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“We finally made it to our discussion of profit. And this is where we get get to make some money for all these widgets. We ve been building. It should clear that the way we compute profit is take our revenue and subtract our costs.
And so that s also how we make our profit function. Now in this case. We re talking about capital p. Of x.
This is not the demand equation. This is a profit function. Capital p. Of x.
And the way we make that function is simply take our revenue function and subtract our cost function we can also talk about marginal profit. Which is the profit..
We ll make by selling that and producing and selling that next item and because we have these rules for derivatives the marginal profit is simply the marginal revenue minus the marginal cost so let s get right into an example from example for. Assume the cost function of c of. X. Equals 100x.
Plus. 200000. And we want to find the profit function the marginal profit function we re gonna plug 2000 into the marginal profit function and we re also going to sketch the graph of the profit now in order to complete this exercise. We really need to know the revenue function from example.
4. So let s add that to the picture and so here s that revenue function from example 4. It was in a different video hopefully you check that one out before you watching this one and so now let s go ahead and complete these. 4 exercises so for part a we re looking for the profit function p of.
X we simply take the revenue function which is. Negative..
002 x squared plus 400 x and we subtract the cost function now listen it s important that we conclude these parentheses. When we start because this negative sign right here is going to distribute to all the terms in the cost function and then we can combine like terms. But i wanted to include these parentheses. So that you understand we re taking the revenue function and subtracting the cost function that subtraction may include distributing the negative to several terms in this case.
Two terms and then we can combine like terms so when we do this there s only one x squared term that s negative. 002 x squared but we do have two x terms 400 x. And negative 100. X.
Squared excuse. Me 100. X. And their sum is 300.
X. And then there s only one constant term in this case..
Now because i distribute that negative sign. It s a minus 200 thousand. And this is our profit function. The marginal profit function then is the derivative of that thing which is going to be.
Negative 004 x plus 300 and then finally to compute the marginal profit at a production level of 2000. Originally plug 2000 into this equation where x is right here and i ve already done that on my calculator and you can verify my calculations. But this turns out to be 220 dollars two hundred twenty dollars per per widget and that s an approximation of the amount of money we ll make if we produce and sell the two thousand first widget. We ll make two hundred twenty dollars.
Which is good news. We re still making money finally. Let s take a look at the graph so part d asked us to graph our profit function. So this is p of x typed.
It in on a computer graph or similar to say. Decimals..
Comm. And you can see that what we re going to end up wanting to locate is the place where we re gonna make the most money our profit. Function maxes out way up here right way up here at over 900000 and it s gonna occur somewhere between. 6000 and 8000 and so as we get more calculus under our belt.
We re gonna learn how to figure out exactly where we re gonna make the most profit for now. Let s just focus on the fact that we can create a profit function. We can do some marginal analysis. We can find the derivative and we can compute the marginal profit at various production.
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