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2 L 272 =Tlc demind cqualion for cettain typr of product is given by lhe fonmula (in dollars) and Ja thc ountiny sold pc nianth d.bzdiere pViE pMCeFind the Margina...

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2 L 272 =Tlc demind cqualion for cettain typr of product is given by lhe fonmula (in dollars) and Ja thc ountiny sold pc nianth d.bzdiere pViE pMCeFind the Marginal Retenue function (in terms quanlin Menh Lab #5 ! ) R-AODae Rlq) =Ilbove-h)tal--Yzove-Ia) R'e)=Ab0e (Lzl Aok-la A- V(i)ze 47 b) Find thc tolal number of scrvets that Pi"oin sold pc nonthin ordcr t0 MxImixc Ihe Iotil TEVLUC _ (show your Wotk)Iuximic total evenuc? What price should be charged

2 L 272 = Tlc demind cqualion for cettain typr of product is given by lhe fonmula (in dollars) and Ja thc ountiny sold pc nianth d.bzd iere p ViE pMCe Find the Marginal Retenue function (in terms quanlin Menh Lab #5 ! ) R-AODae Rlq) =Ilbove-h)tal--Yzove-Ia) R'e)=Ab0e (Lzl Aok-la A- V(i)ze 47 b) Find thc tolal number of scrvets that Pi"oin sold pc nonthin ordcr t0 MxImixc Ihe Iotil TEVLUC _ (show your Wotk) Iuximic total evenuc? What price should be charged



Answers

Use the table to answer the questions below. $$ \begin{array}{|cc|cc|}\hline \text { Quantity } & {} & {} & {} \\ {\text { produced }} & {} & {\text { Total }} & {\text { Marginal }} \\ {\text { and sold }} & {\text { Price }} & {(T R)} & {(M R)} \\ {(Q)} & {(p)} & {} & {(M R)} \\ \hline 0 & {160} & {0} & {-} \\ {2} & {140} & {280} & {130} \\ {4} & {120} & {480} & {90} \\ {6} & {100} & {600} & {50} \\ {8} & {80} & {640} & {10} \\ {10} & {60} & {600} & {-30} \\ \hline\end{array} $$ (a) Use the regression feature of a graphing utility to find a quadratic model that relates the total revenue $(T R)$ to the quantity produced and sold $(Q) .$ (b) Using derivatives, find a model for marginal revenue from the model you found in part (a). (c) Calculate the marginal revenue for all values of $Q$ using your model in part (b), and compare these values with the actual values given. How good is your model?

So for the given problem, we want to find the marginal profit function. So keep in mind a couple of important things. If we're given the revenue function and the cost function, there's a couple of ways we can find the profit function. The first or the marginal profit function first is recognizing when we hear marginal, we should think the derivative because marginal revenue is our prime of X. And marginal cost is C. Prime of X. So the marginal profit function, which would be since we know that the profit function is equal to revenue minus cost. Okay, we know that marginal profit which is p prime of X. That's going to be equal to our prime of X mine fc private max. So we take our marginal revenue and marginal cost and subtract them to get our marginal profit. Another option though, is to take our revenue and cost function and create our profit function. And once we have the profit function, we just differentiate it which still ends up with this which is our marginal profit function.

We have this profit function p which is equal to negative point oh five times X squared plus 20 times X minus 1000. And we're going to find what the additional profit is from Selling 151 units instead of 150 units. So we're going to find P of 150 and p of 151. So this is equal to negative .5 times 150 squared Plus 20 times 150 is equal to 3000 plus 3000 In the -1000. So this would be equal to negative .5 Times 150 Squared plus 2000. So if I plug this into a calculator we have 150 squared times negative five And then plus 2000. And that's equal to $875 to the prophet From selling 150 units is $875. So what the prophet is of selling 151 units. So we have negative .5 Times 1 51 Squared Plus 20 times 1, Which is equal to 3020 minus 1000. It's the sequel to negative .5 Times 1 51 Squared Plus 2020. And so if we do negative, We're sorry, if we do 151 squared and then multiply that by negative five and then we At 2020 we get this is equal to $879.95. So the difference between producing or selling 100 and 51 units 1st, 150 units is about $4.95. Or the difference between 8 75 8 79.95 and 8 75. So we could say that it's $4.95. That's the difference between Selling 151 units 150 units. And now what we're gonna do is we're gonna find are marginal profit function m which is just equal to the first derivative of our profit function. Soapy prime. So I'm here to be equal to the derivative of our profit function, which is equal to negative point five times X squared Plus 20 times x minus 1000 in this derivative. To find, it's pretty straightforward. We're just gonna have to use the power rule on this X squared and X term and then we'll let these constant or this constant here go to zero. So it's going to be equal to negative five times two is negative .1 times X plus 20 and then 1000 goes to zero. So this is our marginal profit function. And we want to find what the marginal profit Function is. When X is equal to 150 so Mm of 150, It's equal to negative .1 times 1 50 plus 20 Which is equal to negative 15 plus 20, Which is equal to $5. So again, here are marginal profit function at X is equal to 1 50 is very close to the difference between producing 1 51 and 1 50 units. It's only five cents off, However, they're still not exactly equal.

Hello. Today we're going to be working on a marginal profit problem. So marginal profit is just an applied version of instantaneous derivatives, instantaneous derivatives being. When you derive an equation and then you plug in an X value into a derivative, you get the slope at that particular X value. So anytime you hear marginal, that's what you're dealing with. So our equation is 36,000. Yes. 36,000. Okay. Plus I believe this is 2000 and 48 times a squared of X minus 1/8 X squared. Uh, it's important to note that the X is the only thing that's squared. It's not the quantity X squared. So in this case when we derive this, uh, the derivative of 36,000 is going to be zero, the derivative of 2000 and 48 times the square root of X. Um we're going to make X to the one half because that's what square is is next opponents. Um, and then when we derive this, we bring the one half down and subtract one afterwards. So this ends up being 2000 and 48 times that one half we brought down. So I'll be over to and then uh X to the one half minus one becomes X. To the negative one half. All right. So then let's just erase this bit really quick to me a little more. Mhm. Easy to read. And then when we do minus 1/8 X squared, what I'm gonna do is I'm going to separate the fractions here. So this is the same thing as 1/8 times. So this is the same thing as 1/8 times. One over X squared. This one over X squared can become X. To the negative too. And so now when we drive this we end up getting and we end up getting uh negative to which I will put where the one was because they're gonna multiply um X to the negative three. So now we do a little bit of This algebra, these negatives will cancel out. And we end up with uh okay, okay 1/4. This X to the negative three can is the same thing as one over X cube. So when you multiply those together you get four X cubed here plus X. To the negative one half is the same thing as one over the square root of X. So this becomes two square root of X. Yeah. Mhm. And then we can reduce down 2000 and 48/2. It's actually get rid of those and that becomes 1000 and 24/1. So this becomes 1000 and 24 over the square root of X. Uh huh. So now we have 1024 over the square root of X. Plus 1/4 X. Cute. We can technically add those together but it doesn't really make that big of a difference because um at the end of the day you're most likely going to plug this into a calculator because what we're gonna do is we're looking for the marginal profit at all of these positions. So the exit 1 51 75 200 so on. Uh So in this case I'm going to go through and and plug in for 1 50 just to show you how to do it and then the rest will essentially be done. Um most likely through calculator or just plug and chug and and going through it. So in this case what we end up having is 1000 and 24 over the square root of 150 plus 1/4 times 100 and 50 cute! Now, just as a general note, this whole section right here um is going to add very little because you're adding one of 150 cube times four, which is gonna be a very large number in the denominator, which means we're going to essentially add almost nothing. So generally speaking, um this section is going to add very little to the left hand section and so um generally speaking, that's uh it's not going to have that much of an effect. So in this case, what we end up having is 1024 divided by the square root of 1 50 ends up being 83.609 and then a whole bunch of numbers afterwards plus 1/4 times 150 cube ends up being so 150 cube ends up being three million. 337th up. Excuse me. Three million. 375,000. Um You multiply that by four you get about 13 point five million. And then you do you do one over. This ends up being 1/13 10.5 million. And when you end up doing that you get 7.4 times 10 to the negative eight. And so in terms of that, um that's going to be points. Was this 1234567 7412345678 So this is going to have almost no effect on that. And so from here we end up just getting 83.609 as our answer for 1 50 0.609 So from there you just go and do the same things with 1 75 on through to 75. And when you do that, you end up getting 77.4. Yeah 07 for 1 75 for 200. You get uh 72 40 eggs 25. You get 68 points 8.267 uh 2 50. You get 64 0.7 sixth three for 2 75. You get 61.7 yeah 50 So as you can see here um as you get larger, the numbers that are getting smaller, which makes sense because um won over four X cubed is only going to get bigger. And excuse me, the denominator of that is only going to get bigger, which means this is just gonna make less and less and less of a difference. And 1000 and 24 divided by the square root of X. This denominator is also going to get bigger um to the point where you're just dividing 1000 and 24 by less of an amount as you go through. So this is going to be your denominator and all of this is your marginal profit at each particular unit. Yeah.

So we're told that this profit function p of X is equal to two x squared, minus five Experts. Six tells us the total profit and hundreds of dollars. After selling ex items, we want to first find the average rate of change for selling 2 to 4 items and then from selling 2 to 3 items. And then we want to find the instantaneous rate of change for the profit, with respect to the number of items produced when X is equal to and they want us to interpret this, and then they want us to find the marginal profit for when X is equal to poor. So it's going to start with find the average rate of change for each of these. So we're just gonna follow that equation that we have in the top left corner here. Let's go ahead and write that equation out first, so it's going to be so instead of efs were gonna peas, so I was going to be p a four minus p of two all over or minus two. So it's going for about what, Pia for people to us. So when we plug in or into P so that before Swerts, 16 times to 32 than five times four is 20 plus six. So that should give us 12. Plus six is a team soapy affords gonna be 18. Then we confined what Pia to is. So we plucked to in. So it's going to give us two squared four times two is eight minus 10 plus six and simplifying that down should give us for soapy of two is four. Now, when we go ahead and simplify this, that should be 14 over to, which is equal to seven. The average rate of change from selling 2 to 4 items is going to be seven. And then what about be what we're going to I need to solve for what PF three is because we already have p of two. So it's good and write the equation also would be p of three minus p of two over three, minus two and just like four p of two ISS for endless federal PS three is up here, so let's go to give us three squared nine times two is 18 and then minus 15 plus six. So that's going to be three plus 609 So P three is nine. So go ahead and plug all that announced. So that would give us five Hoover one, which just simplifies down to five now for finding to instantaneously of change when X is equal to which they tells us the marginal profit. We're going to follow this equation here instead. So let's go ahead and start with first figuring out what is going to be p of war A is first to. So what is P of two plus H Now? You could just put everything in at one time, but I like to kind of space everything out. So let's figure out what this is. So it's going to be, too. And then it's gonna be two plus h squared, so it's gonna be four plus or H plus h squared. And then we multiply that by five would be negative 10 minus five h and then plus six. And then we'd have eight plus eight h plus two h squared. Then we could combine the constants. There we have negative five H minus four, and then we'd have eight plus negative four, which is going to give us four. And then we'd have eight h minus five h, which is going to give us plus three h and then we just have r plus two h squared. And then we can go ahead and do p of two plus h minus. And they would be P ah two. And we have already found out that p of two is four, so we can just go ahead and plug that in. So this would be four plus three h plus two h squared minus for so knows thes forces cancel out with each other. Then we can go ahead and take the limit as h purchase zero Oh, so it B P plus two h minus p of two all over h. And this is our third step and now doing that, so we'd have the limit as h approaches zero, the new mayor would have three h plus two h squared all over H now kind of is like a pro tip. Any time you get to this point here, at least when we're working with polynomial functions, were trying to find the instantaneous rate of change. If you can't have where your H can evenly divide everything, kind of like how we have it here, seeing as we would be left with just two plus h when we don't have the agent the denominator anymore. If for some reason you can't do that, you should go back and check your algebra over here again because you may have mess something up because it should always consult for polynomial. But it was We get to this, we take the limit as h Purchase zero and we know that's the continuous functions without just beat three plus zero, which gives us three. So the instantaneous rate of change is going to be three. Now we want to actually interpret what this means. Well, we have the prophet function, so for you to think about it, the rate of change formula here. So this formula right here. So in the top, we have the change and profit for me, right? This over change in profit in hundreds of dollars over the number of items sold. So we can interpret this in saying that if so, let me maybe walk a little piece of this off right here. So if we so a another item because we're increasing, So if we sell one more item, we expect 300 mawr dollars of profit. So that's essentially what we get for interpreting this. And then for the next one, they don't want us to actually interpret it. They just wanted us to find it at four. Almost forgot to write that for D. Here, this is X is equal to four, so he's gonna follow the same steps that we did before. So first we're going to have that are a is for So we're gonna pee of four plus h spell. Let's just put everything in. It would be too time So four plus H squared, which is going to be 16 plus eight h plus h squared and then it B minus 20 minus five age and then plus six. So simplifying this down, we'd have 32 plus 16 h plus two h squared, and the negative 20 plus six is going to give us negative 14 on B minus five h and then simplifying that down. We should end up with 18 plus 11 h plus two h squared. Then we need to do for plus h r p r for plus h minus p a for and let's see what p afford because I think we found that on the first page here, and we did, and we found that pf four is 18. So we just plugged that into would be 18 plus 11 each plus two h squared. And then we're going tohave minus 18. So the eight teams just cancel out. And then for our last step here, we're going to take the limit as H approaches zero and just going to be a p of four plus h minus P of four all over h. So plug it everything. And no, um, we already had the numerous. Should be 11 h plus two h squared over agent again. Like I was saying before, once we get to this point, we should have at least four polynomial is this age should evenly divide like this, and then we no longer have the agent the denominator, and then we could just apply the limit of this directly, since we know lines are continuous, and doing that would give 11 plus zero, which is just 11. So the instantaneous sort of change for selling four items is going to be 11


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