 Hello and welcome to the session. In this session we discuss the following question which says, The cost of manufacturing of certain items consists of $1000 as overheads, $15 per item as the cost of the material and the labor costs x square upon $10 for x items produced. Find, first, slope of the average cost function, second, slope of the marginal cost function, third, how many items must be produced to have a minimum average cost? Fourth, if Cx is equal to 7 upon 10 into e to the power of minus x upon 7, verify that slope of the AC curve is given by mc minus ac the whole upon x. Before moving on to the solution, let's discuss about the average cost and the marginal cost. First, we have the average cost, it is denoted by AC and this is equal to the total cost of producing and marketing x units of a commodity upon the total quantity of goods produced. So, this is equal to C upon x where x is the quantity of the goods produced. So, average cost AC is equal to the cost function C upon x. Next, we have the marginal cost which is denoted by mc and this is defined as the rate of change of the total cost C with respect to x. So, dc upon dx is mc which is the marginal cost. This is the key idea that we use in this question. Where is no move on to the solution? In the question, we are given the overhead charges, then the cost of the material, the legal cost and they all make together the cost of the manufacturing of the certain item and it is also given that x items are produced. So, on adding the overhead charges, the cost of the material per item and the labor cost, we would get the total cost of manufacturing. So, the total cost is equal to overhead charges which are given as thousand dollars plus the cost of the material that is used now. It is given that fifteen dollars per item is the cost of the material and since there are x items, so fifteen x dollars is the cost of the material used in the production. So, we add here fifteen x plus the labor cost which is x square upon ten. So, this is the total cost we have that x units are produced. Now, in the first part, we are supposed to find the slope of the average cost function. So, first let us see what is average cost. It is given as C upon x where the C is the total cost and x is the quantity of the goods that are produced. So, in the first part we have average cost AC is equal to C upon x where the C is the total cost. So, average cost AC is equal to thousand plus fifteen x plus x square upon ten which is the total cost upon x. Or we can write this as AC is equal to thousand upon x plus fifteen plus x upon ten. Now, the average cost function is equal to V by dx of the average cost that is AC. So, this is equal to d by dx of thousand upon x plus fifteen plus x upon ten which is further equal to minus thousand upon x square plus one upon ten. Or you can say this is equal to one upon ten minus thousand upon x square. So, we have slope of the average cost function is equal to one upon ten minus thousand upon x square. So, this is our answer for the first part. Now, in the second part we need to find the slope of the marginal cost function and we have the marginal cost mc is equal to d by dx of C or dc by dx. So, first of all we find out the marginal cost denoted by mc and this is equal to d by dx of the cost function that is C. Now, this is C thousand plus fifteen x plus x square upon ten as we have C is equal to this. So, mc that is the marginal cost is equal to d by dx of thousand plus fifteen x plus x square upon ten. So, this means we have the marginal cost mc is equal to fifteen plus x upon five. This is the marginal cost. Now, slope of the marginal cost function is equal to d by dx of mc. So, on differentiating mc with respect to x we have one upon five thus slope of the marginal cost function is equal to one upon five. So, this is the answer for the second part of the question. In the next part we are supposed to find that how many items would be produced to have a minimum average cost. Now, let's move on to the next part. Our average cost is thousand upon x plus fifteen plus x upon ten which we have already found out. Now, we have to find out the number of items produced such that we have minimum average cost. So, for this we will apply the conditions of maxima on minimum. So, let's find out d by dx of the average cost which is equal to minus thousand upon x square plus one upon ten. Or you can say that d by dx of ac that is the average cost is equal to one upon ten minus thousand upon x square. For maxima or minimum we take d by dx of average cost as zero that is one upon ten minus thousand upon x square equal to zero which gives us thousand upon x square is equal to one upon ten and from here we have x square is equal to ten thousand. This gives us the value of x as hundred. Now, the number of items produced cannot be negative so we do not take the value of x as minus hundred. So, we take only the positive value of x that is plus hundred. Next we find out d by dx of ac by dx two that is we differentiate d by dx of ac with respect to x. So, we get minus thousand into minus two upon x cube so this is equal to two thousand upon x cube. Let's now find out d two ac by dx two at x equal to hundred. So, this is equal to two thousand upon hundred cube that is hundred into hundred into hundred so further we have one upon five hundred which is greater than zero. Now as d two ac by dx two is greater than zero at x equal to hundred therefore we can say that the average cost is minimum at x equal to hundred. Hence, we can say that hundred items must be produced to have a minimum average cost. So, this answers the third part of the question. Next we have to find that if cx is equal to seven upon ten into e to the power of minus x upon seven verifying that the slope of ac curve that is average cost curve is given by mc minus ac upon x. So, in this we are given cx or c as seven upon ten into e to the power of minus x upon seven. Now, average cost ac is equal to c upon x that is equal to seven upon ten into e to the power of minus x upon seven and this upon x. So, average cost ac is equal to seven upon ten x into e to the power of minus x upon seven. Now the marginal cost mc is equal to dc by dx so on differentiating c with respect to x that is d by dx of seven upon ten into e to the power of minus x upon seven. We have this is equal to seven upon ten into minus one upon seven into e to the power of minus x upon seven. So, we now have minus e to the power of minus x upon seven this upon ten this is the marginal cost. Now, slope of the ac curve that is the average cost curve is given by d by dx of the average cost that is ac. So, we will differentiate ac with respect to x that is d by dx of seven upon ten x into e to the power of minus x upon seven. So, this is equal to x square in the denominator then we have x into d by dx of seven upon ten into e to the power of minus x upon seven minus seven upon ten into e to the power of minus x upon seven into d by dx of x. So, we get this is equal to x into seven upon ten into minus one upon seven into e to the power of minus x upon seven minus seven upon ten into e to the power of minus x upon seven and this whole upon x square. So, further we have minus x upon ten x square into e to the power of minus x upon seven minus seven upon ten x square into e to the power of minus x upon seven. And this is equal to minus e to the power of minus x upon seven upon ten x minus seven upon ten x into e to the power of minus x upon seven into one upon x that is we have written one upon x square as one upon x into one upon x. Now, we have mc is equal to minus e to the power of minus x upon seven upon ten. So, mc upon x would be minus e to the power of minus x upon seven upon ten x. So, from this term we can say that this is mc that is the marginal cost upon x and now this is the average cost. So, average cost upon x is seven upon ten x into e to the power of minus x upon seven into one upon x. So, this term is ac upon x. So, this is equal to mc that is the marginal cost minus the average cost upon x. This is d by dx of the average cost. Hence, we have verified that the slope of the average cost curve is given by marginal cost minus the average cost this one upon x if the cost function is given as seven upon ten into e to the power of minus x upon seven. So, we have proved this also. So, this completes the session. Hope you have understood the solution of this question.