WEBVTT 00:00:02.400 --> 00:00:09.494 Cell phones. They are everywhere. They're always ringing everywhere. 00:00:09.496 --> 00:00:14.705 But how much does a cell phone cost? I don't mean the price. I don't mean the purchase price that you, 00:00:14.707 --> 00:00:20.503 the consumer, pays for it. I mean the cost. Like for example, what do you think it actually costs to 00:00:20.505 --> 00:00:28.901 manufacture this phone, or this phone, or this phone or this phone? There are lots of different styles, 00:00:28.903 --> 00:00:36.101 lots of different plans, and lots of different costs. And what does that have to do with applied calculus? 00:00:36.103 --> 00:00:41.902 Well, what has happened is we now have some tools. We understand limits. We understand continuity. 00:00:41.904 --> 00:00:46.601 We understand the definition of the derivative, and we understand how to find the derivative 00:00:46.603 --> 00:00:52.104 using derivative rules, so we're ready to actually use calculus for the first time. 00:00:52.106 --> 00:01:01.901 The topic for today is marginal analysis. We're going to use a cost function, but what we do here 00:01:01.903 --> 00:01:11.101 can be applied to also revenue functions or to profit functions. So let's take a look at some terms 00:01:11.103 --> 00:01:24.200 before we begin. First, we identify a cost function C of x. Now this is just like f of x, but we use a 00:01:24.202 --> 00:01:41.697 capital C to identify cost. This actually means the cost to produce x phones. 00:01:41.699 --> 00:01:50.006 So if we're talking about 500 phones, then C of x would represent the total cost to produce 500 phones. 00:01:50.008 --> 00:02:15.703 The total cost. Alright, C of x plus 1 then would represent the total cost to produce x plus 1 phones. 00:02:15.705 --> 00:02:22.105 So, suppose I wanted to figure out what it costs just to produce this one phone. 00:02:22.107 --> 00:02:32.304 Well, I would have to calculate the cost of all the x plus 1 phones, which would be this C of x plus 1 00:02:32.306 --> 00:02:40.800 and then the cost to produce all of the x phones and find the difference. 00:02:40.802 --> 00:02:49.300 Now that is possible with algebra. It's boring, it's tedious, and it's long, but it is doable. 00:02:49.302 --> 00:02:52.904 So what does this have to do with calculus, or why is it an improvement? 00:02:52.906 --> 00:02:57.600 Well, let's look at a few more things. Oh, and let's go back and label this. 00:02:57.602 --> 00:03:16.103 This is the cost to produce the x plus 1, or the x plus first, phone. 00:03:16.105 --> 00:03:23.803 So back to calculus. Well, if you remember the definition of the derivative, and we're going to 00:03:23.805 --> 00:03:35.500 use C of x instead of f of x, so we write C prime of x is the limit as h approaches 0, 00:03:35.502 --> 00:03:48.904 but we'll put C of x plus h this time, minus C of x, in place of f of x, divided by h. 00:03:48.906 --> 00:03:54.204 Ok, now you have to think about phones for a few minutes. Now you may have had a situation 00:03:54.206 --> 00:03:59.501 where you have dropped your phone and it is no longer in one piece. In that case you're 00:03:59.503 --> 00:04:04.501 probably aware of the fact that a fractional part of a phone is probably no good. 00:04:04.503 --> 00:04:12.503 So we can't really say that we're going to let h go to zero because we can't really have fractional 00:04:12.505 --> 00:04:21.904 parts of phones that are functional. So really the smallest number that this can change here is 1. 00:04:21.906 --> 00:04:29.105 So what we're going to do is we're going to say that C prime of x equals this definition, and it is 00:04:29.107 --> 00:04:42.802 approximately equal to the limit as h goes to 1, a slight difference, but for our needs this is what 00:04:42.804 --> 00:04:51.903 we're going to look at. So we're going to say copy the function, copy this function, and divide by h. 00:04:51.905 --> 00:04:58.802 So we're going to say that this is approximately equal to this for the sake of a whole phone. 00:04:58.804 --> 00:05:04.700 Well, this limit can be found by direct substitution. We couldn't sub in a 0 here, 00:05:04.702 --> 00:05:08.401 when we were working with the actual definition of derivative. But you see here, 00:05:08.403 --> 00:05:16.200 we can let h be 1, and so when we actually take this limit this part then equals... 00:05:16.202 --> 00:05:23.400 We take the limit. So we sub in a 1 here and a 1 here and what do we get? 00:05:23.402 --> 00:05:35.201 Well, you get C of x plus 1, minus C of x, over 1, and you realize that you've seen that somewhere 00:05:35.203 --> 00:05:43.100 else on the board, and it's up here. Now there's a slight difference between this one and this one 00:05:43.102 --> 00:05:52.001 because this is approximate. But we're talking about maybe hundredths of a cent when we actually 00:05:52.003 --> 00:05:59.001 get into problems with this. So what we're going to do is realize that C prime of x, 00:05:59.003 --> 00:06:05.301 the derivative of the cost function is approximately, let's write that down here, 00:06:05.303 --> 00:06:18.402 is approximately equal to this. So what do we have? 00:06:18.404 --> 00:06:24.800 Well we have a definition that we're going to be able to use here. This is called marginal analysis. 00:06:24.802 --> 00:06:32.199 Let's take a look. It says marginal cost is the approximate. Now why do we have to do that? 00:06:32.201 --> 00:06:38.000 Because we changed the definition here from h going to 0 to 1 because of the whole phone. 00:06:38.002 --> 00:06:48.300 So it's the approximate cost of producing 1 additional unit, or the x plus first, at a production level of x. 00:06:48.302 --> 00:06:58.301 And so, C prime of x is approximately equal to C of x plus 1 minus C of x. Now you can do this, 00:06:58.303 --> 00:07:04.399 but it's so much longer, so you're about to find out now how much easier it is to do this 00:07:04.401 --> 00:07:12.201 marginal analysis on a cost function using calculus. Alright, so let's look at an example. 00:07:12.203 --> 00:07:24.500 Example 1. Talk-Loud Incorporated manufactures cell phones at a cost C in dollars given by this function. 00:07:24.502 --> 00:07:30.598 Now of course someone has to come up with this cost function, ok. Well, this is based on a fixed cost 00:07:30.600 --> 00:07:38.400 of 4000 dollars, and then this is the cost of a phone times the square root of the number of units produced. 00:07:38.402 --> 00:07:44.001 Where does this come from? It comes from data that has just been collected over time by all different 00:07:44.003 --> 00:07:51.000 companies that manufacture cell phones. So here x is the number of phones produced and the question is: 00:07:51.002 --> 00:07:58.498 At a current production level of 400 phones, this must be like a weekly level because it certainly 00:07:58.500 --> 00:08:04.301 wouldn't be a yearly level, find the cost of producing the 401st phone. 00:08:04.303 --> 00:08:11.600 Ok, the first thing you notice about this problem is it doesn't say find the marginal cost. So you have to 00:08:11.602 --> 00:08:19.600 interpret the question and realize that when you're looking for the cost of one additional unit, 00:08:19.602 --> 00:08:31.100 in this case cell phones, that is marginal analysis. Alright, so here's the cost function. 00:08:31.102 --> 00:08:37.301 This is going to be measured in dollars, so it's 600 dollars times the square root of the number 00:08:37.303 --> 00:08:50.700 of phones produced plus 4000, and that's a fixed cost for setup of overhead things like that. 00:08:50.702 --> 00:08:56.100 Ok, now we remember that 00:08:56.102 --> 00:09:07.601 So we're going to use this approximation which means all we have to do is take the derivative. 00:09:07.603 --> 00:09:17.300 Let's rewrite this as a power function, x to the 1/2, so it will be 600 times x to the 1/2. 00:09:17.302 --> 00:09:29.798 Using the derivative rules, 1/2 times 600 is 300 x, then subtract 1, so that's 1/2 minus 1 which is 00:09:29.800 --> 00:09:37.600 negative 1/2, and then the derivative of a constant is 0. Now we're going to have to simplify 00:09:37.602 --> 00:09:42.399 because we're going to use this for something. So we're going to rewrite this function as 00:09:42.401 --> 00:09:52.401 300 over the square root of x. This is x to the negative 1/2, so it's the square root in the denominator. 00:09:52.403 --> 00:10:00.298 Then we evaluate this function. This is where the common error is on this problem. 00:10:00.300 --> 00:10:08.100 A lot of people think ok, well it's the 401st phone so we should sub in at 401. But you have to think about 00:10:08.102 --> 00:10:18.300 the current production level is at 400, and so you're finding the rate of change of the cost function at 400. 00:10:18.302 --> 00:10:31.601 So you take the derivative at 400, so that would be 300 over the square root of 400. So that's 00:10:31.603 --> 00:10:40.600 300 over square root of 400 is 20, and so what do I have? Well I have that, and so I have 00:10:40.602 --> 00:10:51.000 30 divided by 2, and so that's 15. Now this is a real problem, and we have units, and so we put dollars. 00:10:51.002 --> 00:10:57.901 Alright, now let's interpret what this is. This is called marginal analysis, and we have found that 00:10:57.903 --> 00:11:05.301 the cost, the actual cost, not the price but the cost of producing let's say this is the 401st phone 00:11:05.303 --> 00:11:09.601 that was produced in this manufacturing process. 00:11:09.603 --> 00:11:16.600 After 400 were produced, it cost 15 dollars to produce this phone. 00:11:16.602 --> 00:11:21.600 It's kind of amazing when you think about what you actually pay for something like this. 00:11:21.602 --> 00:11:32.301 So we realize here that by using calculus we can shortcut a lot of the tedious work. 00:11:32.303 --> 00:11:41.298 We could have evaluated this function, right here, at 401. That would be this part. 00:11:41.300 --> 00:11:47.800 Then we could have evaluated it at 400 and subtracted, or we could have been smart, 00:11:47.802 --> 00:11:51.701 which is what we did. Take the derivative and evaluate at the current production level. 00:11:51.703 --> 00:11:55.002 One simple calculation. Out comes the number. 00:11:55.003 --> 00:12:01.600 And so that is the first time we've been able to use calculus to actually do something real.