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[{"Name":"Hyperbolic Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Hyperbolic Functions - Introduction - Part A","Duration":"8m 45s","ChapterTopicVideoID":28997,"CourseChapterTopicPlaylistID":293022,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/28997.jpeg","UploadDate":"2022-05-20T06:26:04.5930000","DurationForVideoObject":"PT8M45S","Description":null,"MetaTitle":"Hyperbolic Functions - Introduction - Part A: Video + Workbook | Proprep","MetaDescription":"Hyperbolic Functions - Hyperbolic Functions. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/calculus-i%2c-ii-and-iii/hyperbolic-functions/hyperbolic-functions/vid30549","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.405","Text":"New topic, Hyperbolic Functions,"},{"Start":"00:03.405 ","End":"00:06.735","Text":"which are analogous to circular functions."},{"Start":"00:06.735 ","End":"00:10.595","Text":"These are to the hyperbola what circular functions are to the circle,"},{"Start":"00:10.595 ","End":"00:13.605","Text":"so let\u0027s review circular functions."},{"Start":"00:13.605 ","End":"00:15.930","Text":"These are the trigonometric functions."},{"Start":"00:15.930 ","End":"00:17.189","Text":"But when you say circular,"},{"Start":"00:17.189 ","End":"00:21.000","Text":"it means that the argument is in radians."},{"Start":"00:21.000 ","End":"00:23.890","Text":"The main ones are sine and cosine."},{"Start":"00:23.890 ","End":"00:25.950","Text":"This is how we write them,"},{"Start":"00:25.950 ","End":"00:29.040","Text":"and they\u0027re related to the circle."},{"Start":"00:29.040 ","End":"00:30.600","Text":"A picture will help."},{"Start":"00:30.600 ","End":"00:38.070","Text":"Take a circle of radius 1 and take a point on the circle parameterized by t,"},{"Start":"00:38.070 ","End":"00:40.230","Text":"and there\u0027s 2 ways of looking at t: It\u0027s"},{"Start":"00:40.230 ","End":"00:45.330","Text":"either the arc length or you could say that t is twice the area,"},{"Start":"00:45.330 ","End":"00:49.550","Text":"or the area is half of t. For each such t,"},{"Start":"00:49.550 ","End":"00:54.328","Text":"we get a point on the circle and the x coordinate is cosine t,"},{"Start":"00:54.328 ","End":"00:57.380","Text":"and the y coordinate is sine t. This,"},{"Start":"00:57.380 ","End":"01:00.440","Text":"of course, is the equation of that circle."},{"Start":"01:00.440 ","End":"01:04.790","Text":"It implies that we have the identity that sine^2 plus cosine^2"},{"Start":"01:04.790 ","End":"01:09.318","Text":"equals 1 because this point is on the circle,"},{"Start":"01:09.318 ","End":"01:11.600","Text":"and does other circular functions,"},{"Start":"01:11.600 ","End":"01:15.308","Text":"and they are defined in terms of the sine and cosine,"},{"Start":"01:15.308 ","End":"01:18.394","Text":"and we also have inverse circular functions."},{"Start":"01:18.394 ","End":"01:20.690","Text":"You prefix them with arc,"},{"Start":"01:20.690 ","End":"01:26.200","Text":"or you put a minus 1 to indicate inverse function."},{"Start":"01:26.200 ","End":"01:31.700","Text":"That\u0027s basically that for the circular functions and now the hyperbolic functions."},{"Start":"01:31.700 ","End":"01:34.205","Text":"Instead of starting with a circle,"},{"Start":"01:34.205 ","End":"01:36.905","Text":"we start with a hyperbola."},{"Start":"01:36.905 ","End":"01:41.060","Text":"The main ones are hyperbolic sine and hyperbolic cosine."},{"Start":"01:41.060 ","End":"01:44.000","Text":"Instead of the circle x^2 plus y^2 equals 1,"},{"Start":"01:44.000 ","End":"01:47.480","Text":"we have the hyperbola x^2 minus y^2 equals 1,"},{"Start":"01:47.480 ","End":"01:52.295","Text":"and we represent this right side of the hyperbola."},{"Start":"01:52.295 ","End":"01:53.840","Text":"T is a parameter,"},{"Start":"01:53.840 ","End":"01:56.075","Text":"t is twice the area,"},{"Start":"01:56.075 ","End":"01:58.145","Text":"or the area is half of t,"},{"Start":"01:58.145 ","End":"02:02.510","Text":"and each t gives us cosine hyperbolic sine hyperbolic."},{"Start":"02:02.510 ","End":"02:04.130","Text":"But on the other side,"},{"Start":"02:04.130 ","End":"02:09.505","Text":"we take the area to be negative where t is minus twice the area."},{"Start":"02:09.505 ","End":"02:15.305","Text":"The functions are called hyperbolic sine and hyperbolic cosine,"},{"Start":"02:15.305 ","End":"02:21.760","Text":"and there\u0027s an extra h in the name, h for hyperbolic."},{"Start":"02:21.760 ","End":"02:25.675","Text":"They\u0027re called for short cosh and sinh,"},{"Start":"02:25.675 ","End":"02:28.165","Text":"though I usually just say it in full,"},{"Start":"02:28.165 ","End":"02:30.740","Text":"cosine hyperbolic, sine hyperbolic."},{"Start":"02:30.740 ","End":"02:35.270","Text":"I\u0027ve even heard of this called sinch, various pronunciations."},{"Start":"02:35.270 ","End":"02:38.130","Text":"Anyway, we will focus on the writing."},{"Start":"02:38.380 ","End":"02:40.760","Text":"Here\u0027s the definition."},{"Start":"02:40.760 ","End":"02:46.334","Text":"The cosine hyperbolic is e^x plus e^minus x over 2."},{"Start":"02:46.334 ","End":"02:48.950","Text":"The sine hyperbolic is the same thing,"},{"Start":"02:48.950 ","End":"02:51.540","Text":"but with a minus here."},{"Start":"02:51.820 ","End":"02:55.325","Text":"We\u0027ll prove not the bit about the area,"},{"Start":"02:55.325 ","End":"02:59.804","Text":"that\u0027s less important, but the identity that cosine hyperbolic,"},{"Start":"02:59.804 ","End":"03:04.280","Text":"sine hyperbolic really are on the hyperbola x^2 minus y^2 equals 1,"},{"Start":"03:04.280 ","End":"03:07.070","Text":"which means that we have to show this identity."},{"Start":"03:07.070 ","End":"03:10.105","Text":"Well, I won\u0027t do it now, but we\u0027ll do it a little bit later."},{"Start":"03:10.105 ","End":"03:12.170","Text":"Just like with the circular functions,"},{"Start":"03:12.170 ","End":"03:14.780","Text":"there are 4 others: tangent hyperbolic,"},{"Start":"03:14.780 ","End":"03:17.360","Text":"cotangent hyperbolic, secant hyperbolic,"},{"Start":"03:17.360 ","End":"03:22.380","Text":"cosecant hyperbolic, and they are defined as follows."},{"Start":"03:22.490 ","End":"03:26.405","Text":"We also have inverse hyperbolic functions,"},{"Start":"03:26.405 ","End":"03:28.670","Text":"just like we had with circular."},{"Start":"03:28.670 ","End":"03:29.960","Text":"In the case of circular,"},{"Start":"03:29.960 ","End":"03:33.725","Text":"we had the prefix arc because of the arc length."},{"Start":"03:33.725 ","End":"03:36.125","Text":"Here, we have the prefix just ar,"},{"Start":"03:36.125 ","End":"03:42.170","Text":"and ar is short for area because the parameter is related to the area."},{"Start":"03:42.170 ","End":"03:49.040","Text":"Similarly, all the other 5 hyperbolic inverse functions are"},{"Start":"03:49.040 ","End":"03:55.655","Text":"arcosh or area hyperbolic cosine or inverse hyperbolic cosine,"},{"Start":"03:55.655 ","End":"03:57.850","Text":"and they\u0027re also written this way."},{"Start":"03:57.850 ","End":"04:01.310","Text":"I promised you that we\u0027d proved the identity"},{"Start":"04:01.310 ","End":"04:04.685","Text":"cosine hyperbolic squared minus sine hyperbolic squared is 1."},{"Start":"04:04.685 ","End":"04:06.260","Text":"We\u0027ll do that now."},{"Start":"04:06.260 ","End":"04:10.460","Text":"Note that this is different than the case for circular functions where we"},{"Start":"04:10.460 ","End":"04:14.780","Text":"have a plus if we don\u0027t have the hyperbolic. Let\u0027s start."},{"Start":"04:14.780 ","End":"04:18.980","Text":"Replace each function by its definition,"},{"Start":"04:18.980 ","End":"04:22.325","Text":"the cosine hyperbolic is with a plus here,"},{"Start":"04:22.325 ","End":"04:24.820","Text":"the sine hyperbolic with a minus here."},{"Start":"04:24.820 ","End":"04:27.425","Text":"This is just straightforward algebra."},{"Start":"04:27.425 ","End":"04:29.854","Text":"Squaring means we get a quarter,"},{"Start":"04:29.854 ","End":"04:35.750","Text":"pull that outside the brackets and this thing squared is this."},{"Start":"04:35.750 ","End":"04:37.955","Text":"This squared is this."},{"Start":"04:37.955 ","End":"04:40.565","Text":"Now, when we do this subtraction,"},{"Start":"04:40.565 ","End":"04:42.290","Text":"this will cancel with this,"},{"Start":"04:42.290 ","End":"04:45.350","Text":"and this will cancel with this, but 2 minus,"},{"Start":"04:45.350 ","End":"04:48.145","Text":"minus 2 will give us 4,"},{"Start":"04:48.145 ","End":"04:51.358","Text":"so we get 4 over 4, which is 1,"},{"Start":"04:51.358 ","End":"04:52.520","Text":"and that\u0027s what we had to show,"},{"Start":"04:52.520 ","End":"04:54.590","Text":"that this is equal to 1."},{"Start":"04:54.590 ","End":"04:58.295","Text":"I\u0027ll give you the graphs of all 6 hyperbolic functions now,"},{"Start":"04:58.295 ","End":"04:59.600","Text":"just to give you an idea,"},{"Start":"04:59.600 ","End":"05:03.350","Text":"but we\u0027ll go into that or some of them more later,"},{"Start":"05:03.350 ","End":"05:07.265","Text":"especially the cosine hyperbolic and sine hyperbolic."},{"Start":"05:07.265 ","End":"05:09.245","Text":"I borrowed this picture from somewhere else,"},{"Start":"05:09.245 ","End":"05:12.395","Text":"and this is the order that they are in."},{"Start":"05:12.395 ","End":"05:15.635","Text":"Next, let\u0027s talk about the properties."},{"Start":"05:15.635 ","End":"05:18.260","Text":"Let\u0027s start with the hyperbolic cosine."},{"Start":"05:18.260 ","End":"05:21.275","Text":"Reminder, this is its definition."},{"Start":"05:21.275 ","End":"05:27.710","Text":"First of all, it\u0027s an even function because if I replace x by minus x,"},{"Start":"05:27.710 ","End":"05:31.880","Text":"so just reversing the order of the addition here,"},{"Start":"05:31.880 ","End":"05:33.515","Text":"so it makes no difference,"},{"Start":"05:33.515 ","End":"05:39.440","Text":"it\u0027s also positive because the exponent is always positive."},{"Start":"05:39.440 ","End":"05:40.670","Text":"This plus this is positive,"},{"Start":"05:40.670 ","End":"05:43.070","Text":"and if we divide it by 2, it\u0027s still positive."},{"Start":"05:43.070 ","End":"05:47.270","Text":"You can see that the graph is always positive."},{"Start":"05:47.270 ","End":"05:51.318","Text":"The other property is that it\u0027s defined for all the reals."},{"Start":"05:51.318 ","End":"05:53.780","Text":"There\u0027s no reason why we shouldn\u0027t be able to"},{"Start":"05:53.780 ","End":"05:56.960","Text":"compute the exponential for any real number."},{"Start":"05:56.960 ","End":"06:00.215","Text":"The range, I\u0027ll show you why,"},{"Start":"06:00.215 ","End":"06:02.720","Text":"but it\u0027s from 1 to infinity,"},{"Start":"06:02.720 ","End":"06:05.540","Text":"including the 1, so bigger or equal to 1."},{"Start":"06:05.540 ","End":"06:07.655","Text":"I\u0027ll show you the proof of that now."},{"Start":"06:07.655 ","End":"06:12.315","Text":"Let\u0027s start with the equation y equals cosine hyperbolic x."},{"Start":"06:12.315 ","End":"06:15.105","Text":"Twice y is equal to,"},{"Start":"06:15.105 ","End":"06:17.160","Text":"without the divided by 2 here,"},{"Start":"06:17.160 ","End":"06:18.690","Text":"bring the 2 over to the other side,"},{"Start":"06:18.690 ","End":"06:20.660","Text":"we get 2y is equal to this."},{"Start":"06:20.660 ","End":"06:22.295","Text":"I\u0027ll bring the 2 over,"},{"Start":"06:22.295 ","End":"06:27.665","Text":"multiply by e^x, then collect terms."},{"Start":"06:27.665 ","End":"06:32.240","Text":"I\u0027ll be using an alternative formula for quadratic equations,"},{"Start":"06:32.240 ","End":"06:35.795","Text":"which assumes that there\u0027s a 2 in front of the"},{"Start":"06:35.795 ","End":"06:40.384","Text":"b. I\u0027ll just put a smiley for our variable."},{"Start":"06:40.384 ","End":"06:41.870","Text":"I don\u0027t want to use x again."},{"Start":"06:41.870 ","End":"06:43.610","Text":"If we have, well,"},{"Start":"06:43.610 ","End":"06:48.560","Text":"think of it like ax^2 plus bx plus c equals 0, only here we have a 2."},{"Start":"06:48.560 ","End":"06:51.470","Text":"This makes the formula simpler,"},{"Start":"06:51.470 ","End":"06:53.570","Text":"and it\u0027s useful especially if you have numbers,"},{"Start":"06:53.570 ","End":"06:55.355","Text":"let\u0027s just say it\u0027s an even number."},{"Start":"06:55.355 ","End":"06:56.660","Text":"If you have the 2 here,"},{"Start":"06:56.660 ","End":"06:59.923","Text":"then we don\u0027t need the 4 here and the other 2 here,"},{"Start":"06:59.923 ","End":"07:03.980","Text":"and it\u0027s especially simple if the leading coefficient is 1,"},{"Start":"07:03.980 ","End":"07:07.850","Text":"then it boils down to just this."},{"Start":"07:07.850 ","End":"07:10.010","Text":"What happens in our case,"},{"Start":"07:10.010 ","End":"07:13.175","Text":"e^x is the variable that\u0027s the smiley,"},{"Start":"07:13.175 ","End":"07:18.740","Text":"so we get minus b is plus y and b^2 is y^2,"},{"Start":"07:18.740 ","End":"07:21.265","Text":"minus c is minus 1."},{"Start":"07:21.265 ","End":"07:24.140","Text":"In order for this to make sense or to have a solution,"},{"Start":"07:24.140 ","End":"07:28.040","Text":"we have to have what\u0027s under the square root to be non-negative."},{"Start":"07:28.040 ","End":"07:33.800","Text":"In other words, y^2 has to be bigger or equal to 1 for this to have any solution."},{"Start":"07:33.800 ","End":"07:40.175","Text":"That means that y has to be bigger or equal to 1 because y is positive."},{"Start":"07:40.175 ","End":"07:42.800","Text":"We already said, here it is, positive."},{"Start":"07:42.800 ","End":"07:47.360","Text":"Strictly speaking, we haven\u0027t answered the question of the range,"},{"Start":"07:47.360 ","End":"07:51.125","Text":"what we\u0027ve shown is that if y is in the range,"},{"Start":"07:51.125 ","End":"07:53.300","Text":"then it\u0027s bigger or equal to 1,"},{"Start":"07:53.300 ","End":"07:58.445","Text":"but it doesn\u0027t mean that every y that\u0027s bigger or equal to 1 is in the range."},{"Start":"07:58.445 ","End":"08:01.295","Text":"The range might be from 1-10 or something,"},{"Start":"08:01.295 ","End":"08:07.205","Text":"but we\u0027ll show that when x goes to plus or minus infinity,"},{"Start":"08:07.205 ","End":"08:11.330","Text":"then y goes to infinity, and by continuity,"},{"Start":"08:11.330 ","End":"08:16.490","Text":"the range will be all of the interval from 1 to infinity."},{"Start":"08:16.490 ","End":"08:18.215","Text":"But we won\u0027t do this now,"},{"Start":"08:18.215 ","End":"08:22.760","Text":"we\u0027ll leave this till later to show that the limit at infinity is infinity,"},{"Start":"08:22.760 ","End":"08:25.040","Text":"and minus infinity, it\u0027s also infinity."},{"Start":"08:25.040 ","End":"08:27.095","Text":"We go back to the picture."},{"Start":"08:27.095 ","End":"08:28.400","Text":"As we go further to the right,"},{"Start":"08:28.400 ","End":"08:29.720","Text":"we go up to infinity,"},{"Start":"08:29.720 ","End":"08:31.220","Text":"and continue to the left,"},{"Start":"08:31.220 ","End":"08:33.715","Text":"you also go to infinity."},{"Start":"08:33.715 ","End":"08:39.495","Text":"The derivative of the cosine hyperbolic is sine hyperbolic."},{"Start":"08:39.495 ","End":"08:42.635","Text":"Let\u0027s also leave that till later."},{"Start":"08:42.635 ","End":"08:45.900","Text":"I think it\u0027s time for a break now."}],"ID":30549},{"Watched":false,"Name":"Hyperbolic Functions - Introduction - Part B","Duration":"7m 48s","ChapterTopicVideoID":28998,"CourseChapterTopicPlaylistID":293022,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.530","Text":"We\u0027re back after the break,"},{"Start":"00:01.530 ","End":"00:05.670","Text":"continuing with the properties of the hyperbolic cosine,"},{"Start":"00:05.670 ","End":"00:10.545","Text":"we didn\u0027t quite finish the proofs and we still have a graph to sketch."},{"Start":"00:10.545 ","End":"00:13.169","Text":"You know what, I\u0027ll just give you the graph right away."},{"Start":"00:13.169 ","End":"00:14.955","Text":"We already saw it."},{"Start":"00:14.955 ","End":"00:18.075","Text":"This is the 1/2e^x."},{"Start":"00:18.075 ","End":"00:22.835","Text":"This is 1/2e^minus x and this is the sum of these two."},{"Start":"00:22.835 ","End":"00:26.630","Text":"Close to infinity, it\u0027s very much like 1/2e^x and"},{"Start":"00:26.630 ","End":"00:30.605","Text":"close to minus infinity is like 1/2e^minus x."},{"Start":"00:30.605 ","End":"00:32.210","Text":"Anyway, and this,"},{"Start":"00:32.210 ","End":"00:33.860","Text":"we proved this, this, this,"},{"Start":"00:33.860 ","End":"00:40.210","Text":"we still have to do these 2 limits and the derivative of this one over here."},{"Start":"00:40.210 ","End":"00:42.545","Text":"When x goes to infinity,"},{"Start":"00:42.545 ","End":"00:45.995","Text":"we can just take the 1/2 out the brackets and take the limit of this plus"},{"Start":"00:45.995 ","End":"00:50.510","Text":"this and then we can symbolically just substitute x equals infinity,"},{"Start":"00:50.510 ","End":"00:53.210","Text":"so e^infinity plus e^minus infinity,"},{"Start":"00:53.210 ","End":"00:55.270","Text":"this is infinity, this is 0."},{"Start":"00:55.270 ","End":"00:58.470","Text":"1/2 times infinity is infinity."},{"Start":"00:58.470 ","End":"01:00.855","Text":"So that\u0027s this\u0027s infinity."},{"Start":"01:00.855 ","End":"01:03.070","Text":"For the minus infinity,"},{"Start":"01:03.070 ","End":"01:06.789","Text":"we can use the fact that it\u0027s an even function."},{"Start":"01:06.789 ","End":"01:10.750","Text":"X goes to a or minus a is the same thing."},{"Start":"01:10.750 ","End":"01:14.260","Text":"It\u0027s got the symmetry about the y-axis,"},{"Start":"01:14.260 ","End":"01:16.615","Text":"so it\u0027s also infinity."},{"Start":"01:16.615 ","End":"01:24.025","Text":"Now the derivative cosine hyperbolic is 1/2e^x plus e^minus x. Differentiate that."},{"Start":"01:24.025 ","End":"01:29.725","Text":"This is e^x and this is minus e^minus x because we have the antiderivative."},{"Start":"01:29.725 ","End":"01:33.235","Text":"This, if you look at it, it\u0027s just sine hyperbolic of x."},{"Start":"01:33.235 ","End":"01:35.240","Text":"That\u0027s that part."},{"Start":"01:35.240 ","End":"01:39.144","Text":"Now let\u0027s do some examples from real life."},{"Start":"01:39.144 ","End":"01:43.900","Text":"Turns out that if you suspend a chain free hanging,"},{"Start":"01:43.900 ","End":"01:46.910","Text":"then this forms part of a catenary,"},{"Start":"01:46.910 ","End":"01:49.625","Text":"which is what cosine hyperbolic looks like."},{"Start":"01:49.625 ","End":"01:52.530","Text":"I\u0027ll give you another example."},{"Start":"01:53.140 ","End":"01:57.620","Text":"Because it\u0027s free hanging this suspension bridge."},{"Start":"01:57.620 ","End":"01:59.360","Text":"It\u0027s also a catenary,"},{"Start":"01:59.360 ","End":"02:02.495","Text":"but now an example of not a catenary,"},{"Start":"02:02.495 ","End":"02:05.870","Text":"the Golden Gate Bridge in San Francisco,"},{"Start":"02:05.870 ","End":"02:10.490","Text":"these suspensions are not free hanging."},{"Start":"02:10.490 ","End":"02:15.560","Text":"They pick up the load from here and it\u0027s a constant load for"},{"Start":"02:15.560 ","End":"02:21.425","Text":"horizontal foot or meter and these are actually parabolas,"},{"Start":"02:21.425 ","End":"02:24.340","Text":"pieces of parabola not catenary."},{"Start":"02:24.340 ","End":"02:27.050","Text":"Anyway, that\u0027s just for interest\u0027s sake."},{"Start":"02:27.050 ","End":"02:30.110","Text":"Now that we\u0027ve done the hyperbolic cosine,"},{"Start":"02:30.110 ","End":"02:33.860","Text":"let\u0027s do the hyperbolic sine the properties."},{"Start":"02:33.860 ","End":"02:35.795","Text":"It\u0027s an odd function."},{"Start":"02:35.795 ","End":"02:40.565","Text":"If you replace x by minus x,"},{"Start":"02:40.565 ","End":"02:44.540","Text":"we get minus the original."},{"Start":"02:44.540 ","End":"02:47.000","Text":"I left out the 1/2 here."},{"Start":"02:47.000 ","End":"02:52.070","Text":"You see I\u0027ve replaced x by minus x and minus x is minus minus x,"},{"Start":"02:52.070 ","End":"02:56.545","Text":"which is x and you can just reverse the order and take out a minus."},{"Start":"02:56.545 ","End":"02:59.810","Text":"Also defined for all the real numbers,"},{"Start":"02:59.810 ","End":"03:02.960","Text":"there\u0027s no reason we can\u0027t substitute x is anything."},{"Start":"03:02.960 ","End":"03:06.350","Text":"Once again, the 1/2 doesn\u0027t make any difference here."},{"Start":"03:06.350 ","End":"03:08.930","Text":"The limit at infinity,"},{"Start":"03:08.930 ","End":"03:13.730","Text":"very similar to what we did with the hyperbolic cosine."},{"Start":"03:13.730 ","End":"03:21.890","Text":"We just plug in infinity here and symbolically we get e^infinity minus e^minus infinity."},{"Start":"03:21.890 ","End":"03:26.360","Text":"So this time it\u0027s infinity minus 0 instead of infinity plus 0."},{"Start":"03:26.360 ","End":"03:28.610","Text":"Still infinity."},{"Start":"03:28.610 ","End":"03:32.480","Text":"But the difference is that at minus infinity,"},{"Start":"03:32.480 ","End":"03:34.670","Text":"because this time we don\u0027t have an even function."},{"Start":"03:34.670 ","End":"03:36.395","Text":"We have an odd function."},{"Start":"03:36.395 ","End":"03:38.725","Text":"It\u0027s minus infinity."},{"Start":"03:38.725 ","End":"03:42.290","Text":"Look at the graph now. It\u0027s symmetrical."},{"Start":"03:42.290 ","End":"03:46.655","Text":"It has a rotational symmetry through the origin."},{"Start":"03:46.655 ","End":"03:48.259","Text":"Here it goes to infinity,"},{"Start":"03:48.259 ","End":"03:51.170","Text":"here it goes to minus infinity."},{"Start":"03:51.170 ","End":"03:55.460","Text":"These guidelines are the 1/2e^x,"},{"Start":"03:55.460 ","End":"03:59.125","Text":"and minus 1/2e^minus x."},{"Start":"03:59.125 ","End":"04:02.275","Text":"This is the sum of these 2."},{"Start":"04:02.275 ","End":"04:04.610","Text":"At infinity, it\u0027s very much like"},{"Start":"04:04.610 ","End":"04:09.850","Text":"this function and at minus infinity it\u0027s very much like this one."},{"Start":"04:09.850 ","End":"04:13.260","Text":"The other properties, the range,"},{"Start":"04:13.260 ","End":"04:15.440","Text":"it\u0027s all of the reals."},{"Start":"04:15.440 ","End":"04:20.400","Text":"If you look at it, it goes from minus infinity to infinity and it\u0027s continuous,"},{"Start":"04:20.400 ","End":"04:23.645","Text":"so it goes through all points in between."},{"Start":"04:23.645 ","End":"04:27.530","Text":"But this has proven in one of the exercises,"},{"Start":"04:27.530 ","End":"04:34.490","Text":"the derivative of the sine hyperbolic is cosine hyperbolic is the proof for that."},{"Start":"04:34.490 ","End":"04:37.175","Text":"For a simple, I\u0027ll just leave you to look at it."},{"Start":"04:37.175 ","End":"04:40.610","Text":"It\u0027s increasing, notice."},{"Start":"04:40.610 ","End":"04:44.945","Text":"Because the derivative, which is cosine hyperbolic,"},{"Start":"04:44.945 ","End":"04:48.050","Text":"we showed that this is always bigger or equal to 1,"},{"Start":"04:48.050 ","End":"04:52.055","Text":"it\u0027s positive, the derivative always so increasing just like in"},{"Start":"04:52.055 ","End":"04:56.435","Text":"the picture properties and the graph of the hyperbolic tangent."},{"Start":"04:56.435 ","End":"04:58.670","Text":"Here\u0027s the definition. You know what,"},{"Start":"04:58.670 ","End":"05:01.565","Text":"I\u0027ll show you the graph already start with that."},{"Start":"05:01.565 ","End":"05:05.015","Text":"First property, it\u0027s an odd function."},{"Start":"05:05.015 ","End":"05:08.900","Text":"Proof that tangent hyperbolic is the sine over"},{"Start":"05:08.900 ","End":"05:13.880","Text":"cosine hyperbolic and odd function over an even function is an odd function."},{"Start":"05:13.880 ","End":"05:17.720","Text":"Simple. The domain, all real numbers,"},{"Start":"05:17.720 ","End":"05:22.400","Text":"the hyperbolic cosine is never 0."},{"Start":"05:22.400 ","End":"05:25.700","Text":"It\u0027s always bigger or equal to 1 and both of these are defined everywhere."},{"Start":"05:25.700 ","End":"05:29.805","Text":"So both defined denominator non 0, defined everywhere."},{"Start":"05:29.805 ","End":"05:33.910","Text":"The range from minus 1-1."},{"Start":"05:33.910 ","End":"05:39.800","Text":"This is done in one of the exercises after the tutorial. What else?"},{"Start":"05:39.800 ","End":"05:45.475","Text":"The limit at infinity is 1 that\u0027s also in exercise."},{"Start":"05:45.475 ","End":"05:47.970","Text":"The limit at minus infinity, well,"},{"Start":"05:47.970 ","End":"05:52.115","Text":"we\u0027ll prove that using this, basically,"},{"Start":"05:52.115 ","End":"05:54.050","Text":"because we have an odd function,"},{"Start":"05:54.050 ","End":"05:56.375","Text":"the limit as x goes to minus infinity,"},{"Start":"05:56.375 ","End":"05:59.855","Text":"it\u0027s minus 1 using the odd function."},{"Start":"05:59.855 ","End":"06:03.080","Text":"We can see that here at infinity it goes to 1,"},{"Start":"06:03.080 ","End":"06:05.750","Text":"at minus infinity goes to minus 1."},{"Start":"06:05.750 ","End":"06:10.250","Text":"This part is proven in one of the exercises. What else?"},{"Start":"06:10.250 ","End":"06:13.835","Text":"The derivative is hyperbolic secant^2."},{"Start":"06:13.835 ","End":"06:19.235","Text":"Usually it\u0027s similar give or take pluses and minuses with the circular."},{"Start":"06:19.235 ","End":"06:24.335","Text":"Let\u0027s proved that using the formula for a quotient."},{"Start":"06:24.335 ","End":"06:27.740","Text":"We have the denominator times derivative of"},{"Start":"06:27.740 ","End":"06:31.250","Text":"numerator minus the numerator derivative of denominator."},{"Start":"06:31.250 ","End":"06:33.665","Text":"The derivative of sinh is cosh."},{"Start":"06:33.665 ","End":"06:37.880","Text":"Derivative of cosh is sinh denominator squared."},{"Start":"06:37.880 ","End":"06:44.405","Text":"Now, cosh times cosh is cosh^2 minus sinh^2."},{"Start":"06:44.405 ","End":"06:46.835","Text":"There\u0027s an identity that this equals 1,"},{"Start":"06:46.835 ","End":"06:53.495","Text":"1 over cosh^2 secant hyperbolic^2 as required."},{"Start":"06:53.495 ","End":"07:01.595","Text":"What else? It\u0027s increasing because the derivative is something squared,"},{"Start":"07:01.595 ","End":"07:08.960","Text":"actually positive because the cosh is always positive and 1 over positive is positive."},{"Start":"07:08.960 ","End":"07:13.600","Text":"We see that it really does increase everywhere."},{"Start":"07:13.970 ","End":"07:17.475","Text":"Lastly, it has asymptotes."},{"Start":"07:17.475 ","End":"07:19.140","Text":"Because of these 2 limits,"},{"Start":"07:19.140 ","End":"07:20.240","Text":"1 and minus 1,"},{"Start":"07:20.240 ","End":"07:21.500","Text":"these are asymptotes;"},{"Start":"07:21.500 ","End":"07:24.430","Text":"the right asymptote and the left asymptote."},{"Start":"07:24.430 ","End":"07:30.770","Text":"I\u0027ll just show you that it is used in the real-world, the hyperbolic tangent."},{"Start":"07:30.770 ","End":"07:35.960","Text":"An example of its uses in ocean waves thing is it has a tangent hyperbolic in it."},{"Start":"07:35.960 ","End":"07:37.310","Text":"That\u0027s what I wanted to say,"},{"Start":"07:37.310 ","End":"07:40.505","Text":"that it does have its uses."},{"Start":"07:40.505 ","End":"07:43.430","Text":"It does occur. That\u0027s it for"},{"Start":"07:43.430 ","End":"07:49.200","Text":"hyperbolic tangent and I think it\u0027s time for another break now."}],"ID":30550},{"Watched":false,"Name":"Inverse Hyperbolic Functions","Duration":"6m 33s","ChapterTopicVideoID":28999,"CourseChapterTopicPlaylistID":293022,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.460","Text":"Continuing back from the break."},{"Start":"00:02.460 ","End":"00:07.095","Text":"The next subtopic is Inverse Hyperbolic Functions."},{"Start":"00:07.095 ","End":"00:09.840","Text":"I have encountered them very often."},{"Start":"00:09.840 ","End":"00:11.580","Text":"I guess are not often used,"},{"Start":"00:11.580 ","End":"00:16.695","Text":"but it\u0027s important enough to mention and just talk about them briefly."},{"Start":"00:16.695 ","End":"00:20.745","Text":"These are the inverse functions of the hyperbolic functions."},{"Start":"00:20.745 ","End":"00:24.510","Text":"Of course, hyperbolic functions are not all 1-1."},{"Start":"00:24.510 ","End":"00:26.609","Text":"In the case they\u0027re not,"},{"Start":"00:26.609 ","End":"00:30.270","Text":"we have to restrict the domain for inverting 1."},{"Start":"00:30.270 ","End":"00:34.430","Text":"We\u0027ll see for example that sine hyperbolic is 1-1,"},{"Start":"00:34.430 ","End":"00:39.005","Text":"but cosine hyperbolic isn\u0027t so we have to restrict the domain."},{"Start":"00:39.005 ","End":"00:43.940","Text":"In general, once you\u0027ve done this restriction if necessary"},{"Start":"00:43.940 ","End":"00:48.995","Text":"then you just swap the domain and range to get the inverse function."},{"Start":"00:48.995 ","End":"00:52.885","Text":"It\u0027s true generally for inverse functions not just hyperbolic."},{"Start":"00:52.885 ","End":"00:56.390","Text":"Start with the inverse hyperbolic sinh."},{"Start":"00:56.390 ","End":"00:58.430","Text":"Now, this is 1-1."},{"Start":"00:58.430 ","End":"01:05.030","Text":"Remember it was strictly increasing and the domain and range are both R so it has"},{"Start":"01:05.030 ","End":"01:08.315","Text":"an inverse and the inverse also has a domain and range"},{"Start":"01:08.315 ","End":"01:12.420","Text":"both R. Let\u0027s look at the derivative."},{"Start":"01:12.420 ","End":"01:14.750","Text":"That\u0027s one of the more useful things."},{"Start":"01:14.750 ","End":"01:19.505","Text":"I claim that the derivative is 1 over square root of 1 plus x^2."},{"Start":"01:19.505 ","End":"01:25.010","Text":"Let\u0027s prove that if y is the inverse of hyperbolic sinh then"},{"Start":"01:25.010 ","End":"01:30.430","Text":"x is the hyperbolic sinh of y. I forgot to show the picture."},{"Start":"01:30.430 ","End":"01:32.960","Text":"In general, to get the inverse function,"},{"Start":"01:32.960 ","End":"01:37.250","Text":"you are reflected in the line y=x."},{"Start":"01:37.250 ","End":"01:40.760","Text":"If this blue line is"},{"Start":"01:40.760 ","End":"01:46.840","Text":"the hyperbolic sinh then this red one is the inverse hyperbolic sinh."},{"Start":"01:46.840 ","End":"01:50.525","Text":"I just noticed this is something I\u0027ve seen before."},{"Start":"01:50.525 ","End":"01:54.575","Text":"Not everyone uses ar some use arc,"},{"Start":"01:54.575 ","End":"02:00.650","Text":"the arc, I will omit the c. But just as well we encountered this."},{"Start":"02:00.650 ","End":"02:04.910","Text":"If you see it somewhere in the literature as they"},{"Start":"02:04.910 ","End":"02:10.650","Text":"say which means on the web or somewhere, sorry, back here."},{"Start":"02:10.650 ","End":"02:16.830","Text":"Let\u0027s do dx by dy derivative of sinh is cosine."},{"Start":"02:16.830 ","End":"02:20.580","Text":"Using the identity of this one,"},{"Start":"02:20.580 ","End":"02:25.790","Text":"we can get that the cosine is the square root of 1 plus sinh^2."},{"Start":"02:25.790 ","End":"02:28.340","Text":"It\u0027s positive, so it\u0027s not plus or minus,"},{"Start":"02:28.340 ","End":"02:31.550","Text":"it\u0027s just the positive square root and this is equal"},{"Start":"02:31.550 ","End":"02:35.810","Text":"to square root of 1 plus x^2 because of this."},{"Start":"02:35.810 ","End":"02:40.080","Text":"The reciprocal gives us that dy by dx is 1 over"},{"Start":"02:40.080 ","End":"02:44.140","Text":"the square root of 1 plus x^2 and that\u0027s what we had to show."},{"Start":"02:44.140 ","End":"02:49.195","Text":"One of the uses of this is that it appears in tables of integration."},{"Start":"02:49.195 ","End":"02:52.255","Text":"In physics, you might see an integral of this,"},{"Start":"02:52.255 ","End":"02:56.240","Text":"1 over the square root of x^2 plus a^2."},{"Start":"02:56.450 ","End":"03:01.330","Text":"You would make a substitution that x over a is"},{"Start":"03:01.330 ","End":"03:07.540","Text":"some t and at the end you\u0027ll get the inverse hyperbolic sinh of t,"},{"Start":"03:07.540 ","End":"03:09.290","Text":"which is x over a."},{"Start":"03:09.290 ","End":"03:12.935","Text":"Anyway, it appears in the integration tables."},{"Start":"03:12.935 ","End":"03:17.680","Text":"That\u0027s all I want to say about inverse hyperbolic sinh."},{"Start":"03:17.680 ","End":"03:21.450","Text":"Now let\u0027s go over the inverse hyperbolic cosine."},{"Start":"03:21.450 ","End":"03:25.310","Text":"Cosine is not 1-1."},{"Start":"03:25.310 ","End":"03:28.070","Text":"Here\u0027s the graph."},{"Start":"03:28.070 ","End":"03:31.820","Text":"If you remember, the cosine hyperbolic actually"},{"Start":"03:31.820 ","End":"03:35.495","Text":"is an even function defined everywhere it has the other half,"},{"Start":"03:35.495 ","End":"03:37.960","Text":"but then it wouldn\u0027t be 1-1,"},{"Start":"03:37.960 ","End":"03:40.340","Text":"so if we remove this,"},{"Start":"03:40.340 ","End":"03:45.095","Text":"could also remove this part but it\u0027s more natural to take the positive side."},{"Start":"03:45.095 ","End":"03:51.230","Text":"Then it will be 1-1 and it has an inverse by reflecting through the line y=x."},{"Start":"03:51.230 ","End":"03:53.520","Text":"This is the graph of arc."},{"Start":"03:53.520 ","End":"03:57.615","Text":"Again, I don\u0027t like this c here I think it\u0027s just"},{"Start":"03:57.615 ","End":"04:02.300","Text":"ar cosine hyperbolic from the area without the c. Now if"},{"Start":"04:02.300 ","End":"04:11.855","Text":"the domain is 0 to infinity and the range is from 1 to infinity,"},{"Start":"04:11.855 ","End":"04:13.775","Text":"then when we take the inverse,"},{"Start":"04:13.775 ","End":"04:18.320","Text":"we just reverse it that the domain is 1 to infinity and"},{"Start":"04:18.320 ","End":"04:24.800","Text":"the range is from 0 actually goes up to infinity even though it doesn\u0027t look like."},{"Start":"04:24.800 ","End":"04:28.985","Text":"But it does, just takes a while to get to infinity."},{"Start":"04:28.985 ","End":"04:32.645","Text":"Now the derivative is the most useful thing about it."},{"Start":"04:32.645 ","End":"04:40.235","Text":"I claim that its derivative is 1 over the square root of x^2 minus 1."},{"Start":"04:40.235 ","End":"04:45.875","Text":"Notice that if x is bigger than 1 then it all makes sense."},{"Start":"04:45.875 ","End":"04:49.280","Text":"We get 1 over the square root of something positive."},{"Start":"04:49.280 ","End":"04:51.050","Text":"It\u0027s bigger than 1,"},{"Start":"04:51.050 ","End":"04:52.850","Text":"not bigger or equal to,"},{"Start":"04:52.850 ","End":"04:56.450","Text":"because that it would be 1 over 0."},{"Start":"04:56.450 ","End":"05:03.270","Text":"In fact, it doesn\u0027t have a derivative here because the tangent line is vertical."},{"Start":"05:03.270 ","End":"05:07.330","Text":"It has a derivative for x strictly bigger than 1."},{"Start":"05:07.330 ","End":"05:11.380","Text":"Let\u0027s prove this just like before we have an inverse function,"},{"Start":"05:11.380 ","End":"05:15.545","Text":"we can switch x and y and get rid of this inverse notation."},{"Start":"05:15.545 ","End":"05:18.200","Text":"So far as x is bigger or equal to 1,"},{"Start":"05:18.200 ","End":"05:22.190","Text":"dx by dy is sinh derivative of cosh."},{"Start":"05:22.190 ","End":"05:26.880","Text":"Then using identities, we get this is equal"},{"Start":"05:26.880 ","End":"05:34.320","Text":"to square root of x^2 minus 1 so that dy by dx is 1 over."},{"Start":"05:34.320 ","End":"05:39.510","Text":"But now we have to rule out x=1 for it to make sense."},{"Start":"05:39.510 ","End":"05:46.745","Text":"It\u0027s true for x bigger than 1 and this would give us this integral here."},{"Start":"05:46.745 ","End":"05:50.060","Text":"If you let x over a be t,"},{"Start":"05:50.060 ","End":"05:51.200","Text":"If you make a substitution,"},{"Start":"05:51.200 ","End":"05:58.115","Text":"you\u0027ll get a more general results and from here back to here, when a=1."},{"Start":"05:58.115 ","End":"06:03.080","Text":"That\u0027s all I want to say about inverse hyperbolic functions."},{"Start":"06:03.080 ","End":"06:05.240","Text":"To conclude this topic,"},{"Start":"06:05.240 ","End":"06:11.210","Text":"I just want to show you some more hyperbolic identities just for reference."},{"Start":"06:11.210 ","End":"06:13.429","Text":"One of them we already proved,"},{"Start":"06:13.429 ","End":"06:18.350","Text":"that\u0027s this last one here and one of them,"},{"Start":"06:18.350 ","End":"06:22.135","Text":"which is this one will be in one of the exercises."},{"Start":"06:22.135 ","End":"06:25.280","Text":"The rest of them just have them here for reference."},{"Start":"06:25.280 ","End":"06:28.100","Text":"I don\u0027t want to get into them anymore."},{"Start":"06:28.100 ","End":"06:34.620","Text":"That concludes this clip and the topic of Hyperbolic Functions."}],"ID":30551},{"Watched":false,"Name":"Exercise 1","Duration":"1m 57s","ChapterTopicVideoID":29000,"CourseChapterTopicPlaylistID":293022,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.890","Text":"In this exercise, we\u0027re going to use this identity which we proved in"},{"Start":"00:04.890 ","End":"00:11.205","Text":"the tutorial to prove to other identities as written here in a and b."},{"Start":"00:11.205 ","End":"00:19.830","Text":"Remind you that tangent hyperbolic is defined as sine hyperbolic over cosine hyperbolic,"},{"Start":"00:19.830 ","End":"00:25.200","Text":"and the hyperbolic secant is 1 over the hyperbolic cosine,"},{"Start":"00:25.200 ","End":"00:29.265","Text":"it\u0027s completely analogous to the circular functions."},{"Start":"00:29.265 ","End":"00:37.605","Text":"Anyway, we\u0027ll start with the one that we know and divide both sides by cosh^2."},{"Start":"00:37.605 ","End":"00:48.315","Text":"Here we get 1 and then here we get minus sinh^2 over cosh^2 so it\u0027s 1 over cosh^2,"},{"Start":"00:48.315 ","End":"00:55.070","Text":"sinh over cosh or sine hyperbolic over cosine hyperbolic is tangent hyperbolic,"},{"Start":"00:55.070 ","End":"01:02.300","Text":"tangent sometimes and this is equal to 1 over cosine hyperbolic is secant hyperbolic,"},{"Start":"01:02.300 ","End":"01:08.885","Text":"so secant^2 hyperbolic and that\u0027s what we have to show, that\u0027s all there is to it."},{"Start":"01:08.885 ","End":"01:16.175","Text":"In part b, we just have to note that x can\u0027t be 0 because when x is 0,"},{"Start":"01:16.175 ","End":"01:20.005","Text":"the hyperbolic sine is 0."},{"Start":"01:20.005 ","End":"01:26.880","Text":"Here and here, we\u0027re dividing by hyperbolic sine the denominator,"},{"Start":"01:26.880 ","End":"01:28.200","Text":"so that\u0027s no good,"},{"Start":"01:28.200 ","End":"01:32.180","Text":"so we have to restrict x to be non-zero here."},{"Start":"01:32.180 ","End":"01:39.975","Text":"Other than that, we\u0027ll start from this and divide by sine hyperbolic squared both sides,"},{"Start":"01:39.975 ","End":"01:47.480","Text":"we get this, this over this is the cotangent hyperbolic."},{"Start":"01:47.480 ","End":"01:51.170","Text":"This over this from here is cosecant hyperbolic,"},{"Start":"01:51.170 ","End":"01:53.300","Text":"so it\u0027s squared,"},{"Start":"01:53.300 ","End":"01:58.410","Text":"so we get this is very straightforward. We\u0027re done."}],"ID":30552},{"Watched":false,"Name":"Exercise 2","Duration":"3m 44s","ChapterTopicVideoID":29001,"CourseChapterTopicPlaylistID":293022,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.945","Text":"In this exercise, we\u0027re going to prove that"},{"Start":"00:03.945 ","End":"00:08.820","Text":"the range of the hyperbolic sine is all of the reals."},{"Start":"00:08.820 ","End":"00:10.410","Text":"Maybe a picture."},{"Start":"00:10.410 ","End":"00:17.520","Text":"Here we see if we project this graph onto the y-axis, we cover everything."},{"Start":"00:17.520 ","End":"00:20.520","Text":"It goes down to minus infinity,"},{"Start":"00:20.520 ","End":"00:24.015","Text":"up to plus infinity and everything in-between."},{"Start":"00:24.015 ","End":"00:26.194","Text":"We need a formal proof."},{"Start":"00:26.194 ","End":"00:31.100","Text":"Let y equals sine hyperbolic of x."},{"Start":"00:31.100 ","End":"00:34.550","Text":"We\u0027ll take this equation and show that for any y,"},{"Start":"00:34.550 ","End":"00:37.220","Text":"it has a solution in x."},{"Start":"00:37.220 ","End":"00:39.580","Text":"In other words, you give me any y,"},{"Start":"00:39.580 ","End":"00:42.110","Text":"it will cut the graph somewhere"},{"Start":"00:42.110 ","End":"00:45.560","Text":"so that there is an x such that y is sine hyperbolic of x,"},{"Start":"00:45.560 ","End":"00:47.720","Text":"which means that y is in the range."},{"Start":"00:47.720 ","End":"00:49.325","Text":"Do a bit of algebra here,"},{"Start":"00:49.325 ","End":"00:53.090","Text":"this is half e^x minus e^minus x."},{"Start":"00:53.090 ","End":"01:01.270","Text":"Double both sides, multiply both sides by e^x."},{"Start":"01:01.270 ","End":"01:06.220","Text":"Here\u0027s e^2x here it\u0027s minus 1, collect terms."},{"Start":"01:06.220 ","End":"01:09.635","Text":"This is like a quadratic equation,"},{"Start":"01:09.635 ","End":"01:12.340","Text":"but in the variable e^x."},{"Start":"01:12.340 ","End":"01:17.285","Text":"The version of the quadratic formula,"},{"Start":"01:17.285 ","End":"01:20.990","Text":"where instead of ax^2 plus bx plus c,"},{"Start":"01:20.990 ","End":"01:22.945","Text":"we have a 2 here."},{"Start":"01:22.945 ","End":"01:24.860","Text":"We often use this form."},{"Start":"01:24.860 ","End":"01:29.360","Text":"It\u0027s a bit simpler when the b coefficient is an even number."},{"Start":"01:29.360 ","End":"01:35.090","Text":"The difference between this and the regular form is that there\u0027s normally a 4ac here,"},{"Start":"01:35.090 ","End":"01:39.005","Text":"the 4 disappears and the 2 from the 2a also disappears."},{"Start":"01:39.005 ","End":"01:41.065","Text":"It\u0027s a bit simpler."},{"Start":"01:41.065 ","End":"01:44.540","Text":"Here I borrowed this from the tutorial."},{"Start":"01:44.540 ","End":"01:47.210","Text":"We had this formula instead of x,"},{"Start":"01:47.210 ","End":"01:49.130","Text":"I put a smiley."},{"Start":"01:49.130 ","End":"01:53.180","Text":"It\u0027s even simpler when the a=1."},{"Start":"01:53.180 ","End":"01:57.200","Text":"Then what we have is just minus b plus or"},{"Start":"01:57.200 ","End":"02:01.840","Text":"minus the square root of b^2 minus c. It\u0027s very simple."},{"Start":"02:01.840 ","End":"02:05.415","Text":"B here is minus y,"},{"Start":"02:05.415 ","End":"02:11.120","Text":"so minus minus y is y plus or minus the square root of b^2 is y^2"},{"Start":"02:11.120 ","End":"02:17.640","Text":"and minus c is minus minus 1 is plus 1."},{"Start":"02:17.640 ","End":"02:21.210","Text":"This is what we have now for e^x."},{"Start":"02:21.210 ","End":"02:24.170","Text":"Because e^x is always positive,"},{"Start":"02:24.170 ","End":"02:26.915","Text":"we have to take the plus here."},{"Start":"02:26.915 ","End":"02:30.800","Text":"The square root of y^2 plus 1 is bigger"},{"Start":"02:30.800 ","End":"02:35.240","Text":"than the absolute value of y so subtracting it would make it negative."},{"Start":"02:35.240 ","End":"02:37.180","Text":"The minus is ruled out,"},{"Start":"02:37.180 ","End":"02:39.965","Text":"so e^x has to be this."},{"Start":"02:39.965 ","End":"02:44.990","Text":"This is always positive because even if y is negative,"},{"Start":"02:44.990 ","End":"02:49.040","Text":"the square root of y^2 plus 1 is bigger than the absolute value of y."},{"Start":"02:49.040 ","End":"02:50.855","Text":"It has a solution."},{"Start":"02:50.855 ","End":"02:55.715","Text":"We can take x equals natural log of this."},{"Start":"02:55.715 ","End":"02:57.890","Text":"As I said, because it\u0027s positive,"},{"Start":"02:57.890 ","End":"02:59.269","Text":"it has a solution."},{"Start":"02:59.269 ","End":"03:03.350","Text":"To be precise, we should reverse the order of this."},{"Start":"03:03.350 ","End":"03:08.055","Text":"I want to find a solution to y equals sine hyperbolic of x."},{"Start":"03:08.055 ","End":"03:10.145","Text":"Given y, I want to find x."},{"Start":"03:10.145 ","End":"03:13.370","Text":"Let\u0027s say x equals this."},{"Start":"03:13.370 ","End":"03:19.190","Text":"It\u0027s defined because the argument of the natural log is positive."},{"Start":"03:19.190 ","End":"03:21.635","Text":"If we reverse the steps here,"},{"Start":"03:21.635 ","End":"03:25.645","Text":"we get 2y equals sine hyperbolic of x."},{"Start":"03:25.645 ","End":"03:28.140","Text":"This is our solution."},{"Start":"03:28.140 ","End":"03:30.965","Text":"Reversing the steps, so given any y,"},{"Start":"03:30.965 ","End":"03:37.665","Text":"we have x equals this as a solution and so y is in the range of sine hyperbolic."},{"Start":"03:37.665 ","End":"03:45.240","Text":"That concludes the proof that the range of the sinh function is all of the reals. Done."}],"ID":30553},{"Watched":false,"Name":"Exercise 3","Duration":"2m 17s","ChapterTopicVideoID":29002,"CourseChapterTopicPlaylistID":293022,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.455","Text":"In this exercise, we have to compute a couple of limits."},{"Start":"00:04.455 ","End":"00:10.110","Text":"In part a, we want the limit as x goes to infinity of the hyperbolic tangent."},{"Start":"00:10.110 ","End":"00:14.790","Text":"We expect to get the answer 1."},{"Start":"00:15.050 ","End":"00:18.225","Text":"Definition of the hyperbolic tangent,"},{"Start":"00:18.225 ","End":"00:20.010","Text":"just like with the regular tangent,"},{"Start":"00:20.010 ","End":"00:23.490","Text":"is the sine over the cosine, but hyperbolic."},{"Start":"00:23.490 ","End":"00:28.895","Text":"We can evaluate this using the definitions of the hyperbolic sine and cosine,"},{"Start":"00:28.895 ","End":"00:31.610","Text":"like so, the half cancels."},{"Start":"00:31.610 ","End":"00:35.815","Text":"If we divide top and bottom by e^x,"},{"Start":"00:35.815 ","End":"00:40.325","Text":"then we get this expression with e^minus 2x here."},{"Start":"00:40.325 ","End":"00:43.265","Text":"Now if we let x go to infinity,"},{"Start":"00:43.265 ","End":"00:46.980","Text":"then 2x also goes to infinity."},{"Start":"00:46.980 ","End":"00:51.620","Text":"Here we have e to the minus infinity symbolically,"},{"Start":"00:51.620 ","End":"00:54.650","Text":"and here also e to the minus infinity."},{"Start":"00:54.650 ","End":"00:57.810","Text":"We know that e to the minus infinity is 0,"},{"Start":"00:57.810 ","End":"01:02.120","Text":"so we get 1 minus 0 over 1 plus 0."},{"Start":"01:02.120 ","End":"01:05.845","Text":"The answer is 1 as expected."},{"Start":"01:05.845 ","End":"01:08.649","Text":"Now, on to part b,"},{"Start":"01:08.649 ","End":"01:11.945","Text":"cosine hyperbolic minus sine hyperbolic."},{"Start":"01:11.945 ","End":"01:13.505","Text":"We can use a trick here."},{"Start":"01:13.505 ","End":"01:16.070","Text":"In general, a minus b,"},{"Start":"01:16.070 ","End":"01:19.700","Text":"we can write as a^2 minus b squares over a"},{"Start":"01:19.700 ","End":"01:23.225","Text":"plus b using the difference of squares formula."},{"Start":"01:23.225 ","End":"01:29.513","Text":"That\u0027s an identity that cosine hyperbolic squared minus sine hyperbolic squared is 1."},{"Start":"01:29.513 ","End":"01:32.960","Text":"The reason we needed any tricks at all is because as it is,"},{"Start":"01:32.960 ","End":"01:37.055","Text":"this is an infinity minus infinity limit."},{"Start":"01:37.055 ","End":"01:39.740","Text":"But this way, as you can see, if this is one,"},{"Start":"01:39.740 ","End":"01:42.275","Text":"we\u0027re going to have an infinity plus infinity."},{"Start":"01:42.275 ","End":"01:44.235","Text":"Anyway, let\u0027s continue."},{"Start":"01:44.235 ","End":"01:48.060","Text":"The limit is 1 over the limit of"},{"Start":"01:48.060 ","End":"01:53.430","Text":"cosine hyperbolic plus sine hyperbolic, cosine plus sine."},{"Start":"01:53.430 ","End":"01:56.190","Text":"Each of these, like I said, is infinity."},{"Start":"01:56.190 ","End":"01:59.570","Text":"We have 1 over infinity plus infinity symbolically."},{"Start":"01:59.570 ","End":"02:01.520","Text":"Infinity plus infinity is infinity,"},{"Start":"02:01.520 ","End":"02:04.055","Text":"and 1 over infinity is 0."},{"Start":"02:04.055 ","End":"02:06.385","Text":"This limit is equal to 0."},{"Start":"02:06.385 ","End":"02:10.490","Text":"What it shows is that the cosine and sine hyperbolic,"},{"Start":"02:10.490 ","End":"02:17.970","Text":"are very close to each other when x is very large. We\u0027re done."}],"ID":30554},{"Watched":false,"Name":"Exercise 4","Duration":"2m 35s","ChapterTopicVideoID":29003,"CourseChapterTopicPlaylistID":293022,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.590","Text":"This exercise is on several parts and they\u0027re all about domain and range"},{"Start":"00:04.590 ","End":"00:09.660","Text":"of various hyperbolic and inverse hyperbolic functions,"},{"Start":"00:09.660 ","End":"00:14.235","Text":"and we\u0027ll start with Part a which is the hyperbolic tangent."},{"Start":"00:14.235 ","End":"00:19.185","Text":"Remember it\u0027s the quotient of hyperbolic sine, hyperbolic cosine."},{"Start":"00:19.185 ","End":"00:23.580","Text":"Each of these is defined everywhere and more than that,"},{"Start":"00:23.580 ","End":"00:28.080","Text":"the denominator; the hyperbolic cosine, is never 0."},{"Start":"00:28.080 ","End":"00:32.490","Text":"It\u0027s ranges in fact from 1 to infinity but does not include 0."},{"Start":"00:32.490 ","End":"00:37.182","Text":"There\u0027s no problem for any x of defining the hyperbolic tangent,"},{"Start":"00:37.182 ","End":"00:42.638","Text":"and so we write that the domain is all of R. In Part b"},{"Start":"00:42.638 ","End":"00:46.865","Text":"we\u0027re talking about the hyperbolic cosecant"},{"Start":"00:46.865 ","End":"00:50.945","Text":"which is the reciprocal of the hyperbolic sine."},{"Start":"00:50.945 ","End":"00:55.880","Text":"Now, hyperbolic sine is defined"},{"Start":"00:55.880 ","End":"01:00.680","Text":"everywhere and the range is all of R. In order to take the reciprocal,"},{"Start":"01:00.680 ","End":"01:07.195","Text":"we have to exclude 0 from the range of the hyperbolic sine."},{"Start":"01:07.195 ","End":"01:09.335","Text":"To exclude the 0 there,"},{"Start":"01:09.335 ","End":"01:12.350","Text":"we have to exclude 0 from the domain because"},{"Start":"01:12.350 ","End":"01:17.490","Text":"hyperbolic sine is 0 if and only if the number is 0."},{"Start":"01:18.530 ","End":"01:27.650","Text":"We get that sine hyperbolic takes the reals minus the 0 to the reals minus the 0,"},{"Start":"01:27.650 ","End":"01:29.785","Text":"that\u0027s the sine hyperbolic."},{"Start":"01:29.785 ","End":"01:34.340","Text":"Now, we want to make the jump to the cosecant hyperbolic by taking the reciprocal,"},{"Start":"01:34.340 ","End":"01:41.680","Text":"so it takes the reals minus 0 or the 1 over reals minus 0."},{"Start":"01:41.680 ","End":"01:44.400","Text":"If we take 1 over all the non-zeros,"},{"Start":"01:44.400 ","End":"01:46.555","Text":"we still get all the non-zeros."},{"Start":"01:46.555 ","End":"01:48.800","Text":"Everything is obtainable as the reciprocal of"},{"Start":"01:48.800 ","End":"01:51.220","Text":"something because the reciprocal itself inverse."},{"Start":"01:51.220 ","End":"01:55.340","Text":"We say that the domain and range of the hyperbolic cosecant or"},{"Start":"01:55.340 ","End":"02:00.050","Text":"both are minus 0, non-zero reals."},{"Start":"02:00.050 ","End":"02:07.010","Text":"Now let\u0027s jump from the hyperbolic cosecant to its inverse function."},{"Start":"02:07.010 ","End":"02:12.500","Text":"What we get for the inverse function is just to swap the domain and range,"},{"Start":"02:12.500 ","End":"02:17.960","Text":"but for the cosecant hyperbolic domain and range were both non-zero reals."},{"Start":"02:17.960 ","End":"02:20.120","Text":"If we swap 2 things that are equal around,"},{"Start":"02:20.120 ","End":"02:23.960","Text":"we still get that the domain and range of the hyperbolic;of the"},{"Start":"02:23.960 ","End":"02:28.920","Text":"inverse hyperbolic cosecant of the area hyperbolic cosecant,"},{"Start":"02:28.920 ","End":"02:31.565","Text":"both also non-zero reals."},{"Start":"02:31.565 ","End":"02:35.490","Text":"That said, we\u0027ve concluded this exercise."}],"ID":30555},{"Watched":false,"Name":"Exercise 5","Duration":"4m 28s","ChapterTopicVideoID":28995,"CourseChapterTopicPlaylistID":293022,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.910","Text":"In this exercise, we\u0027re going to compute"},{"Start":"00:02.910 ","End":"00:06.915","Text":"the derivative of 2 inverse hyperbolic functions,"},{"Start":"00:06.915 ","End":"00:11.760","Text":"inverse tangent hyperbolic and inverse secant hyperbolic."},{"Start":"00:11.760 ","End":"00:16.800","Text":"Actually the use of this is in integration is that"},{"Start":"00:16.800 ","End":"00:22.200","Text":"if we\u0027re given these 2 functions and we want to find the indefinite integral,"},{"Start":"00:22.200 ","End":"00:25.800","Text":"then we can use inverse hyperbolic functions."},{"Start":"00:25.800 ","End":"00:28.545","Text":"Let\u0027s get started with part a,"},{"Start":"00:28.545 ","End":"00:34.035","Text":"we let y be inverse hyperbolic tangent,"},{"Start":"00:34.035 ","End":"00:37.474","Text":"and that means by the definition of the inverse,"},{"Start":"00:37.474 ","End":"00:40.820","Text":"that x is hyperbolic tangent of y."},{"Start":"00:40.820 ","End":"00:48.410","Text":"This one is defined from minus 1-1 because the range of hyperbolic tangent is minus 1-1,"},{"Start":"00:48.410 ","End":"00:51.145","Text":"and we invert domain and range."},{"Start":"00:51.145 ","End":"00:52.860","Text":"Differentiate both sides,"},{"Start":"00:52.860 ","End":"00:56.335","Text":"and we\u0027ll use the dx/dy notation."},{"Start":"00:56.335 ","End":"00:58.790","Text":"We already showed that the derivative of"},{"Start":"00:58.790 ","End":"01:02.870","Text":"hyperbolic tangent is the square of hyperbolic secant."},{"Start":"01:02.870 ","End":"01:09.185","Text":"Using identities, this is 1 minus the square of hyperbolic tangent,"},{"Start":"01:09.185 ","End":"01:12.170","Text":"but this is just equal to x."},{"Start":"01:12.170 ","End":"01:14.584","Text":"We have 1 minus x^2."},{"Start":"01:14.584 ","End":"01:20.720","Text":"Now we\u0027ll use the formula that the reciprocal of dx/dy is dy/dx,"},{"Start":"01:20.720 ","End":"01:24.085","Text":"so 1 over 1 minus x^2."},{"Start":"01:24.085 ","End":"01:28.250","Text":"That proves what we had to show in part a."},{"Start":"01:28.250 ","End":"01:31.040","Text":"This is of course defined when x is between minus 1 and 1,"},{"Start":"01:31.040 ","End":"01:32.840","Text":"then x^2 is not equal to 1,"},{"Start":"01:32.840 ","End":"01:34.865","Text":"so we\u0027re not dividing by 0."},{"Start":"01:34.865 ","End":"01:39.660","Text":"Now on to part b, inverse hyperbolic secant."},{"Start":"01:39.660 ","End":"01:43.425","Text":"X is hyperbolic secant of y."},{"Start":"01:43.425 ","End":"01:47.450","Text":"That\u0027s 1 over cosine hyperbolic or cosh."},{"Start":"01:47.450 ","End":"01:56.270","Text":"Then dx/dy is the derivative of this is minus sinh over cosh^2."},{"Start":"01:56.270 ","End":"01:58.685","Text":"We\u0027re using the chain rule."},{"Start":"01:58.685 ","End":"02:02.285","Text":"If it was 1 over x, it would be minus 1 over x^2."},{"Start":"02:02.285 ","End":"02:05.975","Text":"Then we can rewrite this,"},{"Start":"02:05.975 ","End":"02:09.695","Text":"split the denominator into cosh and cosh."},{"Start":"02:09.695 ","End":"02:12.245","Text":"Here we\u0027ll put the minus in front,"},{"Start":"02:12.245 ","End":"02:16.285","Text":"1 and sinh of y."},{"Start":"02:16.285 ","End":"02:24.220","Text":"This is equal to 1 over cosh is secant hyperbolic and this is tangent hyperbolic."},{"Start":"02:24.220 ","End":"02:26.926","Text":"Now we go back from y to x,"},{"Start":"02:26.926 ","End":"02:31.250","Text":"secant hyperbolic y is x here,"},{"Start":"02:31.250 ","End":"02:34.340","Text":"and tangent hyperbolic using identities."},{"Start":"02:34.340 ","End":"02:38.525","Text":"Let\u0027s see, if we go back up. Here it is."},{"Start":"02:38.525 ","End":"02:41.780","Text":"Just put the tangent hyperbolic squared"},{"Start":"02:41.780 ","End":"02:44.945","Text":"on the left and the secant hyperbolic on the right."},{"Start":"02:44.945 ","End":"02:49.370","Text":"You see that we get that the tangent hyperbolic is"},{"Start":"02:49.370 ","End":"02:54.390","Text":"square root of 1 minus secant hyperbolic squared."},{"Start":"02:54.390 ","End":"02:58.410","Text":"We have 1 minus x^2, square root."},{"Start":"02:58.410 ","End":"03:02.835","Text":"Then we can take the reciprocal dx/dy."},{"Start":"03:02.835 ","End":"03:07.155","Text":"We get dy/dx is 1 over this."},{"Start":"03:07.155 ","End":"03:11.895","Text":"Our domain for x was between 0 and 1."},{"Start":"03:11.895 ","End":"03:15.440","Text":"We\u0027re okay. This square root,"},{"Start":"03:15.440 ","End":"03:21.710","Text":"what\u0027s under it will be strictly positive and the denominator is not 0."},{"Start":"03:21.710 ","End":"03:24.080","Text":"That\u0027s what we had to show."},{"Start":"03:24.080 ","End":"03:30.485","Text":"Basically we\u0027re done but don\u0027t go because I wanted to show you an alternative technique."},{"Start":"03:30.485 ","End":"03:34.175","Text":"Instead of using the dx/dy method,"},{"Start":"03:34.175 ","End":"03:37.100","Text":"we can use implicit differentiation."},{"Start":"03:37.100 ","End":"03:41.930","Text":"Take this and y is a function of x and do implicit derivative."},{"Start":"03:41.930 ","End":"03:43.725","Text":"Derivative of x is 1,"},{"Start":"03:43.725 ","End":"03:49.775","Text":"derivative of tangent using the chain rule is secant hyperbolic squared,"},{"Start":"03:49.775 ","End":"03:53.795","Text":"but times the inner derivative, which is y\u0027."},{"Start":"03:53.795 ","End":"03:59.880","Text":"From this we get y\u0027 is 1 over 6 hyperbolic squared of y."},{"Start":"04:00.080 ","End":"04:03.980","Text":"Then using the identity, yes, just here,"},{"Start":"04:03.980 ","End":"04:09.815","Text":"we\u0027ve got this and tangent hyperbolic of y is x."},{"Start":"04:09.815 ","End":"04:12.635","Text":"We get 1 over 1 minus x^2,"},{"Start":"04:12.635 ","End":"04:16.795","Text":"which is the same as what we got for the other technique and that\u0027s good."},{"Start":"04:16.795 ","End":"04:19.000","Text":"Those who don\u0027t like the dx/dy,"},{"Start":"04:19.000 ","End":"04:22.445","Text":"you can use the implicit differentiation technique."},{"Start":"04:22.445 ","End":"04:28.410","Text":"It\u0027s just about the same difficulty. We\u0027re done."}],"ID":30556},{"Watched":false,"Name":"Exercise 6","Duration":"2m 18s","ChapterTopicVideoID":28996,"CourseChapterTopicPlaylistID":293022,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.710","Text":"In this this exercise,"},{"Start":"00:01.710 ","End":"00:04.710","Text":"we\u0027ll prove 1 of the hyperbolic identities."},{"Start":"00:04.710 ","End":"00:11.670","Text":"It is easier to start with the right-hand side and get to the left-hand side."},{"Start":"00:11.670 ","End":"00:14.805","Text":"We start with the plus here,"},{"Start":"00:14.805 ","End":"00:16.680","Text":"instead of the plus minus,"},{"Start":"00:16.680 ","End":"00:19.620","Text":"and later we\u0027ll see what to do if it\u0027s a minus."},{"Start":"00:19.620 ","End":"00:23.250","Text":"Just replace each of the cosine"},{"Start":"00:23.250 ","End":"00:28.185","Text":"hyperbolic by its definition in terms of the exponential and we get this."},{"Start":"00:28.185 ","End":"00:34.095","Text":"Looks very similar, but notice here it\u0027s minus plus and here it\u0027s plus minus."},{"Start":"00:34.095 ","End":"00:40.050","Text":"Multiply out just using binomial times binomial,"},{"Start":"00:40.050 ","End":"00:44.470","Text":"basic algebra, and the denominators are 4(2 times 2)."},{"Start":"00:44.470 ","End":"00:49.220","Text":"Use the rules of exponents to combine like e to the x,"},{"Start":"00:49.220 ","End":"00:53.015","Text":"e to the y is e to the x plus y similarly for the rest."},{"Start":"00:53.015 ","End":"00:56.420","Text":"Notice that, this will"},{"Start":"00:56.420 ","End":"01:00.065","Text":"cancel with this because it\u0027s the same thing with a plus and a minus."},{"Start":"01:00.065 ","End":"01:02.825","Text":"Similarly here we have a minus and a plus."},{"Start":"01:02.825 ","End":"01:06.050","Text":"This simplifies to this."},{"Start":"01:06.050 ","End":"01:08.870","Text":"Here we have the same expression twice."},{"Start":"01:08.870 ","End":"01:11.000","Text":"Instead of 4 in the denominator,"},{"Start":"01:11.000 ","End":"01:14.620","Text":"we can put a 2 in the denominator and just take one of them."},{"Start":"01:14.620 ","End":"01:20.270","Text":"Also replaced the minus x minus y by minus of x plus y."},{"Start":"01:20.270 ","End":"01:26.690","Text":"Now we see that the right-hand side is just the sine hyperbolic of x plus y."},{"Start":"01:26.690 ","End":"01:29.120","Text":"We\u0027ve proven what we had to show."},{"Start":"01:29.120 ","End":"01:32.029","Text":"Just put the left and right reversed."},{"Start":"01:32.029 ","End":"01:34.010","Text":"But the case with a plus."},{"Start":"01:34.010 ","End":"01:37.340","Text":"Now let\u0027s see what happens if we have a minus."},{"Start":"01:37.340 ","End":"01:39.575","Text":"We can write that as a plus,"},{"Start":"01:39.575 ","End":"01:42.475","Text":"but instead of y negative y."},{"Start":"01:42.475 ","End":"01:45.650","Text":"Using the formula we just proved,"},{"Start":"01:45.650 ","End":"01:46.925","Text":"but with a plus,"},{"Start":"01:46.925 ","End":"01:53.210","Text":"we get sine hyperbolic cosine hyperbolic plus cosine hyperbolic sine hyperbolic."},{"Start":"01:53.210 ","End":"01:56.280","Text":"Cosine hyperbolic is an even function,"},{"Start":"01:56.280 ","End":"01:58.440","Text":"so we can just throw out the minus."},{"Start":"01:58.440 ","End":"02:01.440","Text":"But sign hyperbolic is odd."},{"Start":"02:01.440 ","End":"02:07.050","Text":"We keep the minus and the minus will come out in front."},{"Start":"02:07.100 ","End":"02:10.340","Text":"We get exactly what we had to get,"},{"Start":"02:10.340 ","End":"02:12.500","Text":"just like the formula above."},{"Start":"02:12.500 ","End":"02:14.600","Text":"But when we have a minus here,"},{"Start":"02:14.600 ","End":"02:19.500","Text":"we also have a minus here. We\u0027re done."}],"ID":30557}],"Thumbnail":null,"ID":293022}]

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