 Let's look at another example of using the photo one theorem of calculus to calculate the net change that is the the change of our rate of change In this time, let's look at an example where we take marginal costs and compare it to the cost function So if we know that the marginal cost of manufacturing X yards of a certain fabric is given as C prime of X equals 3 minus 0.01 x plus point how many zeros was that zero point zero zero zero zero zero six x squared nailed it And that's going to be measuring dollars Dollars per yard there in terms of our production We want to find the increase of cost from the production level of 2,000 yards to 3,000 or to 4,000 yards So if we know the marginal cost larger cost represents the cost per unit there In this case cost per yard. And so if we were to integrate this that gives us the net change And so this is the fundamental term calculus in action here. So we want to figure out. What's the cost from going from 4,000 yards to? Sorry going from 2,000 yards to 4,000 yards. It's this difference Well, we know from the fundamental theorem of calculus that the integral from 2,000 yards to 4,000 yards if we integrate our marginal cost function will do exactly that So putting in the marginal cost there Make sure you get the right number of zeros. There's a lot of those floating around. We get three minus point zero one X Plus, let's do it zero zero zero zero six X squared DX Well our anti-derivative even with all these zeros here is basically the same as we would for any other polynomial We got three X Minus point zero one X squared over two plus point zero zero zero zero zero six X cubed over three and Evaluate that from two thousand to four thousand. All right. Well, we can simplify some of those coefficients, right? Because we have that two goes into point zero one That'll happen point zero zero five times X squared and then point our three goes in this as well. We're gonna get point zero zero zero zero zero two X cubed evaluate at two thousand and four thousand and So the calculus is really painless in this situation. It turns out again with these type of problems It's always the arithmetic. We're gonna plug in the four thousand first And so when we do that we're gonna get three times four thousand minus point zero Zero five times four thousand squared and Then we get point zero zero zero zero zero two times four thousand Cubed that's the first bits attract from it three times two thousand Minus point zero zero five times the two thousand squared Plus the point zero zero zero zero two Two thousand Cube right there and I did position things so they actually have some like terms combined together if you think of terms of the coefficients So we could just we could compute this as three thousand or three times four thousand minus two thousand We're also gonna get negative point zero zero five. We're gonna times this by four thousand squared minus two thousand squared And then lastly you get point zero zero zero zero zero two times four thousand Cubed minus two thousand cubed. I mean that can help with the arithmetic a little bit Again, there's still a lot going on here four thousand take away two thousand. That's easy enough. That's just two thousand right there Notice you take four squared and you take away from that two squared Four squared is sixteen two squared is four so Sixteen take away four is twelve if you kept track of all of the zeros So you're gonna get a twelve there Four thousand you got your I mean that's there's already got three zeros when you square that you're gonna get six zeros now So this is gonna look like twelve million and then lastly See if I can fit this in here So many zeros This time around we're gonna take Four cubed which is sixty four and subtract from it to cube which is eight sixty four take away eight is fifty six And this time you're gonna have nine zeros behind it. So we got fifty six billion Sweet I filled it in there. And so then again, you can move some of these decimal places around because of the decimals We have there three times two thousand of course is six thousand If you take twelve million times point zero zero five, that's actually gives negative sixty thousand We can move the decimal place a little bit there and then lastly to take the point zero zero zero. Oh I think I forgot a zero there. Whoops a daisy. There should be five zeros there point zero zero zero two times fifty six billion that gives you a hundred and twelve thousand and Then adding all those together we end up with fifty eight thousand and So this will be fifty eight thousand dollars which gives us the cost increase of going from 2,000 yards of fabric to four thousand yards of fabric and so see with this one right here It was the arithmetic that really is the problem use a calculator. Don't be a hero Help you help yourself out there But we can use we can integrate a derivative to try to find the net change of The quantity here in this case it was a change of cost