#Optimization calculus problems plus#

So to figure outĬritical points, we essentially have to find Quantity we need to produce in order to optimize our profit. We will have optimized or we will figure out the Maximum point, then we can say, well, let's produce that many. Of this profit function and are any of those To think about what are the critical points Profit function analytically, the easiest way is

#Optimization calculus problems plus#

To the third plus 6x squared minus 15x plusġ0x, so that is minus 5x. Simplify this as- let's see, we have negative x You subtract a 15x it becomes negative 15x, and then we can Squared, you subtract 6x squared it becomes positive, Ones that would actually you would get from To produce those shoes, you'll have $5,000 in profit. Know, $10,000 of revenue and it costs you $5,000 Your revenue as a function of x minus your cost asĬertain amount and let's say you bring in, I don't Of x for revenue and cost, what is profit as aįunction of x going to be? Well, your profitĪs a function of x is just going to be equal to This is also going to be in thousands of dollars. Thousands of pairs you produce squared plus 15 times Thousands of pairs you produce cubed minus 6 times the And so you hire aīunch of consultants to come up with what yourĬost is as a function of x. Have to build your factory, have to pay yourĮmployees, you have to pay the electricity bill. Now, it would be a niceīusiness if all you had was revenue and no costs. R is equal to 10, but that really means $10,000. Of pairs produced, if x is 1, that means 1,000 pairs produced Per pair for as many pairs as you're willing to give him. How much revenue, which is how much you actually

Now let's thinkĪbout how much money you're going to make per pair. Thousands of pairs of shoes to produce in order to Shoe factory and you're trying to figure out how many

(Another reason can be that Sal doesn't like to do videos of more than ten minutes and this one was I hope this helps as to why Sal 'skipped' this step, even though you are right in pointing out that it could have been included.

With time, the general shape of these equations will pop up in your mind as you do the math. Rather, I think he knew that the general shape of a -x^3 equation tells us that as x increases, y tends towards negative infinity. In fact, I don't think Sal didn't check the end behavior because he thought of it. So the 'bunch of consultants' who came with an equation for the costs couldn't have come up with an equation where the costs are always decreasing with an increase of the production, in the first place ^^Īnd while this law makes intuitive sense (ask to much of somebody or something and you'll kill the goose with the golden eggs), you don't need to take it for granted. It's an economic phenomenon called 'Law of diminishing returns'. Now, since we deal with a factory, there are reasons to believe that past a certain point, the more you add to the production, the less it will yield. It is a good question, and you are mathematically right.