x square plus 3x plus 4. This is the cost function given. Now, in this cost function it is asking you, it is saying that tell me here that this cost function for which value of x will be minimum. So basically we have to minimize this cost function.
So objective function, whichever you want to maximize or minimize, you differentiate that particular function. By differentiating x, keep it equal to zero. When you keep it equal to zero, then what will you get here? You will get a turning point.
At this turning point, either there will be maximum value or minimum value. Now how will we decide that the value of x is minimum or minimum? is maximized or minimum or maximum, then what we have to do is double differentiate this function.
We have to differentiate this function again. By double differentiating this, two options can be made. Two outputs are possible. The first output is that after double differentiating the function, the value that you are getting is either a positive value or a negative value. If the value is positive, that means this is the point of minima.
If the value is negative, then the value is negative. that means the value the point is maxima. So it is opposite. Maximum on negative and Minimum on positive as you have read in AOD.
This is the same thing which I am teaching you again. So now we come to this particular chapter in which we have to talk about Minimization of average cost or total cost and maximization of total revenue or total profit. So who have to minimize? Our cost. Who have to maximize?
The revenue. profit to her company a chati a key profit career we company is in a day to day chati am jada jada no time right the profit function go home go increase gonna now up the key average cost function profit function is about to put them in direct questions app to batata on kick is the regis and questions for solving so question that the example a happy this example to solve for the example make a row ahead the manufacturing cost of an item consist of rupees 6000 as overheads material cost is rupees by per unit and labor cost is x square by 60 for x units produced so first of all what we can do is first of all we can find the value of the cost function first of all we can find the cost function so cost function is fixed cost fixed cost is 6000 plus variable cost variable cost is 5 rupees per unit so if we are saying that X units are produced so the variable cost will be 5x plus labor cost that is x square by 60 so here labor cost is given First of all, we have to find the cost function. It says, find how for how many units must be produced so that the average cost is minimum. So, we have to identify who to minimize first.
So, here who to minimize? Average cost. We have found the cost function. From the cost function, we can find the average cost.
So, how will the average cost function be found? To find the average cost, we divide this whole equation by x. If we divide this whole equation by x, we will get 6000 upon x plus. 5 plus x by 60 so this is the average cost function.
It is saying that how many units should be produced so that our average cost becomes minimum. So how many units should we produce to minimize the average cost? Our objective function is to minimize the average cost.
Let's differentiate this with respect to x. So d by dx of average cost. This will be equal to minus 6000 upon x square plus x upon 60. So this is our differentiation of average cost. Where we have differentiated the objective function.
Now what we have to do is, differentiate and equate it with 0 when we equate it with 0 then the value of x will come from here so the value of x will come from here x value will come 3600 after solving it completely and taking LCM when we do this then 3600 sorry not 3600, it will come 600 let's see how it will come, I will solve it completely so if we take LCM here then it will come x square into 60 So, this will be 16 to 6, 6 is 36, minus 36, 1, 2, 3, 4, plus x square, so here x cube is equal to 0. Now, if we take this whole part over there, then it will be equated with 0. Right, if we take it over here, it will be equated with 0. So, it will be x cube is equal to 36. 100, here x will not come, when we differentiate it, it will be 1 upon 60, 1 by 60 will come, x will not come, so here x square will come, so x square is equal to 36, 1, 2, 3, 4, 4, 0, so x's value will come, its square root, square root, then it will be 600. So the value of x is 600. Means the items that are being produced here, how many are they producing? 600. The average cost on 600 items will be minimum. so the average cost is minimized at x is equal to 600 so the value of x is 600 now what happened after this?
it is saying that find how many units must be produced so that the average cost is minimum so we have removed units produced now we have to minimize the average cost how can we minimize the average cost? so to minimize the average cost how will we know whether it is maximization or minimization? so what we did is we differentiated the function we will differentiate it again so again differentiating we have to write this again differentiating the average cost function if we differentiate it again, it will be d square by dx square of AC that will be equal to, if we differentiate it again, it will be x to the power minus 2 will go ahead, so it will be 12000 upon x cube and this will be 0 so 12000 upon x cube Now if you put the value of x as 600 or if you put any positive value instead of x, then how will this value always come?
It will come positive, that means greater than 0. If the value is greater than 0 when we double differentiate, then it means it is positive. Positive means point of minima. So we will write in the bracket point of minima.
Since double differentiation of the average cost function is a positive value, therefore this is point of minima. Means minimum cost, average cost function is positive. tab aayega jab hum 400 sorry 600 units produce kar rahe hai.
I hope this is clear to each and every one of you. Sabko sab. Iske baat next question dekhte hai. Next question me kya aara hai.
Minimize karna hai. Okay. Total cost function humko diya hua hai yaha pe product ka. Aur is cost function ko humko show karna hai ki kis value pe. For what value of x the cost function will be minimum.
So this is the question. They have given the cost function. They are asking at which value of x.
X ki kaunsi value ke liye. This cost function will be minimum. Okay. So, we have to minimize the cost function.
Whoever we have to minimize, that is our objective function. So, what is the objective function? Cost function. First, let's write what is our cost function. So, our cost function is x cube minus 615 x square by 2 plus 1570x plus 18000. So, this is our cost function.
What we have to do with this cost function? We have to minimize it. So, what does it mean to minimize?
To minimize means, first of all, we will differentiate this. this and put it equal to 0. So, we will differentiate d by dx of cost function Cx which is equals to x cube. 3x square minus 2 will come.
2 from 2 will be cancelled. 615x will come plus 1570 will come. And 18000 is constant.
Its differentiation will be 0. Now, this is a quadratic equation. We will solve this quadratic equation and get the value of x. So, our quadratic equation is here. d by dx of the function.
but to find the turning point we have to put this equal to zero we will equate this with zero after equating with zero what will we get? Very good. We will get two turning points.
When we solve this quadratic equation, we will get two turning points. So, when we solved the quadratic equation, we got two values of x. Which values?
175 and 30. So, these are the two values which you will get on solving this particular quadratic equation. You know how to solve quadratic equations very well. So, from here, you got two values of x by doing dy dx. Now, after this, you will get... value you get means you get turning point after getting turning point what you have to do after getting turning point you have to minimize cost function so to minimize cost function we will differentiate it again so again differentiating to write again differentiating with respect to x.
If we differentiate this with respect to x, then it will be 6x minus 615. This is our double derivative. d by d square by dx square of the cost function. Now we have to find the point of minima.
For the point of minima, we will keep 175 and 30. So d square by d square by dx square of the cost function. So we have to dx at x is equals to 175 or 30. Both values, because there are two values, I have inputted both values. So, when we input both values, how much will it be?
So, definitely, this value will come. On keeping 175, this value will come positive. And when you keep 30 value, then this value will come negative.
Okay? 30 will come negative, because 180 minus 615 will come negative. If you multiply 175 by 6, it will be more than 615. So, this will be positive. value. We had to minimize the cost function.
So when do we minimize? When the positive value comes. So this is the point of minima.
That means if we have to minimize the cost function, the number of units produced should be equal to 175. So next question, now this is a good question. This is also a good question. What is he saying?
Let's see this question. He is saying the demand function for a manufacturer's product is x is equals to, this demand function is given, x is equals to 70 minus 5p. P where P is price per unit, X is the number of units produced. Abhi tak toh yeh samajh mein aagaya hoga.
At what value of X will there be maximum revenue? X ki kaunsi value ke liye revenue maximum hoga. Thik hai?
Toh humko kis ko maximize karna hai yaha pe? Revenue function ko. So that means humko yaha pe sabse pehle kya nikalna padega? Revenue function.
Revenue function kiske barabar hota hai? Price per unit into the number of units sold. Ab yaha pe X diya hai in terms of P. This demand function is in the form of ki X in terms of P. of P but we can change this demand function we can take out P in terms of X so if we take P out in terms of X then what will be the value of P 70-X upon 5 this will be our value of P 70-X upon 5 is the value of P now if we know the value of P and multiply it with X then we will get revenue function so revenue function will be 70X-X square upon 5 so this is the revenue function which we have to maximize ok and also tell the maximum revenue so to maximize what we will do first is we will do second derivative test first we will differentiate it lets differentiate this revenue function with respect to x so what we will get is 14 minus 2x by 5 ok we cancelled it from 5 and it became 14 so we got the differentiation of x minus x square differentiation will be 2x upon 5. So, this is derivative of the revenue function. We have to differentiate it and keep it equal to 0. If we keep it equal to 0, then it will be 14 is equal to 2x upon 5. We cancelled it from 2, it became 7. 7 x 5 is 35. So, x value became 35. That means, where will we get maximum revenue?
When x value becomes 35. When we sell 35 items. Now, maximum or minimum? We don't know yet.
This is basically. the turning point now this turning point can be a point of maxima can also be a point of minima to maximize the minima is to check how to check this function by differentiating it again if we differentiate this function again then what will we get? 14's derivative will be 0 and this will be minus 2 by 5 now this is a constant negative term which is less than 0 if this negative term is less than 0 that means this is point of maxima revenue function which we had to maximize, we have maximized it.
So I hope this is clear to all the students who are watching. We had to get maximum revenue too, so what will we do to get maximum revenue? We will put 35 instead of x. So when we put 35 instead of x, we got the maximum revenue.
How much did we get? We got the maximum revenue of 245 rupees. Next question.
The company charges 700 rupees for a radio set on an order of 60 or less sets. So for sets less than 60, it charges 700 rupees. The charge is reduced by 10 rupees per set for each set ordered in excess of 60. If you order more than 60, then The radio set of 700 is less than 10 rupees. Means you will get 690. Find the largest size order company should allow so as to receive a maximum revenue.
How much maximum order should the company allow? Means how much maximum can they sell these radio sets so that they get maximum revenue? I hope you have understood the question.
So what is this question saying? the company charges Rs 700 for a radio set on order of 60 or less so if you order 60 radio sets 60 or less than 60 so how much will it cost you? Rs 700 per radio for a radio, a radio means single radio if you order greater than 60 if you are ordering greater than 60 radios then it will reduce the radio by Rs 10 it will reduce the radio by Rs 10 means first the company was charging Rs 700 now it will be charging only rupees 690. Okay. It is saying, find the largest size order company should allow so as to receive a maximum revenue.
What should be the largest size for maximum revenue? Okay. So, to answer this question, first of all, let X be the number of sets ordered in excess of 60. It means, the number of sets we ordered in excess of 60 is X. So how many sets we ordered in total? The total sets we ordered were 60 plus x.
If we ordered 60 plus x radios in total, then the revenue function from here will be equal to price per unit into the number of units ordered, price per unit, how much is the price of one unit? 690, a company charges 700 for a radio set, the charge is reduced by 10. for each set ordered in excess of 60. Okay. The charge is reduced by Rs. 10 per set for each set ordered in excess of 60. The amount you will charge above 60, that will be 690. The amount you will charge till 60, that will be 700. So, basically, from here it means that till 60, the price which is being charged per radio, that is 700. and the radius we have done above 60, so how much we have done? X, how much will be the cost of that? 690 into X so if we see this, our revenue function will come, what is it saying in this question?
it is saying, a firm has found from past experience that its profit in terms of number of units X produced is given by, so we have given the profit function, right? where X is lying, from 0 to 35, means the number of items we are producing items are coming between 0 to 35. It says, compute the value of x that maximizes the profit and the profit per unit of the product when this maximum level is achieved. So, we have to maximize the profit function in the first part.
In the second part, we have to get maximum profit. Will you do it? What we have to do?
We have to maximize this profit function. So, what is our profit function? x is equal to minus x cube by 3. plus 729x plus 2700 now if we want to maximize this, what we have to do is differentiate it with respect to x if we differentiate it, then 3 will be 3 to 3, what will be this?
minus x square plus 729 this is here, right? let's equate this with 0, so what is the value of x square? 729, so what will be the value of x from here?
Now, there will be two values of x. If you solve x square here, it will be in plus and minus. Leave the negative value because you cannot produce a negative number of units. You can only produce the maximum number of units. So, what will be 729?
On 29. On 29, you will get 729. So, 29 has come from here. Now, on 29 units, we will have either maximum or minimum profit. Now, how will we know whether it is maximum or minimum? So we will check for maxima and minima.
So we will differentiate it again. Again if we differentiate it, then d square by dx square of the profit function px will give us minus 2x. Now x is positive, so instead of x we will put 29, so it will come as minus 58. So what will come after putting 29? Minus 58. Minus 58 means less than 0, that means point of Maxima.
So this is again the point of Maxima. What is this? It is equal to the point of Maxima.
Yes, 27 will come, sorry, 29 will not come, 27 will come, correct, 27 will come, so we will double 27, 54 will come, minus 54 will come, 27 will come, double will come, correct, in, yes, right, the correct is 27, 27 square is 729, okay, so 54 has come, which is less than 0, which is a negative value, negative means, point of maxima. Now, he was saying what will be the profit? Tell me this also. So, what will be the profit? What will we do?
We will put 27 instead of x in this whole equation. We will get the maximum profit which we can obtain. So, when 27 put key value, then after putting 27, we will get the maximum profit. maximum profit mila will get a million 586 rupees so 586 rupees is the maximum profit yoga hamara maximum profit fair you have a question of a practice Kelly over a practice curly Jega sorry questions if sorry questions are practice car second and both easy and coach be the cut knee here or you could have a good session sorry cheese a happy cover over here yes I did you say I'm the last baby baby shall we start the application of integration integration to commerce and economic so many So I told you that the marginal function of anything, marginal cost function, marginal revenue function, marginal profit function, marginal wherever the word comes, that means d by dx of that particular function.
So if I say marginal cost function, then that means d by dx of cost function. If I say marginal... revenue function that means the d by dx of revenue function if I say marginal profit function that means d by dx of profit function okay so yeah happy hum job be a question is the request I have a co is particular point of time pay a got a co application of integration say I So you will be given some marginal function and from that marginal function you will be asked to find the original function. So you will get a question in such a way that you will be given marginal revenue function and you will be asked to find the revenue function.
You will be given marginal cost function and you will be asked to find the cost function. So basically what we were doing till now, we were given cost function and we used to find marginal function. How did we do it? So we were given cost function and what we used to do to find marginal cost? We used to do DY DX.
So what we used to get? Marginal cost function. Now what will happen to us?
Marginal cost function will be given and we will be asked about cost function. Means vice versa. So how can we do vice versa? So you know integration and differentiation are inverse.
So if we differentiate cost function and find marginal cost function, then we can integrate this marginal cost function again and convert it into cost function. Likewise, if we are given revenue function and we had to convert it into marginal revenue then what we did was we differentiated this revenue function now if we want to convert this marginal revenue back to the revenue function what we have to do is basically we have to integrate this marginal revenue function and we'll go back to the revenue function so this we have to do in this topic Let's see it quickly. So, determination of cost function. I have told you that if we want to define cost function, if we have given marginal cost and we want to find cost function, then we will integrate marginal cost with respect to x and we will come to the answer.
Now, I will explain directly from the question. It is very easy. You don't have to do anything.
The marginal cost function of manufacturing x units of a product is given as 5 plus 16x. Minus 3x squared. The total cost of producing 5 items is Rs. 50. Now this line here.
5 items cost Rs. 500. 6 items cost Rs. 800. Whatever. This type of information will be given. This information will be given to get the value of constant.
You know when we integrate, we get a constant. To get the value of that constant, we will be given some information. And with its help, we can get the value of Kc.
Right. Find the total cost function. So we have given marginal cost function, now we have to find the total cost function.
What will we do? Nothing. We will integrate it on both sides. So when we integrate marginal cost function, we get cost function. If we integrate it with respect to x, then it will be 5x plus 16x square by 2 minus 3x cube by 3. Plus C. In the last, we have put constant plus C.
Now, let's simplify it a little more. So, cost function will be equal to 5x plus 8x square minus x cube plus C. Now, this function of ours is our cost function.
But, but, but, but, brother, in this cost function, this C is your arbitrary constant. What? And we don't need arbitrary constant in our cost function.
Till now, whatever cost function we have studied, average cost function, marginal cost function, revenue function, profit function, none of them had constant. So we have to find the value of this constant. Now to find the value of this constant, you will be given some information. Like here, if you are producing 5 items, then it will cost you Rs. 500. Means, when we keep the value of x as 5, then our cost function, Cx, will be equal to... 500 and here we will keep this value and take the value of c from here.
So cx will be 500. When x is equal to 5, so if we keep x as 5, then 5 into 5 is 25, plus 5 squared is 25, 25 into 8 is 200. minus 25 cube right, no 5 cube, so 5 cube is 125 plus C Now from here we can get the value of C, so how can we get it? Let's do it, 125 out of 200, 75 plus 25 is 100 100 comes here, so the value of C will be 400 So when the value of C becomes 400, we can define the total cost function from here So what was the total cost function? So total cost function will be 5x plus 8x square minus X cube plus 400. So this is our final cost function. So this is the final cost function.
And this is our final cost function. Okay. This is all you have to do in this topic.
It is very easy. You just have to do integration instead of differentiation. That's it. See next question.
It says the marginal cost function of producing X units of a product is given by. So marginal cost function is given by X upon under root of X square. Plus 2500. Find the total cost function and the average cost function.
Total cost function is to be taken out. Average cost function is to be taken out. By total cost function, we can integrate it.
To take out the average cost function, we will divide it by x. If the fixed cost is Rs.1000. Now see, there is a little game here.
Fixed cost is Rs.1000. So first of all, we take out the cost function. So we know how the cost function is taken out. integrate both of them, integrate the marginal cost function with respect to x, we will get our cost function.
So cost function will be equal to integration of x upon under root of x square plus 2500 dx. Now to integrate this, we will add substitution. So in substitution we will write let x square plus 2500 is equal to t square. Let's consider it equal to t square. So the differentiation of x square will be 2x dx is equal to 2T dt.
From here, 2 to 2 will be cancelled. So, x dx will be equal to T dt. So, instead of x dx, what can we write? Instead of x dx, we can write T dt. So, this will be integration of T dt.
upon x square plus 2500 to t square. Under root of t square, which will be equal to t? t to t will be cancelled.
So, cx will be equal to, the cost function will be equal to dt. Dt integration will be t plus c. Now, this c, So C is an arbitrary constant.
Remember this is not a cost function. This is C. So if you want to denote C from any other alphabet, from A, B, K, P, then also you can do it.
So that there is no confusion. So if you want to do it, then let's denote it from a different alphabet. Let's say P.
P is an arbitrary constant. So CX is equal to T plus P. Now, what was the value of t?
In terms of x, x square plus 2500. Under root of. So, Cx came. Cx came under root of x square plus 2500 plus p where p is the arbitrary constant. Now, write it here. p is arbitrary constant.
Okay. Now, we have to find the value of p. To find the value of p, I have given you some information. I have given you some information. I have given you some information.
cost is equal to 1000. Now if fixed cost is equal to 1000, then how can we do it with this? So we know that cost function is made up of two things, fixed cost and variable cost. Now if the value of fixed cost is equal to 1000, what does this mean? This means that fixed cost means such cost which does not depend on the number of items produced.
which you have applied in the beginning, let's say, to buy land, to buy machinery, to set up, that is known as your fixed cost. So, the fixed cost does not depend on the number of items produced. So, if the fixed cost is 1000, it means that this cost function CX, this will be equal to 1000 if X is equal to 0. X is equal to 0 means that if we don't produce even one item, then also our cost, the cost that we have applied, will be equal to what?
It will be equal to fixed cost. We have installed machines, we have bought land, we have installed machines, but we have not produced even one item, then also our cost has been applied. Because that is our fixed cost.
Right? So, when the value of X is 0, then our cost function will be equal to 1000. So, let's keep this value here. So, for X is equal to 0, CX will be equal to 1000. So, CX value is 1000. If we replace x with 0, then it will be 50 plus p. So, the value of p will be 950. So, that means this is your arbitrary constant value. Now, what was he telling us?
He was telling us to find the cost function. Cost function and average cost function. So, what will be the cost function?
Cost function will be cx is equal to under root of x square plus 2500 plus 950. What is this? This is our... Cost function. Okay. Now, cost function is here.
What is being said in the next part? Take out average cost function. You can take out average cost function too.
It is very easy. What will we do to take out average cost function? Divide this whole thing by x. So, this will be under root of x square plus 2500 divided by x plus 950 upon x.
So, this is our average cost function. That's it. You have to do this.
You don't have to do anything more than this. Okay. So, it is very simple.
It is very easy. It is very easy. There is nothing in it.
That's why I say that all students should do Section C. Whether you are in Science, Arts or Commerce, you will do Section C. You will easily gain this number from here. If you want to prepare for Competitive, then you should study Vector 3D. You should study, but you don't need to take any risk here.
When you have an option, you should utilize that option. Okay? Let's go.
Next. The marginal cost of a product is given to be a constant multiple of number of units X produced. Find the total cost function. If fixed cost is Rs. 5000 and the cost of producing 50 units is 5625. Now here we have given two types of information. How will we do it?
Let's see. It says, the marginal cost of a product is given to be a constant multiple of a number of units X produced. So the marginal cost is a constant. multiple of number of units of x produced. Constant multiple of number of units of x produced means the number of units of x produced is a constant multiple of our marginal cost.
So, we have taken the constant here, that is k1. So, we have written here, let k1 is the constant. So, k1 is our constant here, so it has multiple. Multiple means that many times. So, k1 times x.
Right? Now, what is he telling us? He is telling us to find the total cost function. He is talking about finding the cost function.
So, he is talking about finding the cost function, which means we integrate both sides. So, if we do K1x with respect to dx, then if we integrate the marginal cost, then we get the cost function. If we do the integration of K1x, then we get K1x squared by 2 plus one more arbitrary constant. And we have taken this constant, let's say, again as p. We are not taking c because c is already written.
Okay? We have taken it as p. Okay.
Now, this cost function that we got, we have got it. But we are getting two arbitrary constants here. K1 and P.
Because of which, we have to find the value of K and P. If we find the value of K and P, then our cost function will be obtained. Now, to find the value of K and P, there are two unknown, two arbitrary constants, K and P.
So, we have to make two equations. To make two equations, we have to use the information given here. What is it saying? Find the total cost if fixed cost is Rs.5000. Now I have told you that if fixed cost is Rs.5000, then the number of items produced on fixed cost is equal to 0. So if we put 1000 in cost function, 5000, and what will be the value of x?
It will be 0. So if x value is 0, then what will be the value of p? The value of p will be equal to 5000. So the first equation was made from our first information. So as soon as we put the first information, we got the value of whose? p. We got the value of p, now we have to find the value of k1.
Now how will we find the value of k1? So, to produce 50 units, we need to use 5625. So when x is equal to 50, when x is equal to 50, Cx is equals to 5625. So, if we put 50 instead of x, what will happen here? So, let's put 50 instead of x. Cx is 5625 is equals to, let's put 50 instead of x.
So, 50 squared will be 2500. divided by 2 into k1 plus p. How much is the value of p? We have already taken out the value of p as 5000. If we take this 5000 here, then it will be minus 5000. So, it will be 625 is equal to, if we cancel this, then 1 times, two times five times zero times. 1250 is equal to 1250k1. k1 will be 625 divided by 1200 So if we cancel this from 5, then 5, 1, 2, 5. Right?
If we cancel this from 5, then 5, 2, 5, 0. Check. If we cancel this from 25, then 25, 5 times is 125, and 25, 10 times. If we cancel this from 5, then 1 by 2. So from here, what is the value of k1?
The value of k1 is equal to half. What was he telling us? He was telling us to find the cost function. So let's find the cost function. So cost function cx will be.
be equal to K1. What is K1? K1's value is 1 by 2. x square by 2. So, x square by 2 plus p1.
What is the value of p? The value of p is 5000. We will solve it a little more further. So, it will come.
x square by 4 plus 5000. So, this is your net cost function. Total cost function. This is our total cost function. Okay.
So, it is very easy. You just have to understand. And if you understand it once, then it will be done easily.
There is nothing in it. Okay. See the next question. Determination of revenue function. Again, same thing.
If you are given marginal revenue and you are asked to find revenue function, then you integrate marginal revenue and you will get revenue function. Let's see from question, you will understand immediately. Question is, marginal revenue is given where X number of units are produced.
Find the total revenue and the corresponding demand function. Okay, we have to find revenue, revenue function and demand function. Very good. To find the Revenue function, we know that if we integrate the Marginal Revenue with respect to x, we get the Revenue function.
So we integrate both sides with respect to x. So if we integrate both sides, we get 12x-3x3x3 plus 4x2x2 plus constant, which is equal to p. So when we integrate the Marginal Revenue, we get the Revenue function.
So Revenue function Rx will be equal to 12x. minus x cube plus 2x square plus p. This is our revenue function. Now, again the same thing.
We have to remove the value of this arbitrary constant p. Now, to remove its value, you will feel that we have not given anything. Now, see, there is a very good thing in revenue function. What is that? By default, all are in revenue function.
Understand. If, Our number of units sold is equal to zero. What is revenue function? Revenue function means, the amount of money we earned by selling X number of units. That is your revenue.
The revenue of a company is the amount of money it earns by selling its products. So if it is selling X number of products, then the revenue that is coming is what our revenue function is giving us. So if we did not sell even one item, we have produced lakhs of items, but we did not sell even one item, we did not sell even one item. So how much revenue will we have? So in this case, our revenue will be equal to 0. You just have to remember this by default and your work will be done.
What you have to remember is to understand. When we will not sell even one item, then the revenue will be 0. So when we will keep the value of x as 0, then rx will also become 0. So we keep the value of x as 0. rx will also become 0. So rx is 0, x value is 0. So what is the value of p here? P value is equal to 0. Now you will say that sir this will always happen.
No, it is not like that. In every question it will not happen that constant 0 will come. It is possible that value will change somewhere.
Here all were in terms of x. That is why this constant 0 has come. Otherwise it is not necessary that 0 only comes.
So we have taken out P value of 0. So finally our revenue function is done. 12x minus x cube plus 2x square. This is our final revenue function.
Now, as soon as our revenue function comes, what do we have to find from here? It says also find the demand function. So, how do we find the demand function? Demand function is from p. So, p is equal to Rx upon x demand function is in p terms so Rx upon x Rx was this, we will divide it by x so it will be 12 minus x square plus 2x so this is our yeah yeah Mara demand function so demand function of price P with respect to with in relation to X take a yeah yeah Mara demand function take both us on a next question they think or care of the marginal revenue function of a commodity is given as marginal revenue the away carrot find the total revenue and the corresponding demand function same to same as equation so category nearby job email revenue the other a hummus can integrate call it a hair with respect to X If we integrate it with respect to x, then we will get the revenue function.
This will be 6 upon x minus 3 with a negative sign minus 4x. So, here you can integrate it with substitution. I know that 1 upon x. the differentiation is minus 1 upon x square so if 1 upon x is the whole square of x minus 3 then if we integrate it, it will be the opposite minus 1 upon x, so minus 6 upon x minus 3 minus 4x plus p now what is p here?
p is an arbitrary constant, write it here P is your arbitrary constant. So that the teacher who is checking, he should not be confused that this is not a P profit function. So write it. Consider it P, K, C, or anything. Let's consider it K, which is easier.
K is our arbitrary constant. Okay. Now the same thing.
When our X value is 0, then our revenue will be 0. With the help of this, we will get K value. So if we get K value here, then it will be 0 is equal to minus 6 upon minus 3. minus 0 plus k. So, this will be equal to 2. So, k is equal to minus 2. Do you see?
So, here when we kept x's value as 0, revenue function became 0. But here k's value is minus 2. In the previous question, 0 was coming. So, you don't think that 0 will come every time. So, you have to keep this and check it. You have to check it. Okay?
k's value is minus 2. So, as soon as k's value came, we got our revenue function. So, revenue function rx is equal to minus 6 upon x minus 3 minus 4x minus 2. This is our finally revenue function. Now, as soon as our revenue function comes, it is asking us to find the demand function. Demand function is very easy.
We know that demand function P is equal to Rx upon x. So, if we find P from here, then demand function will be minus 6 upon x into x minus 3 minus 4 minus 2 upon x. This is our demand function. That's it.
Finished. Pura chapter hi khatam. Baha isyada kuch hai hi nahi iss chapter me. Itna easy chapter hai.
Aur araam se aap yaha se number score kar sakte ho. That's it. Now these are the last questions. DIY. Do it yourself.
Yala questions aap sare kar lije. In questions ke maine answers bhi diye hain. Congratulations for completing the chapter.
Keep learning with Yash sir. Theek hai. So next class me melenge. Next chapter leke aata ho aapke le. Leech takke le.
Take care. God bless you all. Bye bye.
Jai Hind. Padte rahi hai. Mojh karte rahi hai. Aur mehnat karte rahi hai.
Bye bye. Take care.