WEBVTT 00:00:05.000 --> 00:00:07.700 Okay, what's the first thing we always do? 00:00:09.720 --> 00:00:10.620 Direct substitution. 00:00:14.820 --> 00:00:26.400 Okay, zero squared times sine of, what does one over zero go to? 00:00:30.400 --> 00:00:37.923 Any number in terms of limits over zero goes to either positive or negative 00:00:37.923 --> 00:00:38.673 infinity. 00:00:41.040 --> 00:00:43.880 We don't know until we do it from the left or from the right. 00:00:47.200 --> 00:00:48.100 Let's just call it infinity. 00:00:50.640 --> 00:00:55.520 And what does sine of infinity or sine of a very large number do? 00:00:56.540 --> 00:00:57.440 Go to. 00:00:58.140 --> 00:00:59.040 What's that? 00:01:02.220 --> 00:01:08.260 Let's, uh, so this goes to one of these. 00:01:12.440 --> 00:01:14.720 What does sine do as x gets larger and larger and larger? 00:01:17.460 --> 00:01:19.200 Think of the sine function, what does it do? 00:01:20.160 --> 00:01:21.440 It just keeps going, oscillating. 00:01:22.760 --> 00:01:23.960 It does not approach a number. 00:01:25.260 --> 00:01:26.160 Okay? 00:01:26.560 --> 00:01:29.740 Since sine does not approach a number, this does not exist. 00:01:34.420 --> 00:01:37.820 So zero times does not exist, I have no idea what that means. 00:01:39.360 --> 00:01:43.635 Okay, so the function value definitely doesn't exist zero, there's a hole there, or 00:01:43.635 --> 00:01:44.385 there's 00:01:43.940 --> 00:01:46.834 There may be a hole or some other behavior at zero; direct substitution has not 00:01:46.834 --> 00:01:47.584 settled it. 00:01:49.220 --> 00:01:51.640 Okay, so anyways, direct substitution does not work. 00:01:53.260 --> 00:01:54.160 So, 00:01:56.180 --> 00:01:57.520 we need another trick. 00:01:59.940 --> 00:02:03.220 It's one of our limit laws. Why don't you take out that limit laws. 00:02:03.220 --> 00:02:05.900 Okay, the very last one is called the squeeze theorem. 00:02:33.160 --> 00:02:34.560 And 00:02:34.560 --> 00:02:37.740 has two hypotheses. 00:02:38.320 --> 00:02:40.260 Let me look at what a hypothesis is in math. 00:02:42.200 --> 00:02:46.060 So, most theorems in math are an if 00:02:47.080 --> 00:02:50.240 something is true, then 00:02:50.240 --> 00:02:55.160 something is true. So all the things that must be true are called the hypotheses. 00:02:59.780 --> 00:03:01.240 You don't have to write that down. 00:03:01.920 --> 00:03:05.892 Okay, so this has two hypotheses, two things that have to be true for the conclusion 00:03:05.892 --> 00:03:06.642 to 00:03:06.140 --> 00:03:07.040 be true. 00:03:08.000 --> 00:03:16.500 Okay, so first off, we have to have three functions. 00:03:18.400 --> 00:03:21.909 The middle function must be always greater than some function and always less than 00:03:21.909 --> 00:03:22.659 some 00:03:22.160 --> 00:03:23.060 other function. 00:03:24.440 --> 00:03:30.314 and the limit as x approaches some number of this one and this one must equal the 00:03:30.314 --> 00:03:31.064 same 00:03:30.660 --> 00:03:31.560 number. 00:03:33.580 --> 00:03:41.920 So to use the squeeze theorem, we have to take our function and make sure it's bound 00:03:41.920 --> 00:03:47.480 just oscillates back and forth, they're always bound between two other functions. 00:03:49.540 --> 00:03:53.926 What's the largest, no matter what x is, what's the largest sine of some angle will 00:03:53.926 --> 00:03:54.676 be? 00:03:55.440 --> 00:03:56.340 1. 00:03:57.100 --> 00:03:58.380 Right, so here's the graph. 00:03:59.400 --> 00:04:00.300 Is this right? 00:04:00.960 --> 00:04:06.620 The maximum value here is at 1, and the minimum value is negative 1. 00:04:06.620 --> 00:04:26.404 So, this function, which is y equals 1 and y equals negative 1, it's bound between 00:04:26.404 --> 00:04:27.640 those. 00:04:27.640 --> 00:04:35.720 So we can always write this inequality. 00:04:37.300 --> 00:04:42.367 This is always less than or equal to one and always greater than or equal to 00:04:42.367 --> 00:04:43.117 negative one. 00:04:46.300 --> 00:04:50.060 The expression is at most one and at least negative one. 00:04:50.980 --> 00:04:53.780 You can have all kinds of functions inside the sine function. 00:04:53.780 --> 00:04:58.721 When we have sine of something we always know it's bound between negative one and 00:04:58.721 --> 00:04:59.471 positive one. 00:04:59.380 --> 00:05:05.220 And the only thing we're missing is this guy right here, okay, so we take this 00:05:19.600 --> 00:05:25.020 And we multiply all three sides by x squared. 00:05:33.460 --> 00:05:37.120 We multiply an inequality by a function or a number. 00:05:37.480 --> 00:05:39.140 When do we reverse the sign? 00:05:42.140 --> 00:05:43.760 When do we reverse the inequality? 00:05:44.520 --> 00:05:45.420 If it's negative. 00:05:45.360 --> 00:05:47.180 will x squared ever be negative? 00:05:48.280 --> 00:05:49.760 No, so we don't need to do that. 00:05:50.860 --> 00:05:53.960 If we did, if we just multiply by x, for example, 00:05:54.900 --> 00:05:57.760 this will still work out, but you have to do separate cases, 00:05:58.140 --> 00:05:59.260 but let's not worry about that. 00:05:59.780 --> 00:06:01.320 Okay, so now we got this inequality. 00:06:01.320 --> 00:06:04.800 And now this is that middle function here. 00:06:07.800 --> 00:06:13.360 And now we simply take the limit as x approaches zero of all three sides. 00:06:15.220 --> 00:06:18.420 We go the limit as x approaches zero. 00:06:18.420 --> 00:06:25.160 The limit as x approaches zero. The limit as x approaches zero. 00:06:25.160 --> 00:06:28.480 And what is the limit as x approaches zero of negative x squared? 00:06:30.280 --> 00:06:31.180 Zero. 00:06:31.860 --> 00:06:32.820 Just direct substitution. 00:06:33.140 --> 00:06:34.320 It's a continuous polynomial. 00:06:35.200 --> 00:06:36.100 That's zero. 00:06:37.680 --> 00:06:39.840 The limit as x approaches zero of x squared? 00:06:41.900 --> 00:06:43.320 Zero direct substitution. 00:06:48.400 --> 00:06:54.760 So this limit must be less than or equal to zero and greater than or equal to zero. 00:06:54.960 --> 00:06:56.240 So what must this limit equal? 00:06:56.960 --> 00:06:57.860 Zero. 00:06:58.600 --> 00:07:02.020 So we often write by the squeeze theorem. 00:07:08.220 --> 00:07:17.400 The limit as x approaches 0 of x squared sine of 1 over x equals 0. 00:07:26.300 --> 00:07:27.940 And when do we use the squeeze theorem? 00:07:29.720 --> 00:07:36.080 only special cases typically involving a sine or a cosine. 00:07:38.440 --> 00:07:45.260 Because we can always bound sine between one and negative one, same with cosine. 00:07:47.060 --> 00:07:47.960 Does that make sense? 00:07:48.660 --> 00:07:49.560 You just have to... 00:07:50.240 --> 00:07:51.360 How do I know to do that? 00:07:51.420 --> 00:07:55.380 Well, if everything else doesn't work, try it. 00:07:55.380 --> 00:08:02.380 And typically it involves a trig function. Let me graph these functions. 00:08:02.380 --> 00:08:03.280 x squared. 00:08:13.880 --> 00:08:16.860 Oh, geez, what did I do? To the power of sine of x? 00:08:16.860 --> 00:08:19.240 That's kind of cool. That's not the function. 00:08:23.920 --> 00:08:25.100 There it is. 00:08:27.420 --> 00:08:31.862 The function x squared times sine of one over x is always bounded between those two 00:08:31.862 --> 00:08:32.612 functions. 00:08:33.980 --> 00:08:36.100 If I zoom out. 00:08:41.060 --> 00:08:48.040 If you zoom in you see green functions are functions. 00:08:48.700 --> 00:08:51.640 You get squeezed in from both sides. 00:08:51.880 --> 00:08:53.040 That's why we call this squeeze theorem. 00:08:53.040 --> 00:08:55.700 sometimes called referred to as the sandwich theorem. 00:08:56.660 --> 00:08:59.540 That function's a sandwich between two functions. 00:09:03.880 --> 00:09:05.060 But there it is. 00:09:07.980 --> 00:09:10.040 Okay, so that's the squeeze theorem. 00:09:10.040 --> 00:09:14.120 Is this function continuous or over the numbers? 00:09:14.120 --> 00:09:27.989 No. No. That's discontinuous x equals four. Is it continuous x equals seven? Yes. 00:09:27.989 --> 00:09:28.980 How 00:09:28.980 --> 00:09:33.000 last year or the year before, how do we define what it meant to be continuous? 00:09:40.280 --> 00:09:41.900 There's no break in the function. 00:09:43.700 --> 00:09:47.787 Some of your teachers might have said if you can write the whole function without 00:09:47.787 --> 00:09:48.537 lifting 00:09:48.060 --> 00:09:49.800 up your pencil, it's continuous. 00:09:51.520 --> 00:09:52.800 That's not really a good definition. 00:09:53.760 --> 00:09:55.320 So we have a mathematical definition. 00:09:56.300 --> 00:10:01.160 And they didn't define it for you before because you didn't know what a limit was. 00:10:03.380 --> 00:10:07.800 So, the first thing we are going to do is define what it means to be continuous at a 00:10:07.800 --> 00:10:08.700 number. 00:10:10.680 --> 00:10:15.300 And here is the definition. 00:10:17.160 --> 00:10:18.340 Then we will all explain. 00:10:18.340 --> 00:10:19.940 I want you to write it down on the off-sides. 00:10:19.940 --> 00:10:25.280 Okay, so the definition, a function is continuous 00:10:25.280 --> 00:10:28.580 at some number, not on a set of numbers. 00:10:29.280 --> 00:10:34.220 At a single point, if, has to be defined, 00:10:35.400 --> 00:10:38.800 the limit must exist as x approaches that number. 00:10:39.840 --> 00:10:41.280 Really is all we care about. 00:10:43.900 --> 00:10:46.400 The limit as x approaches that number 00:10:46.400 --> 00:10:47.880 must equal the function value. 00:10:47.880 --> 00:10:53.620 When the limit equals the function value, the function is continuous at that value. 00:10:57.100 --> 00:10:59.260 It is continuous at that value. 00:11:04.300 --> 00:11:06.060 So here 00:11:06.060 --> 00:11:09.780 is continuous at 7 because the limit 00:11:09.780 --> 00:11:15.440 as X approaches 7 equals F of 7. 00:11:19.400 --> 00:11:20.300 It's 00:11:19.700 --> 00:11:23.040 discontinuous at 4 because the limit 00:11:23.040 --> 00:11:27.800 as X approaches 4 does not exist. 00:11:27.800 --> 00:11:29.800 So it definitely cannot be continuous. 00:11:35.860 --> 00:11:40.000 And then the next thing is it's continuous on an interval. 00:11:40.220 --> 00:11:41.120 What's an interval? 00:11:43.900 --> 00:11:46.660 It's a set of numbers between two numbers. 00:11:48.340 --> 00:11:50.580 We usually use interval notation. 00:11:51.660 --> 00:11:53.340 This case on an open interval. 00:11:54.440 --> 00:11:58.040 if it's continuous at every value on that interval. 00:12:01.360 --> 00:12:03.560 If it's continuous over all real numbers, 00:12:03.680 --> 00:12:05.020 we say it's continuous everywhere. 00:12:09.440 --> 00:12:12.720 So this function is continuous on this open interval 00:12:14.600 --> 00:12:16.640 and on this open interval. 00:12:20.240 --> 00:12:23.560 We say it's continuous on this set of numbers. 00:12:24.620 --> 00:12:29.265 It's continuous on every value inside this open interval, every value inside this 00:12:29.265 --> 00:12:30.015 open interval. 00:12:29.980 --> 00:12:35.800 What is the limit as x approaches 4 from the left? 00:12:36.020 --> 00:12:39.200 Let's call this 5. 00:12:39.200 --> 00:12:50.920 The limit as x approaches 4 from the left? 5. And what is f of 4? 5. 00:12:52.200 --> 00:12:59.300 Okay, if these are the same numbers, the function is definitely discontinuous at 4, 00:12:59.920 --> 00:13:05.940 however we say it's continuous from the left, because we went from the left. 00:13:08.720 --> 00:13:14.360 So it is continuous from the left because the function value equals that limit from 00:13:14.360 --> 00:13:15.260 the left. 00:13:15.540 --> 00:13:18.990 It is discontinuous from the right because the limit exists from the right but is 00:13:18.990 --> 00:13:19.740 not 00:13:19.220 --> 00:13:20.160 equal to the function value. 00:13:21.960 --> 00:13:32.220 So the definition being continuous from the left to the right is what I just said. 00:13:32.220 --> 00:13:34.040 No, that's not it. 00:13:35.420 --> 00:13:37.827 The function is continuous from the left at x equals a if the limit as x approaches 00:13:37.827 --> 00:13:38.960 a from the left equals f of a. 00:13:40.820 --> 00:14:01.000 f of x continuous on the left if the limit as x approaches at x equals a if f of x. 00:14:01.000 --> 00:14:03.420 equals f of a. 00:14:05.000 --> 00:14:07.200 f of a. 00:14:16.620 --> 00:14:19.680 The function is continuous from the right at x equals a if the limit as x approaches 00:14:19.680 --> 00:14:21.120 a from the right equals f of a. 00:14:22.700 --> 00:14:25.360 from the right 00:14:29.500 --> 00:14:38.820 at x equals a if the limit as x approaches a from the right 00:14:40.260 --> 00:14:42.400 equals F of. 00:14:42.400 --> 00:14:44.220 Let's call this fx. 00:15:04.520 --> 00:15:13.980 What is the limit as x approaches negative 2 of f of x? 00:15:19.320 --> 00:15:24.520 as x approaches 2, is it equal to 1? 00:15:25.780 --> 00:15:27.460 No. The answer is no. Why not? 00:15:29.660 --> 00:15:31.280 Because this limit 00:15:31.280 --> 00:15:34.560 as x approaches negative 2 00:15:34.560 --> 00:15:38.360 from the left does not exist. 00:15:39.460 --> 00:15:41.320 You can't approach it from the left. 00:15:42.540 --> 00:15:44.980 So this does not exist. 00:15:49.680 --> 00:15:53.720 The limit as x approaches negative 2 from the right, 00:15:54.620 --> 00:15:56.060 what does that equal? One. 00:15:57.940 --> 00:15:58.840 Okay. 00:16:03.560 --> 00:16:06.640 And what is the limit as x approaches 00:16:06.640 --> 00:16:10.180 4 from the left of f? 00:16:11.100 --> 00:16:20.260 5. The limit as x approaches 4 from the right does not exist. 00:16:20.260 --> 00:16:24.320 And the limit as x approaches 4. 00:16:25.340 --> 00:16:26.280 Same as x. 00:16:28.500 --> 00:16:30.580 Does not exist. 00:16:34.260 --> 00:16:35.160 OK. 00:16:42.840 --> 00:16:46.160 So is this function discontinuous at negative 2? 00:16:50.360 --> 00:16:51.760 It's a tricky question. 00:16:53.120 --> 00:16:54.020 How about a zero? 00:16:54.040 --> 00:16:54.940 Is it continuously zero? 00:16:56.140 --> 00:17:02.780 Is it continuous on every, or on this open interval? 00:17:04.400 --> 00:17:05.300 For sure. 00:17:06.740 --> 00:17:19.422 Ok, so, it's discontinuous at negative 2 because the limit, this limit does not 00:17:19.422 --> 00:17:22.140 equal f of 00:17:22.140 --> 00:17:23.080 negative 2. 00:17:26.020 --> 00:17:28.520 However, that's an end point, right? 00:17:28.860 --> 00:17:29.860 We have two end points. 00:17:31.020 --> 00:17:34.360 So, we define what it means to be continuous. 00:17:34.720 --> 00:17:42.056 We actually say this is continuous on the closed interval, even though it's not 00:17:42.056 --> 00:17:42.806 continuous 00:17:42.580 --> 00:17:43.480 at negative two. 00:17:44.140 --> 00:17:52.752 But we say it's continuous on the closed interval if the left endpoint is continuous 00:17:52.752 --> 00:17:53.900 from the 00:17:54.640 --> 00:17:55.740 Where do we have? 00:17:57.380 --> 00:18:03.586 If it's continuous from the right of the left endpoint, and if it's continuous from 00:18:03.586 --> 00:18:04.336 the 00:18:04.000 --> 00:18:09.860 left, the right endpoint, we say it's continuous on the closed interval. 00:18:13.200 --> 00:18:23.620 Okay, so this is the actual, so a function is continuous on the closed interval a to 00:18:23.620 --> 00:18:24.520 b. 00:18:25.400 --> 00:18:27.940 It has to be continuous everywhere inside the interval. 00:18:32.520 --> 00:18:38.848 And the limit as x approaches a, the left endpoint, from the right is equal to the 00:18:38.848 --> 00:18:39.598 function 00:18:39.220 --> 00:18:43.960 and B, the right endpoint, continuous from the left. 00:18:46.320 --> 00:18:50.360 Let me say it's continuous on that closed interval. 00:18:50.360 --> 00:18:52.260 Why do endpoint conditions matter? 00:18:57.380 --> 00:18:58.280 Well... 00:19:03.900 --> 00:19:05.620 Are we done, right? 00:19:22.220 --> 00:19:24.280 This is the square root of x, right? 00:19:28.820 --> 00:19:30.780 Is it continuous at x equals zero? 00:19:30.780 --> 00:19:38.680 No, because the limit as x approaches zero from the left doesn't exist. 00:19:39.780 --> 00:19:45.920 However, what interval is it continuous on? 00:19:51.780 --> 00:20:02.449 including zero because it's a left endpoint and as x approaches zero from the right 00:20:02.449 --> 00:20:03.199 does 00:20:03.160 --> 00:20:04.180 equal the function value. 00:20:04.940 --> 00:20:10.696 So it is continuous on this closed interval which is not talking about continuous at 00:20:10.696 --> 00:20:11.446 a 00:20:11.080 --> 00:20:11.980 point. 00:20:13.380 --> 00:20:17.898 It's as long as the left endpoint is continuous from the right, the right endpoint 00:20:17.898 --> 00:20:18.648 is continuous 00:20:18.500 --> 00:20:19.400 from the left. 00:20:20.020 --> 00:20:21.380 It's continuous on this interval. 00:20:21.380 --> 00:20:24.840 So what is the domain of this? 00:20:29.920 --> 00:20:31.180 Including zero, correct? 00:20:32.240 --> 00:20:39.380 Okay, so this is continuous on its domain. 00:20:42.760 --> 00:20:46.080 So in fact, every function we know 00:20:52.080 --> 00:20:55.120 is continuous on its domain. 00:20:56.100 --> 00:20:58.140 Every polynomial, every rational function, 00:20:58.260 --> 00:20:59.820 every radical function, every trigonometric, 00:20:59.820 --> 00:21:05.480 every exponential, every logarithmic function is continuous on its domain. 00:21:06.700 --> 00:21:08.760 If you know the domain, you know where it's continuous. 00:21:17.540 --> 00:21:22.761 You don't have to write, well, you can just say all functions we know are continuous 00:21:22.761 --> 00:21:23.740 on its domain. 00:21:23.740 --> 00:21:25.900 And then we've got a couple rules. 00:21:27.240 --> 00:21:29.300 Well, not rules, but properties. 00:21:32.780 --> 00:21:36.840 Any number times a continuous function is continuous. 00:21:39.540 --> 00:21:43.000 Seven times the square root of x is continuous on its domain. 00:21:44.220 --> 00:21:46.380 Any number times a continuous function is continuous. 00:21:46.380 --> 00:21:50.660 The sum or difference of continuous functions is continuous. 00:21:53.340 --> 00:22:01.640 Let's say g of x is sine of x plus x cubed or something. 00:22:02.300 --> 00:22:06.948 We know this is continuous because we know this is continuous and this is 00:22:06.948 --> 00:22:07.698 continuous. 00:22:07.880 --> 00:22:10.980 And the sum of continuous functions is continuous. 00:22:12.140 --> 00:22:14.440 The product of continuous functions is continuous. 00:22:18.280 --> 00:22:23.460 cosine of x times e to the x is continuous function. 00:22:24.140 --> 00:22:26.260 Trigonometric functions and exponential functions are continuous on their domains. 00:22:27.380 --> 00:22:29.220 All exponential functions are continuous. 00:22:29.580 --> 00:22:32.200 The product of continuous functions are continuous. 00:22:35.260 --> 00:22:44.780 The quotient is continuous is whenever g of any value is not equal to zero. 00:22:45.620 --> 00:22:46.820 And it won't be continuous there. 00:22:48.820 --> 00:22:53.952 But in scalar multiple any coefficient times continuous function, sum or difference, 00:22:53.952 --> 00:22:54.702 product 00:22:54.380 --> 00:22:57.980 of continuous functions and quotient are continuous. 00:22:57.980 --> 00:23:01.000 Where is this continuous? 00:23:07.540 --> 00:23:09.160 Is x to the fifth plus one continuous? 00:23:10.320 --> 00:23:11.220 No. 00:23:12.400 --> 00:23:15.300 It's a polynomial for sure continuous everywhere. 00:23:17.060 --> 00:23:18.260 Is e to the x continuous? 00:23:20.920 --> 00:23:22.900 Yeah, all exponential functions are continuous everywhere. 00:23:25.360 --> 00:23:27.160 Will e to the x ever be zero? 00:23:31.640 --> 00:23:34.680 All exponential functions always give you a positive value. 00:23:34.680 --> 00:23:37.000 since this will never be zero, 00:23:38.000 --> 00:23:39.980 this is definitely a continuous function 00:23:40.840 --> 00:23:42.240 over all real numbers. 00:23:43.720 --> 00:23:45.920 So we're looking for whether a function is continuous 00:23:45.920 --> 00:23:50.520 or not, especially with the quotients. 00:23:51.700 --> 00:23:53.480 You have to check, is this continuous? 00:23:53.660 --> 00:23:54.560 Where is this continuous? 00:23:54.640 --> 00:23:55.540 Where is this continuous? 00:23:55.980 --> 00:23:57.700 And then a special case to quotients, 00:23:58.340 --> 00:23:59.500 we gotta make sure it's not zero. 00:24:00.120 --> 00:24:02.600 Whenever the denominator is zero, it's not continuous. 00:24:02.600 --> 00:24:04.520 It's discontinuous. 00:24:07.220 --> 00:24:09.980 And then one more property. 00:24:12.100 --> 00:24:17.540 The composition of continuous functions are continuous. 00:24:22.300 --> 00:24:24.880 Just write that. 00:24:25.080 --> 00:24:27.580 The composition of continuous functions are continuous. 00:24:27.580 --> 00:24:31.680 F composed with G as long as F and G are continuous 00:24:32.520 --> 00:24:34.160 F of G is continuous 00:24:34.160 --> 00:24:35.060 Example... 00:24:34.600 --> 00:24:43.098 I have no idea what this function looks like, but do we know whether it is 00:24:43.098 --> 00:24:44.160 continuous or 00:24:44.160 --> 00:24:45.060 not? 00:24:47.560 --> 00:24:50.000 Is sine of x continuous over all real numbers? 00:24:51.340 --> 00:24:55.540 Yes, is 2 to the x continuous over all real numbers? Yes. 00:24:56.620 --> 00:24:58.380 So I plugged in sine of x 00:24:58.980 --> 00:25:00.400 into this function 00:25:01.300 --> 00:25:03.880 with guarantees continuous over all real numbers. 00:25:05.620 --> 00:25:06.520 Okay. 00:25:05.720 --> 00:25:08.140 Okay, let's do some examples. 00:25:08.140 --> 00:25:11.640 Let's see how far we go. 00:25:11.640 --> 00:25:13.820 I think we did 00:25:13.820 --> 00:25:15.280 a similar example. 00:25:15.440 --> 00:25:16.660 Let's do this. 00:25:25.280 --> 00:25:26.180 Okay. 00:25:27.780 --> 00:25:30.920 Actually, let me do this plus one. 00:25:33.520 --> 00:25:41.060 And then, e to the x, get the x to the first one. 00:25:41.060 --> 00:25:45.600 So where, or on what, where is this continuous? 00:25:56.100 --> 00:25:57.380 So how are we going to do this? 00:26:03.660 --> 00:26:07.860 So, we have to check, so any piecewise functions, we have to check each piece 00:26:07.860 --> 00:26:08.610 separately. 00:26:10.280 --> 00:26:11.180 Okay. 00:26:13.840 --> 00:26:26.609 So if x is greater than or equal, let's go with greater than zero, greater than pi, 00:26:26.609 --> 00:26:27.360 then 00:26:27.360 --> 00:26:31.480 f of x equals sine of x, right? 00:26:36.440 --> 00:26:37.520 Is sine of x continuous? 00:26:39.620 --> 00:26:40.520 Definitely. 00:26:41.540 --> 00:26:44.120 OK, so we definitely know it's continuous from pi to infinity. 00:26:48.340 --> 00:27:12.283 For zero less than or equal to x and x less than pi, f of x equals x squared plus 00:27:12.283 --> 00:27:13.480 one. 00:27:18.240 --> 00:27:31.407 Then if x is less than zero, f of x equals e to the x, which is an exponential 00:27:31.407 --> 00:27:32.157 function, 00:27:32.840 --> 00:27:33.740 continuous everywhere. 00:27:35.680 --> 00:27:38.760 We know it's continuous here. 00:27:39.300 --> 00:27:45.455 So we check each piece. As long as they are continuous, it's continuous on the open 00:27:45.455 --> 00:27:46.205 interval. 00:27:47.580 --> 00:27:53.460 So really all we have to do is check at pi and at zero. 00:28:03.520 --> 00:28:05.580 at x equals zero 00:28:09.140 --> 00:28:11.200 okay this is where it changes 00:28:12.440 --> 00:28:16.560 from this function to this function 00:28:18.520 --> 00:28:21.700 we have to do the limit from the left and the limit from the right separately 00:28:21.700 --> 00:28:28.880 So what is the limit as x approaches... 00:28:28.880 --> 00:28:31.500 uh... sorry, let's go to pi first, my bad. 00:28:33.020 --> 00:28:34.680 pi from the left. 00:28:43.540 --> 00:28:45.160 So from the left, 00:28:45.740 --> 00:28:47.240 f of x is equal to this. 00:28:48.300 --> 00:28:49.540 So this is the limit 00:28:49.540 --> 00:28:54.820 as x approaches pi from the left of x squared plus one. 00:28:56.700 --> 00:29:01.660 And what does that equal? Direct substitution of pi squared plus one. 00:29:02.880 --> 00:29:03.780 Sum number. 00:29:07.760 --> 00:29:13.360 The limit as x approaches pi from the right. 00:29:22.160 --> 00:29:28.320 limit as x approaches pi from the right. 00:29:28.900 --> 00:29:32.400 So f of x is equal to sine of x. 00:29:32.560 --> 00:29:40.740 which equals sine of pi which equals 00:29:44.640 --> 00:29:47.460 zero thank you okay 00:29:50.160 --> 00:29:52.960 so what is the limit as x approaches pi 00:29:57.610 --> 00:30:01.130 It does not exist. 00:30:08.930 --> 00:30:13.290 Is this function continuous from the left or the right or neither at pi? 00:30:19.150 --> 00:30:20.790 What is f of pi? 00:30:25.670 --> 00:30:30.050 Pi is here, so it's sine of pi, which is zero. 00:30:33.710 --> 00:30:35.990 So is it continuous from the left or the right? 00:30:38.230 --> 00:30:39.130 At pi. 00:30:40.870 --> 00:30:44.010 From the right. So, final with this. 00:30:50.610 --> 00:30:52.250 Our function looks like this. 00:30:53.930 --> 00:30:55.650 So it is continuous on the right. 00:30:59.590 --> 00:31:00.490 What is it? 00:31:00.270 --> 00:31:03.010 x squared plus one. Looks like that. 00:31:03.890 --> 00:31:09.152 sorry, open circle here, is discontinuous on the left, and it's definitely 00:31:09.152 --> 00:31:09.902 discontinuous 00:31:09.590 --> 00:31:10.490 at pi. 00:31:12.570 --> 00:31:14.210 Now let's check at zero. 00:31:15.490 --> 00:31:25.748 The limit as x approaches zero from the left, the limit as x approaches zero from 00:31:25.748 --> 00:31:27.030 the left, 00:31:27.030 --> 00:31:33.010 From the left, f of x is right here. 00:31:33.990 --> 00:31:35.890 x squared plus 1. 00:31:41.150 --> 00:31:45.290 Direct substitution plus 1. 00:31:49.950 --> 00:31:50.850 Question? 00:31:54.110 --> 00:31:55.750 Oh, I'm sorry. 00:31:55.750 --> 00:32:03.979 I was looking at the wrong thing. This is 0 from the left. This is e to the x. My 00:32:03.979 --> 00:32:04.729 bad. 00:32:11.410 --> 00:32:22.750 Sorry, this is e to the x, which is e to the 0, which is 1. So the limit as x 00:32:22.750 --> 00:32:23.500 approaches 00:32:23.290 --> 00:32:33.010 0 from the right, the limit as x approaches 0 from the right of, this was the x 00:32:33.010 --> 00:32:33.760 squared 00:32:33.550 --> 00:32:47.779 plus 1, which is 0 squared plus 1, which is 1. So, the limit as x approaches 0 is 1. 00:32:47.779 --> 00:32:48.529 And 00:32:48.490 --> 00:32:59.970 f of zero is this guy, zero squared plus one. 00:33:04.490 --> 00:33:13.550 So it's definitely continuous at zero. 00:33:13.550 --> 00:33:18.890 Ok, any questions on this? 00:33:19.590 --> 00:33:22.970 So for piecewise we just gotta check, the main thing we gotta do is check where the 00:33:22.970 --> 00:33:24.650 two functions meet. 00:33:25.210 --> 00:33:27.590 And then of course inside you have to check where is the continuous. 00:33:27.590 --> 00:33:29.570 I chose three continuous functions. 00:33:29.570 --> 00:33:33.830 OK, the intermediate value theorem. 00:33:35.730 --> 00:33:37.570 Last quick, this is a quick topic. 00:33:38.430 --> 00:33:39.730 Let me find it. 00:33:39.730 --> 00:33:40.670 Shh. 00:33:40.670 --> 00:33:43.110 Okay, write this down and then we'll talk later. 00:33:43.110 --> 00:33:44.510 Okay. 00:33:44.510 --> 00:33:47.430 Okay, so this is called an existence theorem. 00:33:48.690 --> 00:33:55.470 We're gonna have three, sort of four different existence theorems in calculus. 00:33:56.930 --> 00:34:00.090 All it does is tell you a value exists. 00:34:01.050 --> 00:34:03.090 It doesn't help us find the value at all. 00:34:03.570 --> 00:34:05.990 It just guarantees for it to exist. 00:34:06.630 --> 00:34:10.142 This is an existence theorem: it guarantees a value exists without telling us how to 00:34:10.142 --> 00:34:10.892 find it. 00:34:10.930 --> 00:34:12.030 Anybody remember this? 00:34:13.470 --> 00:34:18.290 But technically we don't really have it till this class. 00:34:18.370 --> 00:34:20.010 Why is that? 00:34:26.930 --> 00:34:28.770 Because we don't need it, somebody said. 00:34:30.990 --> 00:34:31.890 This is not true. 00:34:31.850 --> 00:34:33.330 We definitely need it. 00:34:48.150 --> 00:34:53.610 Here we go. 00:34:55.410 --> 00:34:57.510 What about this has to do with calculus? 00:35:00.750 --> 00:35:03.490 I don't see anything about limits in here. 00:35:05.170 --> 00:35:06.110 Those are good. 00:35:06.350 --> 00:35:07.690 Well, their sort is. 00:35:10.490 --> 00:35:11.390 OK. 00:35:12.470 --> 00:35:22.974 The only hypothesis, hypothesis C, is f has to be a continuous function on a closed 00:35:22.974 --> 00:35:23.724 interval. 00:35:27.330 --> 00:35:31.650 And truly we didn't define what it means to be continuous till just today. 00:35:32.750 --> 00:35:36.781 Before your teacher said, well you can draw it without lifting up your pencil, it's 00:35:36.781 --> 00:35:37.531 continuous. 00:35:38.150 --> 00:35:45.486 But truly right, it's continuous on a closed interval, means the limit as x 00:35:45.486 --> 00:35:46.236 approaches 00:35:47.590 --> 00:35:50.370 some other value, let me call it C, 00:35:51.350 --> 00:35:56.468 Continuity on the closed interval includes continuity from the right at a and from 00:35:56.468 --> 00:35:57.930 the left at b. 00:35:58.590 --> 00:36:00.430 for every value on the open interval. 00:36:01.430 --> 00:36:05.530 And it's continuous from the left at b, 00:36:05.790 --> 00:36:06.970 and it's continuous from the right at a. 00:36:07.890 --> 00:36:10.370 Okay, we now know the actual definition. 00:36:10.950 --> 00:36:14.670 So that's why it's part of this class, okay? 00:36:14.670 --> 00:36:28.062 But all this thing says is, as long as you have a continuous function, let's put in 00:36:28.062 --> 00:36:28.850 some 00:36:28.850 --> 00:36:40.210 numbers for class one and eight or something, two and ten. 00:36:40.210 --> 00:36:48.912 as long as we have a continuous function, let n be any number between f of a and f 00:36:48.912 --> 00:36:49.662 of 00:36:49.370 --> 00:36:59.030 b. What is f of b? 10. f of a is 2. 00:37:01.650 --> 00:37:06.730 n can be any number between 10 and 2. Any number here. 00:37:16.430 --> 00:37:23.707 And then we are guaranteed, actually as long as f of a is not equal f of b, that's 00:37:23.707 --> 00:37:24.457 the 00:37:24.090 --> 00:37:25.790 one thing, it's definitely not equal. 00:37:27.210 --> 00:37:35.990 There must exist a value c, there must be a c for any number, let's call this number 00:37:35.990 --> 00:37:51.590 n there must be a C in the open interval such that F of C equals n. So for example, 00:37:58.010 --> 00:38:04.490 5, are we guaranteed to have a value between 1 and 8? 00:38:05.110 --> 00:38:06.450 Yes, we are guaranteed. 00:38:06.730 --> 00:38:15.559 Because 5 is between 10 and 2, we are guaranteed a value in 1 to 8, the open 00:38:15.559 --> 00:38:16.309 interval. 00:38:18.010 --> 00:38:20.030 such an F of C equals five. 00:38:21.510 --> 00:38:24.030 Our function takes on every single value 00:38:24.030 --> 00:38:26.470 and sometimes it can take on it twice, right? 00:38:26.490 --> 00:38:30.210 So in fact, there are three different C values, 00:38:31.390 --> 00:38:32.930 C one, C two, C three, 00:38:34.070 --> 00:38:36.390 but this theorem intermediate value theorem 00:38:36.390 --> 00:38:38.670 just guarantees there's at least one. 00:38:40.530 --> 00:38:42.530 When is this used? 00:38:42.530 --> 00:38:50.530 And typically something like this. 00:38:54.720 --> 00:39:00.260 Show where is a 00:39:04.480 --> 00:39:12.660 Show that f has a zero between zero and two. 00:39:28.900 --> 00:39:30.980 Now what's a zero of a function? 00:39:34.860 --> 00:39:37.340 A zero of a function is an x value. 00:39:38.460 --> 00:39:53.160 Okay, a zero is an x value that makes f of x equal to zero. 00:39:57.080 --> 00:40:01.240 So to show that, we're going to use the intermediate value term. 00:40:01.920 --> 00:40:12.080 Since f is a polynomial, it is continuous on the closed interval from zero to two. 00:40:12.300 --> 00:40:13.460 And why is that? 00:40:20.980 --> 00:40:23.520 How do we know this is continuous on the closed interval? 00:40:25.620 --> 00:40:35.220 because it's a polynomial, are continuous everywhere. 00:40:41.720 --> 00:40:45.760 And what is f of zero equal to? 00:40:47.380 --> 00:40:50.140 Zero cubed minus four, negative four. 00:40:50.140 --> 00:40:58.560 F of 2 is 2 cubed minus 4 is 4. 00:41:01.900 --> 00:41:23.412 By the Intermediate Value Theorem, there exists a c in the open interval from zero 00:41:23.412 --> 00:41:26.280 to two. 00:41:26.280 --> 00:41:31.520 the open interval such that 00:41:34.280 --> 00:41:38.080 f of c equals zero because zero 00:41:38.080 --> 00:41:43.140 is between negative four 00:41:43.140 --> 00:41:44.040 and four 00:41:53.660 --> 00:41:59.280 This function looks something like this, right? 00:42:01.360 --> 00:42:06.200 This was negative four. 00:42:09.240 --> 00:42:10.700 This was positive four. 00:42:12.600 --> 00:42:16.480 Our function since it's continuous it takes on every single value here. 00:42:19.220 --> 00:42:21.860 And this would be our value of C right there. 00:42:22.320 --> 00:42:23.580 Such a F of C is zero. 00:42:24.660 --> 00:42:25.560 Okay. 00:42:25.680 --> 00:42:28.910 We're only going to use it a few times but sometimes we're going to have to mention 00:42:28.910 --> 00:42:29.660 the 00:42:29.100 --> 00:42:30.080 intermediate value theorem. 00:42:31.480 --> 00:42:32.380 And so on. 00:42:32.180 --> 00:42:33.180 Okay we're going to stop here.