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MATH 138 01 Spring 2020 Week 4 Section 9.7 Homework: Score: 0 of 1 pt 9.7.17 by C(x) = 283 0.3x and R(x) = 2x- 0.03X _ cost and revonue Qiven Find the marginal pro...

Question

MATH 138 01 Spring 2020 Week 4 Section 9.7 Homework: Score: 0 of 1 pt 9.7.17 by C(x) = 283 0.3x and R(x) = 2x- 0.03X _ cost and revonue Qiven Find the marginal profit function PoF-Dier your answer In the answer box and then cllck = Check parte Answer: ehowing

MATH 138 01 Spring 2020 Week 4 Section 9.7 Homework: Score: 0 of 1 pt 9.7.17 by C(x) = 283 0.3x and R(x) = 2x- 0.03X _ cost and revonue Qiven Find the marginal profit function PoF-D ier your answer In the answer box and then cllck = Check parte Answer: ehowing



Answers

Find the marginal profit for producing units. (The profit is measured in dollars.) $$ P=-0.25 x^{2}+2000 x-1,250,000 $$

So we're giving this profit function P and we want to find the marginal profit function. So the marginal profit function M. Is just equal to the derivative of our profit function. Or P. Print. So we can say M. Is equal to the derivative -125 times X squared Plus 12.2 times X 25,000. The first thing we can do is we can split up this derivative since we have the derivative of one term plus another term minus another term. We can split that up into the derivative of the first term plus the derivative of the second term minus the derivative of the third term. So this is equal to the derivative of negative point who 25 x squared plus the derivative 12.2 x. And then minus the derivative of 25,000. So the derivative of all constants goes to zero. So this derivative here, it's just gonna be equal to zero. So we can cross it out or not deal with it. And the next thing that we want to do is we want to bring out these constants here that are multiplied by our X squared and are extra. We can actually bring those out front and then multiplying by the derivative of our just resulting term, which is gonna be X squared and X. So this is equal to negative point oh 25 times the derivative of X squared plus 12.2 times the derivative of X. And so the derivative, the derivative of X square. We can use the power rule to figure that out. And the power rule is if we ever have a derivative of X racism power and and that's just equal to n times X. To the n minus one power. So here and is equal to two and here and is equal to one. So are marginal profit function. M It's going to be equal to negative point oh 25 times two X since we use this power role here and then we have plus 12.2 times one times X. To the zero since and was equal to one for the derivative of just X. So we can simplify this. And our marginal profit function M is going to be equal to negative .05. When we multiply by two times x plus 12.2 since one times X zero is just one. So this is our marginal profit function.

You guys did this? No problem. 26 in this problem, we need to find the marginal profit function. He had the cost function. And revenue functions are given. We know that when we subtract the cost function from the revenue function, we get the profit function and when we do this calculation we get that profit function is 3.9 X minus 0.2 X squared minus 1 45. Then we need to find the marginal profit function. We know that the marginal profit function is the first derivative of the profit function, which we did not S P prime X. And when we take the first derivative of the profit function and profit function, we just calculated as 3.9 x -0.02 exists where -145. And when we take this derivative we will get the marginal profit function Here 3.9 is a constant. It will take out. Then we need to take the derivative of X. Then we have this negative sign here. Then we have 0.02 which is again a constant. It will come out and then we will take the derivative of X square. Then we have 145 which is a constant. We need to take the derivative of that. Now uh from the differentiation rule we know that when we take the derivative of extra depart and we get an extra double N -1. And is the power of fix here and in this case we have a part of 14 X and X is where the power is too. And we take the derivative of X, we get one extra depart one minus one. And when we take the derivative of X is where we get to extradite about two minus one. Then the situation comes out to 3.9 multiplied by extra deep. Our zero. When a 0.02 multiplied by 26 zero. Because when we take the derivative of a constant we get to zero And extra power zero is visible to one. Therefore 3.9 multiplied by one, gives us 3.9 In this minus sign. Then 0.02 multiplied by 26 gives us 0.04 X. Their follower marginal profit function is 3.9 -0.04 X.

You guys leads to problem 18. In this problem we need to find the marginal profit function. And the cost function is given the revenue function is given. We know that when we sub strapped cost function from revenue function, we get profit function. Therefore we need to substitute these situations here. Now the revenue function minus cost function. And when we do the calculation we get the profit function be excess 26.5 X minus 0.3 X squared minus 4500. Then we need to find the marginal profit function. We know marginal profit function is the forest relative of the profit function which we can do notice. P prime X. And when we take the derivative of them profit function, which we just calculated we will get the marginal profit function. Had 26.5 is a constant. Therefore it will come out of the derivative. You can dress it out. Then we need to take the derivative of eggs. Then you have my minus sign Against 0.03 is constant. It will come out. Then we need to take the derivative of excess square here And then we have 4500. It is a constant From the differentiation rules, we know that when we take the derivative of extra bar and we get an extra bar N -1 -1 here. In his and his power. And in this situation we have The power as one here. Therefore when we take the derivative we get 11 deployed by extra 2.1 -1. And here we have to as the power. Therefore the derivative we will get as Two multiplied by X 3.2 -1. And when we do this calculation we will get 26.5 multiplied by extra power zero. My no 0.03 multiplied by 26. And then when we take the derivative of a constant, we get zero And then extra zero means one anything to the power zero is one. Therefore 26.5 multiplied way one gives us 26.5 Minour here, we have 0.03 multiplied by two. When we do this calculation we will get to 0.066. Therefore our revenue, our marginal profit function is 26.5 My NAN 0.06 X.

So if we want to find cost function, given that the marginal pass function is C prime of X is equal to one point two x times the natural rate of one point two, and that if we produce two units, is going to cost us nine dollars and forty cents. So we might want to write down what some of these things mean first. So the marginal cost functions, remember Marginal just needs a very small change on something. So in this case, a very small change of our cost function and what this is wanting us to find or what it means, or to be a marginal cost function in terms of calculus would be the derivatives. So very small change and the way we do it, very small changes is with the derivative, so I can selection before start. I also need to know that. So if I find my cost function, C and I plugged in to Intuit, I should get nine dollars in forty four cents. So what is it telling us is, since the marginal cost function is the derivative of the cost function, I need to integrate it and plug and see it too and salt for my integration constant. So let's go ahead and write down. See, your prime of books is equal to one point two times on that log of one point two. And like I was saying, if we want to find the cost function we want to integrate this with respect to X, so integrating each side with respect to ACS so integrating the marginal cost function are the derivative of the cost function will get the CFX And then if I want to take the anti derivative of an exponential, remember that we're told that anti derivative Oh, these eh with the power k x, the X were case and Constant is going to be a k of X for a to the k X power over Okay, times the natural log day plus, um, constants. So when he teams this fact here so first we'll use the fact that scale er's can be pulled out of an interval. So it would be natural log of one point two one point two x And then I put my journal back down and now I can use the fact that we stated earlier to be so. This will be one point two acts over. So in this case, my caves in one times the natural log of my base, which is one point two and then I add on my constant up on this case, all use D for my constant since we already have a sea floating around so you can go ahead and simple fighters down. So I have a natural log during the numerator and the denominator. So those cancel out and I'll just be left with one point two x plus de is equal to my cost function. So this is just one possible cost function out of many. And if we combine this with the fact that if I produced two units is going to cost nine dollars important for sense and that all possible cost functions just given that marginal or in this form, I can end up getting that. So I need nine point four four two equal photo right function first, a one point two on the premises there, plus the and then since that sea of two I plug to him to hear and doing that on the nine point four four and then one point two squared is one point four four rusty and then solving for Dallas truck That over I get d is so using the fact that he is equal to eight and this is the general cost function I end up with a sea of X is equal to one point two x plus leaked. And so this here will be our cost function given that to unit cell for nine dollars and forty four cents.


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