[{"Name":"Introduction to Graphs of a Function","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Connection between Graphs of F and F`","Duration":"17m 5s","ChapterTopicVideoID":6240,"CourseChapterTopicPlaylistID":4030,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6240.jpeg","UploadDate":"2019-11-14T07:17:56.1600000","DurationForVideoObject":"PT17M5S","Description":null,"MetaTitle":"Relation between Graphs of F and F`: Video + Workbook | Proprep","MetaDescription":"Graphs of a Function and its Derivative - Introduction to Graphs of a Function. 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/graphs-of-a-function-and-its-derivative/introduction-to-graphs-of-a-function/vid6240","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.340","Text":"In this clip, I\u0027m going to be talking about"},{"Start":"00:02.340 ","End":"00:05.170","Text":"the connection between the graph of a function,"},{"Start":"00:05.170 ","End":"00:10.410","Text":"and I\u0027d like to emphasize the word graph between that,"},{"Start":"00:10.410 ","End":"00:12.299","Text":"and the graph of its derivative,"},{"Start":"00:12.299 ","End":"00:15.285","Text":"I also want to emphasize the word graph."},{"Start":"00:15.285 ","End":"00:19.320","Text":"It\u0027s not a concept we\u0027ve encountered much so far,"},{"Start":"00:19.320 ","End":"00:21.525","Text":"this graph of a derivative,"},{"Start":"00:21.525 ","End":"00:25.560","Text":"so let\u0027s start right away with an example."},{"Start":"00:25.560 ","End":"00:30.780","Text":"The example I\u0027m going to take will be the following function."},{"Start":"00:30.780 ","End":"00:41.904","Text":"Let\u0027s take F of x to be equal to x cubed minus x."},{"Start":"00:41.904 ","End":"00:44.180","Text":"If I asked you to sketch its graph,"},{"Start":"00:44.180 ","End":"00:45.470","Text":"you\u0027d probably know what to do."},{"Start":"00:45.470 ","End":"00:48.020","Text":"You might find the intersection with the axis and"},{"Start":"00:48.020 ","End":"00:51.950","Text":"find that it cuts the x-axis at minus 1, 0 and 1."},{"Start":"00:51.950 ","End":"00:54.515","Text":"The y-axis only at the origin."},{"Start":"00:54.515 ","End":"00:58.280","Text":"You could find intervals where it increases,"},{"Start":"00:58.280 ","End":"01:02.975","Text":"decreases, concave, convex, and so on."},{"Start":"01:02.975 ","End":"01:06.500","Text":"I\u0027ll save you the trouble and I\u0027ll do"},{"Start":"01:06.500 ","End":"01:09.670","Text":"the sketch for you because that\u0027s not what we\u0027re discussing here,"},{"Start":"01:09.670 ","End":"01:11.810","Text":"is how to sketch."},{"Start":"01:11.940 ","End":"01:16.200","Text":"Here it is, the graph of this function."},{"Start":"01:16.200 ","End":"01:21.785","Text":"Now, if I differentiate this function or take its derivative,"},{"Start":"01:21.785 ","End":"01:23.525","Text":"then what I\u0027ll get,"},{"Start":"01:23.525 ","End":"01:30.360","Text":"I\u0027ll get f prime of x. F prime of x,"},{"Start":"01:30.360 ","End":"01:32.240","Text":"well, it\u0027s a function in its own right,"},{"Start":"01:32.240 ","End":"01:33.770","Text":"so I can also give it a name,"},{"Start":"01:33.770 ","End":"01:36.515","Text":"let\u0027s call it f of x,"},{"Start":"01:36.515 ","End":"01:39.560","Text":"is equal to, it\u0027s a polynomial, we know how to do this,"},{"Start":"01:39.560 ","End":"01:44.305","Text":"it\u0027s 3x squared minus 1."},{"Start":"01:44.305 ","End":"01:48.350","Text":"Now, other than the fact that it\u0027s a derivative of this,"},{"Start":"01:48.350 ","End":"01:50.510","Text":"it\u0027s a function, as I said, in its own right."},{"Start":"01:50.510 ","End":"01:52.190","Text":"We can sketch its graph,"},{"Start":"01:52.190 ","End":"01:56.010","Text":"and that will look something like this."},{"Start":"01:56.180 ","End":"02:01.200","Text":"Here it is. We know that it\u0027s a parabola."},{"Start":"02:01.200 ","End":"02:02.720","Text":"It\u0027s upward facing."},{"Start":"02:02.720 ","End":"02:06.970","Text":"It cuts the y-axis at minus 1."},{"Start":"02:06.970 ","End":"02:09.410","Text":"It\u0027s symmetrical about the y-axis."},{"Start":"02:09.410 ","End":"02:15.200","Text":"It\u0027s an even function. This is what it looks like. Let me label them."},{"Start":"02:15.200 ","End":"02:18.860","Text":"This one is the graph of"},{"Start":"02:18.860 ","End":"02:27.530","Text":"the function and the other one"},{"Start":"02:27.530 ","End":"02:33.420","Text":"is the graph of the derivative,"},{"Start":"02:33.530 ","End":"02:37.330","Text":"the derivative function that is."},{"Start":"02:38.120 ","End":"02:42.820","Text":"Oops, I made a spelling mistake, there."},{"Start":"02:43.030 ","End":"02:49.764","Text":"This tutorial deals with the connection between the graph of the function,"},{"Start":"02:49.764 ","End":"02:51.170","Text":"this is the function F,"},{"Start":"02:51.170 ","End":"02:53.080","Text":"and the graph of the derivative,"},{"Start":"02:53.080 ","End":"02:56.780","Text":"the derivative because it\u0027s f prime."},{"Start":"02:56.850 ","End":"03:02.965","Text":"Now, we happen to have been given the equation of the function,"},{"Start":"03:02.965 ","End":"03:06.720","Text":"and we could compute the equation of the derivative and sketch them both,"},{"Start":"03:06.720 ","End":"03:10.960","Text":"but in general, what we\u0027re going to do is"},{"Start":"03:10.960 ","End":"03:16.340","Text":"discuss the connection between the graphs when we don\u0027t have the equations."},{"Start":"03:16.340 ","End":"03:18.940","Text":"We\u0027ll actually do things both ways."},{"Start":"03:18.940 ","End":"03:20.215","Text":"A question might be,"},{"Start":"03:20.215 ","End":"03:26.815","Text":"given the graph of a function to sketch its derivative and also the other way round."},{"Start":"03:26.815 ","End":"03:31.220","Text":"Now, the connection between these 2 graphs is"},{"Start":"03:31.220 ","End":"03:35.135","Text":"better explained when we don\u0027t draw them separately,"},{"Start":"03:35.135 ","End":"03:40.649","Text":"but superimpose them on 1 set of axis as follows."},{"Start":"03:40.670 ","End":"03:43.835","Text":"Here they are superimposed."},{"Start":"03:43.835 ","End":"03:45.905","Text":"I just want to label them."},{"Start":"03:45.905 ","End":"03:49.205","Text":"This one was the original function."},{"Start":"03:49.205 ","End":"03:58.310","Text":"This was y equals F of x. I used capital F, doesn\u0027t matter,"},{"Start":"03:58.310 ","End":"04:06.545","Text":"just to distinguish it from the derivative which was y equals f of x,"},{"Start":"04:06.545 ","End":"04:12.500","Text":"which is actually the derivative of F of x."},{"Start":"04:12.500 ","End":"04:15.530","Text":"The blue one is the function,"},{"Start":"04:15.530 ","End":"04:17.765","Text":"the red one is the derivative."},{"Start":"04:17.765 ","End":"04:23.210","Text":"I\u0027d like to make a recommendation as far as the use of colors goes."},{"Start":"04:23.210 ","End":"04:25.895","Text":"When you do the exercises,"},{"Start":"04:25.895 ","End":"04:28.160","Text":"you should probably stick to the same colors."},{"Start":"04:28.160 ","End":"04:30.350","Text":"It doesn\u0027t have to be blue and red,"},{"Start":"04:30.350 ","End":"04:33.995","Text":"but whatever color it is that you used for the original function,"},{"Start":"04:33.995 ","End":"04:38.305","Text":"stick with it and whatever choice you use for the derivative,"},{"Start":"04:38.305 ","End":"04:39.740","Text":"stick with that too."},{"Start":"04:39.740 ","End":"04:45.800","Text":"In this clip, we\u0027re going to use blue for the original and red for the derivative."},{"Start":"04:45.800 ","End":"04:51.005","Text":"At this point, I\u0027m not actually interested in what these functions are."},{"Start":"04:51.005 ","End":"04:53.390","Text":"This was x cubed minus x,"},{"Start":"04:53.390 ","End":"04:55.760","Text":"and this was 3x squared minus 1,"},{"Start":"04:55.760 ","End":"04:58.115","Text":"but that doesn\u0027t matter at this stage."},{"Start":"04:58.115 ","End":"05:02.810","Text":"We\u0027re going to talk in general and I wanted to explore the connection in"},{"Start":"05:02.810 ","End":"05:08.250","Text":"general between a function and its derivative as far as their graphs go."},{"Start":"05:08.250 ","End":"05:16.325","Text":"Let\u0027s recall some basic calculus of what the derivative means."},{"Start":"05:16.325 ","End":"05:21.890","Text":"Remember that the derivative is actually the slope of the function,"},{"Start":"05:21.890 ","End":"05:23.540","Text":"that\u0027s what it represents."},{"Start":"05:23.540 ","End":"05:25.970","Text":"When a function is increasing,"},{"Start":"05:25.970 ","End":"05:30.350","Text":"its derivative is positive and when a function is decreasing,"},{"Start":"05:30.350 ","End":"05:32.255","Text":"its derivative is negative."},{"Start":"05:32.255 ","End":"05:36.420","Text":"Let me first write that down and then I\u0027ll illustrate further."},{"Start":"05:38.230 ","End":"05:46.500","Text":"When the original function is increasing,"},{"Start":"05:50.600 ","End":"05:58.600","Text":"the derivative f prime is positive."},{"Start":"05:59.420 ","End":"06:04.915","Text":"It works the other way round too so I\u0027ll just write and vice versa,"},{"Start":"06:04.915 ","End":"06:08.240","Text":"which means the other way round."},{"Start":"06:08.730 ","End":"06:16.375","Text":"Similarly, for decreasing, when our original function,"},{"Start":"06:16.375 ","End":"06:18.490","Text":"which is here F,"},{"Start":"06:18.490 ","End":"06:21.980","Text":"is decreasing,"},{"Start":"06:26.220 ","End":"06:29.310","Text":"let me just move this down a bit,"},{"Start":"06:29.310 ","End":"06:36.040","Text":"then f prime is negative,"},{"Start":"06:36.920 ","End":"06:39.530","Text":"and the other way around too,"},{"Start":"06:39.530 ","End":"06:40.970","Text":"when the derivative is negative,"},{"Start":"06:40.970 ","End":"06:42.245","Text":"the function is decreasing."},{"Start":"06:42.245 ","End":"06:46.130","Text":"Again, I\u0027ll write vice versa,"},{"Start":"06:46.130 ","End":"06:48.700","Text":"which means the other way round too."},{"Start":"06:48.700 ","End":"06:51.260","Text":"If f prime is negative,"},{"Start":"06:51.260 ","End":"06:54.065","Text":"f is decreasing and so on."},{"Start":"06:54.065 ","End":"06:57.419","Text":"Let\u0027s see this in more detail."},{"Start":"06:57.460 ","End":"07:00.200","Text":"The thing that I\u0027m interested in,"},{"Start":"07:00.200 ","End":"07:02.270","Text":"as far as the red function goes,"},{"Start":"07:02.270 ","End":"07:07.730","Text":"the derivative is where it\u0027s positive and where it\u0027s negative."},{"Start":"07:07.730 ","End":"07:13.205","Text":"Positive means above the x-axis and negative is below the y-axis."},{"Start":"07:13.205 ","End":"07:19.160","Text":"These point here, and there\u0027s 2 of them in this particular case,"},{"Start":"07:19.160 ","End":"07:20.600","Text":"although in general you can\u0027t say,"},{"Start":"07:20.600 ","End":"07:23.690","Text":"they might not be any and there might be very many."},{"Start":"07:23.690 ","End":"07:26.419","Text":"But here are the places where"},{"Start":"07:26.419 ","End":"07:32.359","Text":"the derivative function changes from positive to negative or from negative to positive."},{"Start":"07:32.359 ","End":"07:35.840","Text":"Here it\u0027s positive, here it\u0027s negative, here it\u0027s positive."},{"Start":"07:35.840 ","End":"07:39.170","Text":"Let me label them just to be more precise here."},{"Start":"07:39.170 ","End":"07:40.850","Text":"I\u0027ll label this one as a,"},{"Start":"07:40.850 ","End":"07:42.440","Text":"this value of x, and this one,"},{"Start":"07:42.440 ","End":"07:49.880","Text":"I\u0027ll label b and I\u0027ll draw some dotted lines through and I\u0027ll also"},{"Start":"07:49.880 ","End":"07:54.140","Text":"just emphasize the place where"},{"Start":"07:54.140 ","End":"08:00.285","Text":"the dotted lines cut the blue function, here and here."},{"Start":"08:00.285 ","End":"08:02.315","Text":"Now, let\u0027s examine these."},{"Start":"08:02.315 ","End":"08:03.845","Text":"Let\u0027s look at the first one."},{"Start":"08:03.845 ","End":"08:07.345","Text":"When F is increasing,"},{"Start":"08:07.345 ","End":"08:09.260","Text":"the original function is increasing,"},{"Start":"08:09.260 ","End":"08:11.705","Text":"the derivative function is positive."},{"Start":"08:11.705 ","End":"08:15.020","Text":"Perhaps I should have written these in color. You know what?"},{"Start":"08:15.020 ","End":"08:21.590","Text":"Not a bad idea. Looking at the first statement, we can see it."},{"Start":"08:21.590 ","End":"08:26.060","Text":"If we look at this part here of the blue function,"},{"Start":"08:26.060 ","End":"08:31.795","Text":"F, is increasing up to this point where x is a."},{"Start":"08:31.795 ","End":"08:34.940","Text":"At the same time, the red function,"},{"Start":"08:34.940 ","End":"08:37.100","Text":"the derivative is positive,"},{"Start":"08:37.100 ","End":"08:39.275","Text":"it\u0027s above the axis."},{"Start":"08:39.275 ","End":"08:43.210","Text":"Similarly, to the right of b,"},{"Start":"08:43.210 ","End":"08:46.490","Text":"the derivative function is positive,"},{"Start":"08:46.490 ","End":"08:48.830","Text":"it\u0027s above the x-axis."},{"Start":"08:48.830 ","End":"08:54.310","Text":"This function from here onwards is increasing."},{"Start":"08:54.310 ","End":"08:57.080","Text":"The same is true for the second statement."},{"Start":"08:57.080 ","End":"09:00.230","Text":"If we look between a and b,"},{"Start":"09:00.230 ","End":"09:08.345","Text":"we see that the red function is negative and the blue function is decreasing."},{"Start":"09:08.345 ","End":"09:13.280","Text":"Actually, one of the connections is that we are"},{"Start":"09:13.280 ","End":"09:18.080","Text":"interested in different things that are related ie,"},{"Start":"09:18.080 ","End":"09:21.410","Text":"with the case of the original function,"},{"Start":"09:21.410 ","End":"09:27.725","Text":"what we\u0027re interested is where it\u0027s increasing and where it\u0027s decreasing."},{"Start":"09:27.725 ","End":"09:30.920","Text":"As here, increasing, decreasing, increasing."},{"Start":"09:30.920 ","End":"09:34.280","Text":"But in the case of the red function,"},{"Start":"09:34.280 ","End":"09:39.360","Text":"the derivative, what we\u0027re interested is where it\u0027s positive and where it\u0027s negative."},{"Start":"09:39.360 ","End":"09:42.095","Text":"As here, positive, negative,"},{"Start":"09:42.095 ","End":"09:45.005","Text":"positive. That\u0027s the relation."},{"Start":"09:45.005 ","End":"09:50.300","Text":"Positivity or negativity of the red function coincides"},{"Start":"09:50.300 ","End":"09:54.854","Text":"with increasing or decreasing for the original function"},{"Start":"09:54.854 ","End":"09:58.990","Text":"So far, we\u0027ve done 2 points out of 3."},{"Start":"09:58.990 ","End":"10:00.745","Text":"There is a third one."},{"Start":"10:00.745 ","End":"10:05.544","Text":"Notice that the first one relates to the derivative being positive."},{"Start":"10:05.544 ","End":"10:08.680","Text":"The second is when the derivative is"},{"Start":"10:08.680 ","End":"10:13.810","Text":"negative and there was going to be a third case when it\u0027s 0."},{"Start":"10:13.810 ","End":"10:17.020","Text":"Also as far as the function goes,"},{"Start":"10:17.020 ","End":"10:20.215","Text":"we\u0027ve considered increasing and we\u0027ve considered decreasing."},{"Start":"10:20.215 ","End":"10:23.695","Text":"But what about maximum and minimum?"},{"Start":"10:23.695 ","End":"10:28.225","Text":"Let me just write it down and then we\u0027ll explain it in the picture."},{"Start":"10:28.225 ","End":"10:38.860","Text":"The thing is this, and that is that when F has an extremum,"},{"Start":"10:38.860 ","End":"10:44.500","Text":"and remember that extremum is the common word for maximum or minimum."},{"Start":"10:44.500 ","End":"10:47.590","Text":"What do we say about F prime in this?"},{"Start":"10:47.590 ","End":"10:51.325","Text":"F prime changes sign,"},{"Start":"10:51.325 ","End":"10:53.815","Text":"all will be explained,"},{"Start":"10:53.815 ","End":"11:00.770","Text":"changes sign, and another way of saying that is crosses the x axis."},{"Start":"11:04.740 ","End":"11:07.710","Text":"When f has an extremum,"},{"Start":"11:07.710 ","End":"11:10.890","Text":"F prime changes sign across the x-axis and as before,"},{"Start":"11:10.890 ","End":"11:13.770","Text":"we\u0027ll write the words and vice versa."},{"Start":"11:13.770 ","End":"11:15.120","Text":"It\u0027s a 2 way process,"},{"Start":"11:15.120 ","End":"11:17.740","Text":"if you know one you know the other."},{"Start":"11:17.900 ","End":"11:20.510","Text":"But I would like to,"},{"Start":"11:20.510 ","End":"11:22.420","Text":"at the beginning of my explanation,"},{"Start":"11:22.420 ","End":"11:27.670","Text":"point out that I didn\u0027t say F prime is 0 like I started to say,"},{"Start":"11:27.670 ","End":"11:31.060","Text":"that\u0027s not good enough because it could"},{"Start":"11:31.060 ","End":"11:35.020","Text":"go from positive to 0 and then bounce back to positive again."},{"Start":"11:35.020 ","End":"11:38.170","Text":"For example, it actually has to be 0"},{"Start":"11:38.170 ","End":"11:41.530","Text":"in a way that it\u0027s positive on one side and negative on the other,"},{"Start":"11:41.530 ","End":"11:43.330","Text":"like at a or b."},{"Start":"11:43.330 ","End":"11:47.305","Text":"Now it\u0027s negative on the left and positive on the right and in each of these cases,"},{"Start":"11:47.305 ","End":"11:49.510","Text":"it actually crosses the x axis,"},{"Start":"11:49.510 ","End":"11:51.775","Text":"it doesn\u0027t just touch the x-axis,"},{"Start":"11:51.775 ","End":"11:53.170","Text":"it actually crosses it."},{"Start":"11:53.170 ","End":"11:55.195","Text":"You can choose either a formulation,"},{"Start":"11:55.195 ","End":"11:57.355","Text":"either to say that F prime,"},{"Start":"11:57.355 ","End":"12:04.750","Text":"that it changes sign or you could think of it as crosses the x-axis."},{"Start":"12:04.750 ","End":"12:09.280","Text":"In the case of the function itself, like here,"},{"Start":"12:09.280 ","End":"12:13.420","Text":"we have a maximum,"},{"Start":"12:13.420 ","End":"12:15.714","Text":"it\u0027s an extremum, but it\u0027s a maximum."},{"Start":"12:15.714 ","End":"12:18.865","Text":"Here we have an extremum and it happens to be a minimum."},{"Start":"12:18.865 ","End":"12:21.805","Text":"Of course it could be several of these or none of these,"},{"Start":"12:21.805 ","End":"12:24.700","Text":"so that\u0027s the relationship."},{"Start":"12:24.700 ","End":"12:26.545","Text":"You can see by the colors,"},{"Start":"12:26.545 ","End":"12:30.430","Text":"the relationship between f and F prime."},{"Start":"12:30.430 ","End":"12:33.639","Text":"Increasing means this is positive,"},{"Start":"12:33.639 ","End":"12:36.835","Text":"this is decreasing, same thing as this is negative."},{"Start":"12:36.835 ","End":"12:39.969","Text":"The extremum means this one changes sign,"},{"Start":"12:39.969 ","End":"12:43.375","Text":"or the alternative crosses the x-axis."},{"Start":"12:43.375 ","End":"12:47.380","Text":"I think we\u0027re about ready to do some exercises."},{"Start":"12:47.380 ","End":"12:50.035","Text":"Let me think of anything else I want to say."},{"Start":"12:50.035 ","End":"12:54.025","Text":"Yes, indeed, I do have something to say."},{"Start":"12:54.025 ","End":"12:59.350","Text":"First of all, a typical exercise will be given the blue one,"},{"Start":"12:59.350 ","End":"13:02.080","Text":"find the red one, or given the red one, find the blue one."},{"Start":"13:02.080 ","End":"13:06.055","Text":"Meaning we\u0027ll be given an exercise where we\u0027re told,"},{"Start":"13:06.055 ","End":"13:07.375","Text":"we\u0027re given the graph,"},{"Start":"13:07.375 ","End":"13:10.930","Text":"just a picture of the function itself and"},{"Start":"13:10.930 ","End":"13:15.560","Text":"we have to sketch roughly its derivative or vice versa."},{"Start":"13:16.320 ","End":"13:19.270","Text":"One technique that helps,"},{"Start":"13:19.270 ","End":"13:22.690","Text":"which I must show you although you do not have to use it."},{"Start":"13:22.690 ","End":"13:25.540","Text":"It\u0027s just optional for those who like tables,"},{"Start":"13:25.540 ","End":"13:29.020","Text":"or at least this particular table I\u0027m going to show you is the table method and"},{"Start":"13:29.020 ","End":"13:33.670","Text":"you will see it\u0027s very familiar from investigation of functions."},{"Start":"13:33.670 ","End":"13:36.265","Text":"What we do is we make a table."},{"Start":"13:36.265 ","End":"13:41.440","Text":"The table will typically have 3 rows,"},{"Start":"13:41.440 ","End":"13:45.160","Text":"where the top one I will label x,"},{"Start":"13:45.160 ","End":"13:47.200","Text":"the second one, y prime,"},{"Start":"13:47.200 ","End":"13:50.155","Text":"which is F prime,"},{"Start":"13:50.155 ","End":"13:54.550","Text":"when I say F prime that\u0027s like y prime."},{"Start":"13:54.550 ","End":"13:58.120","Text":"Y prime, y prime."},{"Start":"13:58.120 ","End":"14:02.290","Text":"This F is y and f y, y,"},{"Start":"14:02.290 ","End":"14:04.720","Text":"y and F prime is y prime, y prime,"},{"Start":"14:04.720 ","End":"14:09.475","Text":"y prime, which is also little f there."},{"Start":"14:09.475 ","End":"14:13.030","Text":"So x, y prime and y. I\u0027ll"},{"Start":"14:13.030 ","End":"14:16.615","Text":"demonstrate it in our particular example though each example is different."},{"Start":"14:16.615 ","End":"14:25.990","Text":"But here we mark the points where the derivative is 0, crosses the x-axis."},{"Start":"14:25.990 ","End":"14:32.785","Text":"So here we have points of interest are a and the other point of interest is B."},{"Start":"14:32.785 ","End":"14:34.405","Text":"At these 2 points,"},{"Start":"14:34.405 ","End":"14:36.564","Text":"the derivative F prime,"},{"Start":"14:36.564 ","End":"14:39.775","Text":"y prime is 0."},{"Start":"14:39.775 ","End":"14:44.020","Text":"In-between it is to the left of a,"},{"Start":"14:44.020 ","End":"14:48.475","Text":"we have that the red function is positive,"},{"Start":"14:48.475 ","End":"14:50.515","Text":"that\u0027s the y prime."},{"Start":"14:50.515 ","End":"14:54.590","Text":"Actually I should have written these in color also. Hang on."},{"Start":"14:54.590 ","End":"14:57.240","Text":"It looks better in color and you know what,"},{"Start":"14:57.240 ","End":"14:59.370","Text":"all these little green ys,"},{"Start":"14:59.370 ","End":"15:01.185","Text":"I can get rid of those."},{"Start":"15:01.185 ","End":"15:06.090","Text":"We just remember that y means f and y prime is F prime. Where were we?"},{"Start":"15:06.090 ","End":"15:11.205","Text":"We were saying that the red function is 0 here and here at a and b."},{"Start":"15:11.205 ","End":"15:15.125","Text":"To the left of a, it\u0027s above the axis so positive."},{"Start":"15:15.125 ","End":"15:16.990","Text":"Between a and b,"},{"Start":"15:16.990 ","End":"15:18.190","Text":"it\u0027s below the axis,"},{"Start":"15:18.190 ","End":"15:20.050","Text":"and hence it\u0027s negative,"},{"Start":"15:20.050 ","End":"15:22.640","Text":"and to the right of b,"},{"Start":"15:25.110 ","End":"15:31.030","Text":"it\u0027s again above the axis so it\u0027s positive and that\u0027s as far as"},{"Start":"15:31.030 ","End":"15:39.220","Text":"the derivative F prime goes and as far as the function itself up to the point a,"},{"Start":"15:39.220 ","End":"15:41.539","Text":"this is like the interval,"},{"Start":"15:42.690 ","End":"15:48.430","Text":"it means like less than a and this is between a and b,"},{"Start":"15:48.430 ","End":"15:50.740","Text":"let\u0027s say like x less than a."},{"Start":"15:50.740 ","End":"15:54.310","Text":"Here we might have a less than x,"},{"Start":"15:54.310 ","End":"16:02.360","Text":"less than b and here x is greater than b. Yeah, it\u0027s okay."},{"Start":"16:02.490 ","End":"16:08.695","Text":"Here the less than a, we are increasing,"},{"Start":"16:08.695 ","End":"16:16.600","Text":"the blue function is increasing and so I put an upwards arrow up into the right."},{"Start":"16:16.600 ","End":"16:20.350","Text":"Here between this point and this point"},{"Start":"16:20.350 ","End":"16:24.640","Text":"that function is decreasing and so I put a down arrow,"},{"Start":"16:24.640 ","End":"16:26.560","Text":"and to the right of b,"},{"Start":"16:26.560 ","End":"16:29.860","Text":"x bigger than b, it\u0027s again increasing."},{"Start":"16:29.860 ","End":"16:32.770","Text":"Just at these 2 points, at a and b,"},{"Start":"16:32.770 ","End":"16:35.380","Text":"I have an extremum, this is a maximum and this is a minimum."},{"Start":"16:35.380 ","End":"16:38.845","Text":"When we see up and then down, that\u0027s max,"},{"Start":"16:38.845 ","End":"16:42.340","Text":"maximum and down and then up,"},{"Start":"16:42.340 ","End":"16:45.340","Text":"like here, that\u0027s a minimum."},{"Start":"16:45.340 ","End":"16:48.430","Text":"Now, in any individual case will either have"},{"Start":"16:48.430 ","End":"16:51.820","Text":"this row and deduce this row or the other way around."},{"Start":"16:51.820 ","End":"16:54.100","Text":"As I say, it\u0027ll be some exercises, where we\u0027re given the blue,"},{"Start":"16:54.100 ","End":"16:55.840","Text":"have to find the red and in some cases,"},{"Start":"16:55.840 ","End":"16:58.585","Text":"we\u0027ll be given the red and having to find the blue."},{"Start":"16:58.585 ","End":"17:02.020","Text":"I think I\u0027ve gone on long enough,"},{"Start":"17:02.020 ","End":"17:05.360","Text":"it\u0027s time to do some exercises."}],"ID":6240},{"Watched":false,"Name":"Example 1","Duration":"4m 54s","ChapterTopicVideoID":6228,"CourseChapterTopicPlaylistID":4030,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.680","Text":"In this example, we\u0027re given the graph of function f of x."},{"Start":"00:04.680 ","End":"00:06.555","Text":"That\u0027s this 1 in blue,"},{"Start":"00:06.555 ","End":"00:13.095","Text":"and we have to sketch roughly the graph of the derivative y equals f prime of x,"},{"Start":"00:13.095 ","End":"00:14.835","Text":"and I\u0027m going to do that 1 in red."},{"Start":"00:14.835 ","End":"00:20.100","Text":"Now, the main thing to know about the original function is where is it increasing,"},{"Start":"00:20.100 ","End":"00:22.110","Text":"where is it decreasing,"},{"Start":"00:22.110 ","End":"00:24.405","Text":"and where are its extrema?"},{"Start":"00:24.405 ","End":"00:28.320","Text":"Let\u0027s take a look. We start off and we\u0027re decreasing."},{"Start":"00:28.320 ","End":"00:30.945","Text":"We come to some minimum here,"},{"Start":"00:30.945 ","End":"00:33.120","Text":"let\u0027s say it\u0027s this point here,"},{"Start":"00:33.120 ","End":"00:35.635","Text":"and that\u0027s an extremum, it\u0027s a minimum."},{"Start":"00:35.635 ","End":"00:39.380","Text":"Then we start increasing again until we get"},{"Start":"00:39.380 ","End":"00:43.220","Text":"to say this point here and there we have a maximum,"},{"Start":"00:43.220 ","End":"00:46.330","Text":"and from there on we are decreasing."},{"Start":"00:46.330 ","End":"00:50.210","Text":"Now, in terms of the derivative function,"},{"Start":"00:50.210 ","End":"00:53.180","Text":"what this means is that where I have the extremum,"},{"Start":"00:53.180 ","End":"00:54.965","Text":"the derivative is 0,"},{"Start":"00:54.965 ","End":"00:58.580","Text":"so I have already a point on the derivative function."},{"Start":"00:58.580 ","End":"01:00.790","Text":"I also have another 0 here,"},{"Start":"01:00.790 ","End":"01:04.105","Text":"and I know that between these 2,"},{"Start":"01:04.105 ","End":"01:05.840","Text":"the original function is increasing,"},{"Start":"01:05.840 ","End":"01:07.280","Text":"so the derivative is positive,"},{"Start":"01:07.280 ","End":"01:10.640","Text":"so I have to draw some positive function that starts here and ends here."},{"Start":"01:10.640 ","End":"01:12.560","Text":"Let\u0027s draw something like this."},{"Start":"01:12.560 ","End":"01:15.385","Text":"We\u0027re not expected to be very precise at this stage."},{"Start":"01:15.385 ","End":"01:17.390","Text":"Now, because of the extremum,"},{"Start":"01:17.390 ","End":"01:20.050","Text":"maximum or minimum, we\u0027re going to cross the axis."},{"Start":"01:20.050 ","End":"01:22.105","Text":"Already I know that I\u0027m crossing it."},{"Start":"01:22.105 ","End":"01:27.020","Text":"It also makes sense because from here and to the left we are decreasing,"},{"Start":"01:27.020 ","End":"01:29.165","Text":"so we have to remain negative."},{"Start":"01:29.165 ","End":"01:33.025","Text":"Thread function is going to be negative and I\u0027ll continue it a bit."},{"Start":"01:33.025 ","End":"01:36.200","Text":"Likewise, to the right of this point,"},{"Start":"01:36.200 ","End":"01:40.265","Text":"perhaps I should have given these points some names and why don\u0027t we do that?"},{"Start":"01:40.265 ","End":"01:44.870","Text":"Let\u0027s call this point a and this point b."},{"Start":"01:44.870 ","End":"01:51.560","Text":"In fact, it would be nice to maybe even draw some dotted lines to indicate ranges."},{"Start":"01:51.560 ","End":"01:53.180","Text":"We have up to a,"},{"Start":"01:53.180 ","End":"01:58.800","Text":"we have a between a and b and from b onwards."},{"Start":"01:58.800 ","End":"02:01.695","Text":"Back to the sketching and let me just finish this,"},{"Start":"02:01.695 ","End":"02:03.370","Text":"to the right of this b,"},{"Start":"02:03.370 ","End":"02:08.675","Text":"the function is decreasing so the derivative remains negative."},{"Start":"02:08.675 ","End":"02:10.655","Text":"This is precise enough,"},{"Start":"02:10.655 ","End":"02:12.110","Text":"good enough at this stage,"},{"Start":"02:12.110 ","End":"02:17.500","Text":"just label it that this is the function f prime of x,"},{"Start":"02:17.500 ","End":"02:20.120","Text":"and basically, we\u0027re done except"},{"Start":"02:20.120 ","End":"02:23.275","Text":"that this is the kind of question you could see on an exam."},{"Start":"02:23.275 ","End":"02:25.850","Text":"Then on the exam if you just leave it like this,"},{"Start":"02:25.850 ","End":"02:27.650","Text":"but don\u0027t give any explanations,"},{"Start":"02:27.650 ","End":"02:28.820","Text":"then you\u0027ll lose points,"},{"Start":"02:28.820 ","End":"02:30.530","Text":"so you have a choice of 2 things,"},{"Start":"02:30.530 ","End":"02:33.290","Text":"you can either summarize what we just did in words."},{"Start":"02:33.290 ","End":"02:37.850","Text":"Something like we look at the blue function and we see where it\u0027s decreasing,"},{"Start":"02:37.850 ","End":"02:39.835","Text":"increasing and where the extrema are."},{"Start":"02:39.835 ","End":"02:42.400","Text":"At this extremum, we label it a,"},{"Start":"02:42.400 ","End":"02:44.490","Text":"at this extremum we label it b."},{"Start":"02:44.490 ","End":"02:51.574","Text":"To the left of a the function is decreasing so the derivative is below the axis,"},{"Start":"02:51.574 ","End":"02:53.960","Text":"say something like that to the right of b,"},{"Start":"02:53.960 ","End":"02:55.625","Text":"and between a and b,"},{"Start":"02:55.625 ","End":"02:58.640","Text":"the function is increasing so its derivative is"},{"Start":"02:58.640 ","End":"03:02.750","Text":"positive and it crosses the axis at a and b because of the extrema,"},{"Start":"03:02.750 ","End":"03:04.700","Text":"and thus we get the shape that we get."},{"Start":"03:04.700 ","End":"03:06.185","Text":"But I\u0027m going to use plan B,"},{"Start":"03:06.185 ","End":"03:07.385","Text":"which is the table."},{"Start":"03:07.385 ","End":"03:09.150","Text":"I prefer the table method,"},{"Start":"03:09.150 ","End":"03:10.785","Text":"less writing to do."},{"Start":"03:10.785 ","End":"03:12.779","Text":"Let\u0027s make a table,"},{"Start":"03:12.779 ","End":"03:15.120","Text":"and this just always looks the same."},{"Start":"03:15.120 ","End":"03:18.034","Text":"There\u0027s always 3 rows to the table."},{"Start":"03:18.034 ","End":"03:20.930","Text":"There\u0027s a row for x,"},{"Start":"03:20.930 ","End":"03:25.730","Text":"there\u0027s a row for y prime or f prime of x,"},{"Start":"03:25.730 ","End":"03:28.730","Text":"so let\u0027s do that 1 in red,"},{"Start":"03:28.730 ","End":"03:30.655","Text":"left prime of x,"},{"Start":"03:30.655 ","End":"03:35.765","Text":"and to a blue 1 also for the original function itself,"},{"Start":"03:35.765 ","End":"03:38.600","Text":"which is f of x, or you can write it as y."},{"Start":"03:38.600 ","End":"03:42.335","Text":"Now, the important values are the extrema."},{"Start":"03:42.335 ","End":"03:45.980","Text":"In this case, they are the a and the b in the picture."},{"Start":"03:45.980 ","End":"03:47.750","Text":"Don\u0027t assume you\u0027ll always have 2 of them,"},{"Start":"03:47.750 ","End":"03:49.655","Text":"you could have any number including non,"},{"Start":"03:49.655 ","End":"03:53.240","Text":"and therefore we look at the blue function because we started off,"},{"Start":"03:53.240 ","End":"03:54.710","Text":"that\u0027s what we know."},{"Start":"03:54.710 ","End":"03:56.390","Text":"In the blue function,"},{"Start":"03:56.390 ","End":"04:00.290","Text":"we know that at a and b we have the extrema."},{"Start":"04:00.290 ","End":"04:06.215","Text":"In this case, a happens to be a minimum and b happens to be a maximum,"},{"Start":"04:06.215 ","End":"04:09.215","Text":"and between the minimum and the maximum,"},{"Start":"04:09.215 ","End":"04:12.320","Text":"the function is increasing."},{"Start":"04:12.320 ","End":"04:17.044","Text":"After the maximum it is decreasing and before the minimum,"},{"Start":"04:17.044 ","End":"04:19.715","Text":"it is also decreasing."},{"Start":"04:19.715 ","End":"04:24.695","Text":"If we translate this into what\u0027s happening with the f prime,"},{"Start":"04:24.695 ","End":"04:30.185","Text":"extrema are places where the derivative is 0, where it\u0027s decreasing,"},{"Start":"04:30.185 ","End":"04:36.630","Text":"it is negative, increasing positive, decreasing negative."},{"Start":"04:36.630 ","End":"04:44.240","Text":"It\u0027s this row here that helps us to draw the graph because we know that we have a 0 here,"},{"Start":"04:44.240 ","End":"04:48.035","Text":"a 0 here, negative here, positive here,"},{"Start":"04:48.035 ","End":"04:50.240","Text":"and negative here, and basically,"},{"Start":"04:50.240 ","End":"04:52.495","Text":"that\u0027s all there is to it."},{"Start":"04:52.495 ","End":"04:55.569","Text":"I\u0027m done on this example."}],"ID":6241},{"Watched":false,"Name":"Example 2","Duration":"2m 12s","ChapterTopicVideoID":6229,"CourseChapterTopicPlaylistID":4030,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.580","Text":"Here\u0027s another example, given the graph of a function here in blue."},{"Start":"00:04.580 ","End":"00:07.140","Text":"We want to draw the graph of its derivative function,"},{"Start":"00:07.140 ","End":"00:08.475","Text":"and we\u0027ll do that in red."},{"Start":"00:08.475 ","End":"00:10.605","Text":"But since this is a second example,"},{"Start":"00:10.605 ","End":"00:12.210","Text":"I\u0027m going to do it a bit quicker."},{"Start":"00:12.210 ","End":"00:16.410","Text":"What I\u0027m going to do is look for the areas of increasing, decreasing, and extremum."},{"Start":"00:16.410 ","End":"00:19.425","Text":"Here it\u0027s increasing, here we have a maximum."},{"Start":"00:19.425 ","End":"00:22.245","Text":"Here it\u0027s decreasing, here we have a minimum."},{"Start":"00:22.245 ","End":"00:23.535","Text":"Here it is increasing,"},{"Start":"00:23.535 ","End":"00:24.830","Text":"here we have a maximum,"},{"Start":"00:24.830 ","End":"00:26.405","Text":"and here it\u0027s decreasing."},{"Start":"00:26.405 ","End":"00:28.100","Text":"Let\u0027s label these points."},{"Start":"00:28.100 ","End":"00:31.805","Text":"I\u0027m just going to say that this corresponds to here."},{"Start":"00:31.805 ","End":"00:40.920","Text":"There\u0027s a point a, this point b and this one c. Now I\u0027m going to make the table."},{"Start":"00:40.920 ","End":"00:46.235","Text":"Notice increase maximum, decrease minimum, increase maximum, decrease."},{"Start":"00:46.235 ","End":"00:50.420","Text":"We have x, we have f prime of x and f of x,"},{"Start":"00:50.420 ","End":"00:52.940","Text":"which you could call y prime and y if you like."},{"Start":"00:52.940 ","End":"00:57.365","Text":"The important points, which are a, b,"},{"Start":"00:57.365 ","End":"01:02.030","Text":"and c. What we can say is that at point a,"},{"Start":"01:02.030 ","End":"01:04.675","Text":"we have a maximum."},{"Start":"01:04.675 ","End":"01:07.095","Text":"Up to a, we are increasing."},{"Start":"01:07.095 ","End":"01:08.715","Text":"At a, we have a maximum,"},{"Start":"01:08.715 ","End":"01:11.490","Text":"between a and b, we are decreasing."},{"Start":"01:11.490 ","End":"01:13.875","Text":"At b, we have a minimum."},{"Start":"01:13.875 ","End":"01:17.655","Text":"Between b and c, we are increasing."},{"Start":"01:17.655 ","End":"01:24.455","Text":"At c, we have a maximum and then from c and beyond, we are decreasing."},{"Start":"01:24.455 ","End":"01:28.005","Text":"These symbols translate for f prime,"},{"Start":"01:28.005 ","End":"01:30.795","Text":"increasing means f prime positive,"},{"Start":"01:30.795 ","End":"01:33.600","Text":"extremum means that it\u0027s 0."},{"Start":"01:33.600 ","End":"01:37.605","Text":"Decreasing means negative, extremum means 0,"},{"Start":"01:37.605 ","End":"01:41.390","Text":"increasing is positive, extremum is 0,"},{"Start":"01:41.390 ","End":"01:42.950","Text":"and this is negative."},{"Start":"01:42.950 ","End":"01:45.230","Text":"As far as the derivative goes,"},{"Start":"01:45.230 ","End":"01:50.080","Text":"we note that we have a 0 here and here 0."},{"Start":"01:50.080 ","End":"01:54.140","Text":"At these points, we have 0 and between them we have positive, negative,"},{"Start":"01:54.140 ","End":"01:58.159","Text":"positive, negative so I need the red to be here positive,"},{"Start":"01:58.159 ","End":"02:02.300","Text":"so here negative and then again going to 0,"},{"Start":"02:02.300 ","End":"02:08.190","Text":"and then positive and then to 0 and then negative."},{"Start":"02:08.190 ","End":"02:13.060","Text":"The general shape and that\u0027s it. We are done."}],"ID":6242},{"Watched":false,"Name":"Example 3","Duration":"3m 49s","ChapterTopicVideoID":6230,"CourseChapterTopicPlaylistID":4030,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.305","Text":"This exercise is different than the previous two."},{"Start":"00:04.305 ","End":"00:07.875","Text":"In here we have the derivative function given"},{"Start":"00:07.875 ","End":"00:11.610","Text":"and we have to sketch from it the function itself."},{"Start":"00:11.610 ","End":"00:13.410","Text":"We\u0027re going to draw the blue one."},{"Start":"00:13.410 ","End":"00:16.275","Text":"Notice that there\u0027s an extra piece of information here."},{"Start":"00:16.275 ","End":"00:19.080","Text":"We\u0027re told that when we draw the blue function,"},{"Start":"00:19.080 ","End":"00:22.185","Text":"that f of 0 equals 0,"},{"Start":"00:22.185 ","End":"00:25.080","Text":"which means that I can already start drawing"},{"Start":"00:25.080 ","End":"00:28.620","Text":"the original function because it passes through the origin,"},{"Start":"00:28.620 ","End":"00:30.220","Text":"so it\u0027s going to go through here."},{"Start":"00:30.220 ","End":"00:32.600","Text":"It turns out that you have to have an extra piece of"},{"Start":"00:32.600 ","End":"00:35.810","Text":"information when you\u0027re working backwards from the red to the blue."},{"Start":"00:35.810 ","End":"00:38.580","Text":"Otherwise, there\u0027s just no way to know where to start from."},{"Start":"00:38.580 ","End":"00:40.010","Text":"This is like an anchor."},{"Start":"00:40.010 ","End":"00:45.980","Text":"It\u0027s similar in many ways to the other exercise where we go from the blue to the red."},{"Start":"00:45.980 ","End":"00:47.105","Text":"In this case also,"},{"Start":"00:47.105 ","End":"00:48.874","Text":"I\u0027m going to make a table."},{"Start":"00:48.874 ","End":"00:51.090","Text":"But the table works differently."},{"Start":"00:51.090 ","End":"00:54.125","Text":"In this case and what interests us is,"},{"Start":"00:54.125 ","End":"00:55.760","Text":"where is it positive,"},{"Start":"00:55.760 ","End":"00:56.810","Text":"where is is negative,"},{"Start":"00:56.810 ","End":"00:59.990","Text":"and where is it 0, and especially where does it cross the axis?"},{"Start":"00:59.990 ","End":"01:01.445","Text":"I\u0027ll mark those points here."},{"Start":"01:01.445 ","End":"01:04.955","Text":"Here I have a 0, here I have a 0."},{"Start":"01:04.955 ","End":"01:08.300","Text":"I give them letters a and this is b."},{"Start":"01:08.300 ","End":"01:10.985","Text":"It\u0027s positive between a and b,"},{"Start":"01:10.985 ","End":"01:16.490","Text":"negative to the left of a and negative to the right of b."},{"Start":"01:16.490 ","End":"01:19.625","Text":"What we\u0027re going to expect from the blue function is for it to be"},{"Start":"01:19.625 ","End":"01:23.480","Text":"decreasing up to this dotted line."},{"Start":"01:23.480 ","End":"01:25.250","Text":"Then between these two,"},{"Start":"01:25.250 ","End":"01:30.230","Text":"it\u0027s going to be increasing and then it\u0027s going to be decreasing again."},{"Start":"01:30.230 ","End":"01:34.115","Text":"Here because the derivative\u0027s going from negative to positive,"},{"Start":"01:34.115 ","End":"01:36.140","Text":"we\u0027re going to expect a minimum here,"},{"Start":"01:36.140 ","End":"01:38.945","Text":"and we\u0027re going to expect maximum somewhere here."},{"Start":"01:38.945 ","End":"01:41.075","Text":"Let\u0027s draw a table right away."},{"Start":"01:41.075 ","End":"01:49.070","Text":"X, here we\u0027re going to have f prime of x and the original function f of x,"},{"Start":"01:49.070 ","End":"01:51.020","Text":"that\u0027s what we\u0027re looking for. We\u0027re given the red."},{"Start":"01:51.020 ","End":"01:54.860","Text":"Now, the important points are a and b."},{"Start":"01:54.860 ","End":"01:56.195","Text":"This time we know,"},{"Start":"01:56.195 ","End":"01:59.115","Text":"here is 0, here is 0."},{"Start":"01:59.115 ","End":"02:04.235","Text":"Not only is it 0 it\u0027s actually crossing because before a it\u0027s negative,"},{"Start":"02:04.235 ","End":"02:06.305","Text":"between these two it\u0027s positive,"},{"Start":"02:06.305 ","End":"02:08.135","Text":"and after that it\u0027s negative."},{"Start":"02:08.135 ","End":"02:10.895","Text":"What that means as far as the original function,"},{"Start":"02:10.895 ","End":"02:12.365","Text":"we have a decreasing,"},{"Start":"02:12.365 ","End":"02:13.895","Text":"we have an increasing,"},{"Start":"02:13.895 ","End":"02:15.545","Text":"we have a decreasing."},{"Start":"02:15.545 ","End":"02:18.080","Text":"Therefore, here we have not only do we"},{"Start":"02:18.080 ","End":"02:20.900","Text":"know it\u0027s an extremum we actually know it\u0027s a minimum."},{"Start":"02:20.900 ","End":"02:23.000","Text":"Here we know it\u0027s a maximum."},{"Start":"02:23.000 ","End":"02:26.450","Text":"We\u0027re all ready to do the sketch and you\u0027ll see in"},{"Start":"02:26.450 ","End":"02:29.840","Text":"the process why we need this extra anchor."},{"Start":"02:29.840 ","End":"02:35.565","Text":"I want to draw something that is decreasing to a minimum to the left of x equals a."},{"Start":"02:35.565 ","End":"02:37.665","Text":"There\u0027s so many ways I could do that."},{"Start":"02:37.665 ","End":"02:41.330","Text":"If I don\u0027t have something to start off with, I just can\u0027t do it."},{"Start":"02:41.330 ","End":"02:45.110","Text":"We start off with here and we\u0027re going to be decreasing to a minimum,"},{"Start":"02:45.110 ","End":"02:48.215","Text":"or even go a little bit beyond just to indicate the minimum,"},{"Start":"02:48.215 ","End":"02:51.035","Text":"and we\u0027re decreasing all the time."},{"Start":"02:51.035 ","End":"02:54.610","Text":"Now between a and b,"},{"Start":"02:54.610 ","End":"02:57.200","Text":"the derivative is positive,"},{"Start":"02:57.200 ","End":"03:01.010","Text":"so the function is increasing and it\u0027s increasing to a maximum."},{"Start":"03:01.010 ","End":"03:03.290","Text":"Now I don\u0027t know exactly how,"},{"Start":"03:03.290 ","End":"03:08.550","Text":"but it increases somehow and reaches a maximum."},{"Start":"03:08.550 ","End":"03:12.545","Text":"I\u0027ll go a little bit beyond just to show that we have a maximum."},{"Start":"03:12.545 ","End":"03:14.875","Text":"Here we had some minimum."},{"Start":"03:14.875 ","End":"03:19.175","Text":"Similarly here I\u0027ll roll a little tangent to indicate that we have a maximum,"},{"Start":"03:19.175 ","End":"03:22.010","Text":"and then we\u0027re decreasing something like this."},{"Start":"03:22.010 ","End":"03:23.915","Text":"I don\u0027t know how far it decreases."},{"Start":"03:23.915 ","End":"03:26.450","Text":"That\u0027s really about as much as you would be expected."},{"Start":"03:26.450 ","End":"03:29.070","Text":"You can\u0027t get very precise in this kind of question."},{"Start":"03:29.070 ","End":"03:30.450","Text":"That way summarize again."},{"Start":"03:30.450 ","End":"03:33.680","Text":"We are decreasing down to a minimum at point a."},{"Start":"03:33.680 ","End":"03:37.640","Text":"Then we\u0027re increasing to a maximum above x equals b,"},{"Start":"03:37.640 ","End":"03:40.205","Text":"and then we\u0027re decreasing again."},{"Start":"03:40.205 ","End":"03:43.520","Text":"This would be the function f of x."},{"Start":"03:43.520 ","End":"03:48.870","Text":"We got the blue from the red and we are done."}],"ID":6243},{"Watched":false,"Name":"Example 4","Duration":"2m 44s","ChapterTopicVideoID":6231,"CourseChapterTopicPlaylistID":4030,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.880","Text":"This question is similar to the previous one,"},{"Start":"00:02.880 ","End":"00:04.680","Text":"so I\u0027ll do it a bit faster."},{"Start":"00:04.680 ","End":"00:08.805","Text":"Once again, we\u0027re given the red function, meaning the derivative,"},{"Start":"00:08.805 ","End":"00:11.415","Text":"and we have to sketch from it"},{"Start":"00:11.415 ","End":"00:15.825","Text":"the original function from which it was derived, that\u0027s f of x."},{"Start":"00:15.825 ","End":"00:18.480","Text":"As before, I explained that we have to get an extra piece"},{"Start":"00:18.480 ","End":"00:20.984","Text":"of information to where to start the function."},{"Start":"00:20.984 ","End":"00:24.570","Text":"In this case, we\u0027re also given that it goes through the origin."},{"Start":"00:24.570 ","End":"00:27.300","Text":"Already we have a piece of blue to"},{"Start":"00:27.300 ","End":"00:30.395","Text":"start off with that the original function is going to go through."},{"Start":"00:30.395 ","End":"00:33.660","Text":"As before, we\u0027re going to be concerned,"},{"Start":"00:33.660 ","End":"00:35.280","Text":"because this is the derivative,"},{"Start":"00:35.280 ","End":"00:41.355","Text":"with positivity and negativity and places where it changes sign or crosses the axis."},{"Start":"00:41.355 ","End":"00:45.590","Text":"I can see that here\u0027s a place where I cross the axis,"},{"Start":"00:45.590 ","End":"00:48.065","Text":"here\u0027s a place where it crosses the axis."},{"Start":"00:48.065 ","End":"00:51.435","Text":"That\u0027s given letters a, b,"},{"Start":"00:51.435 ","End":"00:55.910","Text":"and c. If I look at this function from left to right,"},{"Start":"00:55.910 ","End":"00:59.465","Text":"I see that it is positive and then 0,"},{"Start":"00:59.465 ","End":"01:01.230","Text":"then negative, then 0,"},{"Start":"01:01.230 ","End":"01:03.545","Text":"then positive, then 0, then negative."},{"Start":"01:03.545 ","End":"01:05.990","Text":"I\u0027ll just go ahead and start with the table."},{"Start":"01:05.990 ","End":"01:09.260","Text":"We have the first row being x,"},{"Start":"01:09.260 ","End":"01:12.890","Text":"the next row is f prime of x,"},{"Start":"01:12.890 ","End":"01:16.775","Text":"and the last row will be f of x."},{"Start":"01:16.775 ","End":"01:19.430","Text":"In x, we put in all the interesting points,"},{"Start":"01:19.430 ","End":"01:24.455","Text":"meaning the places where essentially the function is 0 at a,"},{"Start":"01:24.455 ","End":"01:28.910","Text":"b and I have c. And I know that f prime here is 0,"},{"Start":"01:28.910 ","End":"01:33.375","Text":"0 and 0 plus, minus, plus, minus."},{"Start":"01:33.375 ","End":"01:36.575","Text":"For the derivative means increasing for the function."},{"Start":"01:36.575 ","End":"01:38.660","Text":"Similarly, negative is decreasing."},{"Start":"01:38.660 ","End":"01:41.555","Text":"Here we\u0027re increasing and here we\u0027re decreasing."},{"Start":"01:41.555 ","End":"01:45.590","Text":"When we go from plus to minus, it\u0027s a maximum."},{"Start":"01:45.590 ","End":"01:47.435","Text":"Here we have a minimum,"},{"Start":"01:47.435 ","End":"01:50.360","Text":"and here we have a maximum."},{"Start":"01:50.360 ","End":"01:55.010","Text":"What I want to do now is just use the blue and start sketching."},{"Start":"01:55.010 ","End":"01:57.110","Text":"We have an anchor point and I usually like to"},{"Start":"01:57.110 ","End":"02:00.185","Text":"start in the interval where I have this anchor."},{"Start":"02:00.185 ","End":"02:02.570","Text":"And this lies between a and b."},{"Start":"02:02.570 ","End":"02:07.385","Text":"Between a and b, we\u0027re decreasing from a maximum to a minimum."},{"Start":"02:07.385 ","End":"02:13.205","Text":"What I have to do is to have the function decreasing from a maximum to a minimum."},{"Start":"02:13.205 ","End":"02:16.505","Text":"Let\u0027s start with some maximum, let\u0027s say here."},{"Start":"02:16.505 ","End":"02:18.080","Text":"Then go through here,"},{"Start":"02:18.080 ","End":"02:22.250","Text":"and then go on here to a minimum between b and"},{"Start":"02:22.250 ","End":"02:26.645","Text":"c. We are increasing from a minimum up to some maximum."},{"Start":"02:26.645 ","End":"02:29.430","Text":"Beyond c, we are decreasing."},{"Start":"02:29.430 ","End":"02:32.225","Text":"To the left of a we are increasing."},{"Start":"02:32.225 ","End":"02:37.220","Text":"All I need to do now is label it that this function is the original function f of x."},{"Start":"02:37.220 ","End":"02:41.310","Text":"You see how this condition came to be used."},{"Start":"02:41.310 ","End":"02:44.380","Text":"We\u0027re done."}],"ID":6244},{"Watched":false,"Name":"Exercise 1","Duration":"3m 36s","ChapterTopicVideoID":6232,"CourseChapterTopicPlaylistID":4030,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.895","Text":"In this exercise, we\u0027re given the graph of a derivative of a function."},{"Start":"00:05.895 ","End":"00:08.880","Text":"We actually don\u0027t know what f of x looks like,"},{"Start":"00:08.880 ","End":"00:11.455","Text":"and we\u0027re just given its derivative."},{"Start":"00:11.455 ","End":"00:15.255","Text":"We have to answer questions about the original function f,"},{"Start":"00:15.255 ","End":"00:17.085","Text":"even though we don\u0027t have its graph."},{"Start":"00:17.085 ","End":"00:22.035","Text":"We have to basically show 3 things about the original function f which we don\u0027t have,"},{"Start":"00:22.035 ","End":"00:24.395","Text":"and that is the extrema,"},{"Start":"00:24.395 ","End":"00:29.525","Text":"the intervals where the function is increasing and where it\u0027s decreasing."},{"Start":"00:29.525 ","End":"00:31.085","Text":"Now in the tutorial,"},{"Start":"00:31.085 ","End":"00:33.620","Text":"we learned that these 3 concepts are all"},{"Start":"00:33.620 ","End":"00:37.430","Text":"related in the original function which we use to draw it in blue,"},{"Start":"00:37.430 ","End":"00:39.335","Text":"and the derived function."},{"Start":"00:39.335 ","End":"00:44.630","Text":"Extrema are places where the derivative crosses the axis."},{"Start":"00:44.630 ","End":"00:46.250","Text":"This is of interest to me."},{"Start":"00:46.250 ","End":"00:48.905","Text":"It\u0027s where the axis is crossed."},{"Start":"00:48.905 ","End":"00:52.130","Text":"This is where the axis is crust also."},{"Start":"00:52.130 ","End":"00:55.445","Text":"These will correspond to the extrema,"},{"Start":"00:55.445 ","End":"01:00.680","Text":"the place where the derivative is positive corresponds to the increasing."},{"Start":"01:00.680 ","End":"01:04.160","Text":"In this part here it\u0027s positive"},{"Start":"01:04.160 ","End":"01:08.345","Text":"and the original function is decreasing when the derivative is negative."},{"Start":"01:08.345 ","End":"01:14.440","Text":"Here it\u0027s positive, here it\u0027s negative and here it\u0027s negative, and here it\u0027s 0."},{"Start":"01:14.440 ","End":"01:16.835","Text":"Once we\u0027ve got this information,"},{"Start":"01:16.835 ","End":"01:18.710","Text":"we make a table."},{"Start":"01:18.710 ","End":"01:21.305","Text":"There\u0027s a variation on the theme of a table,"},{"Start":"01:21.305 ","End":"01:25.360","Text":"and that is to just draw a number line."},{"Start":"01:25.360 ","End":"01:27.410","Text":"On this number line,"},{"Start":"01:27.410 ","End":"01:30.035","Text":"I mark the important points,"},{"Start":"01:30.035 ","End":"01:33.680","Text":"which in this case are the places where the derivative"},{"Start":"01:33.680 ","End":"01:38.195","Text":"crosses the axis and that will be 1 and 5."},{"Start":"01:38.195 ","End":"01:42.139","Text":"I want to know about the function f,"},{"Start":"01:42.139 ","End":"01:46.280","Text":"which is the original function and I want to write some things about f prime"},{"Start":"01:46.280 ","End":"01:50.570","Text":"from which I conclude about F. Now as far as f prime is concerned,"},{"Start":"01:50.570 ","End":"01:55.835","Text":"what we said is that here it is negative."},{"Start":"01:55.835 ","End":"01:58.815","Text":"At this point, it\u0027s 0."},{"Start":"01:58.815 ","End":"02:03.530","Text":"From 1-5 it\u0027s positive and here it\u0027s negative below the axis,"},{"Start":"02:03.530 ","End":"02:05.660","Text":"above the axis, below the axis."},{"Start":"02:05.660 ","End":"02:09.290","Text":"Now the interpretation for the original function f,"},{"Start":"02:09.290 ","End":"02:15.740","Text":"which we don\u0027t have is that derivative negative means the function is decreasing."},{"Start":"02:15.740 ","End":"02:21.499","Text":"Here it is increasing because it\u0027s plus,"},{"Start":"02:21.499 ","End":"02:26.500","Text":"and here it is decreasing because the derivative is a minus."},{"Start":"02:26.500 ","End":"02:29.510","Text":"Between the decreasing and increasing,"},{"Start":"02:29.510 ","End":"02:32.930","Text":"we have an extremum of type minimum."},{"Start":"02:32.930 ","End":"02:37.420","Text":"This 0.1 is actually a mean"},{"Start":"02:37.420 ","End":"02:42.920","Text":"and this point where x is 5 is between increasing and decreasing."},{"Start":"02:42.920 ","End":"02:46.545","Text":"It\u0027s an extremum of type maximum."},{"Start":"02:46.545 ","End":"02:51.470","Text":"Remember, we wanted extrema increasing and decreasing."},{"Start":"02:51.470 ","End":"02:55.040","Text":"In fact, the extrema even split up into minimum and maximum."},{"Start":"02:55.040 ","End":"03:03.115","Text":"Let\u0027s see what we have is a minimum at x equals 1."},{"Start":"03:03.115 ","End":"03:09.605","Text":"We have a maximum when x equals 5,"},{"Start":"03:09.605 ","End":"03:14.390","Text":"we have increasing between 1 and 5."},{"Start":"03:14.390 ","End":"03:19.280","Text":"I\u0027ll write it as 1 less than x, less than 5."},{"Start":"03:19.280 ","End":"03:27.645","Text":"That\u0027s the increasing interval and decreasing that would be less than 1 or bigger than 5."},{"Start":"03:27.645 ","End":"03:37.750","Text":"I would say that x is less than 1 or x is bigger than 5. We are done."}],"ID":6245},{"Watched":false,"Name":"Exercise 2","Duration":"3m 2s","ChapterTopicVideoID":6233,"CourseChapterTopicPlaylistID":4030,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"This exercise is very similar to the previous one."},{"Start":"00:03.900 ","End":"00:08.370","Text":"Again, we have to determine properties of the function f of x,"},{"Start":"00:08.370 ","End":"00:13.800","Text":"which we don\u0027t have by means of the graph of its derivative f prime of x,"},{"Start":"00:13.800 ","End":"00:15.495","Text":"which we do have."},{"Start":"00:15.495 ","End":"00:17.340","Text":"In the previous case,"},{"Start":"00:17.340 ","End":"00:19.020","Text":"we had to find, what was it?"},{"Start":"00:19.020 ","End":"00:22.230","Text":"Extremum, increasing and decreasing and"},{"Start":"00:22.230 ","End":"00:26.010","Text":"the corresponding things here that we have to find out are"},{"Start":"00:26.010 ","End":"00:33.960","Text":"the inflection points and where the function is concave up and where it\u0027s concave down."},{"Start":"00:33.960 ","End":"00:36.690","Text":"Now, each of these properties in"},{"Start":"00:36.690 ","End":"00:41.610","Text":"the original function has an equivalent property in the derived function."},{"Start":"00:41.610 ","End":"00:48.465","Text":"Inflection points, they would be extrema of the derivative,"},{"Start":"00:48.465 ","End":"00:54.880","Text":"and there\u0027s only 1 extremum here and that would be this here where x is 3."},{"Start":"00:54.880 ","End":"00:56.730","Text":"That is the extremum,"},{"Start":"00:56.730 ","End":"00:59.195","Text":"it\u0027s a maximum of the derivative,"},{"Start":"00:59.195 ","End":"01:05.540","Text":"the concave up for the original function f corresponds"},{"Start":"01:05.540 ","End":"01:12.560","Text":"to the increasing and that would be increasing everywhere up to 3."},{"Start":"01:12.560 ","End":"01:19.960","Text":"In fact, just write the word increasing here and decreasing here."},{"Start":"01:19.960 ","End":"01:26.645","Text":"The extremum, I\u0027ll write max here and that\u0027s really all I need to know."},{"Start":"01:26.645 ","End":"01:29.180","Text":"These points I\u0027m not using here."},{"Start":"01:29.180 ","End":"01:33.590","Text":"I just used the same diagram as in the previous exercise,"},{"Start":"01:33.590 ","End":"01:35.030","Text":"but here I don\u0027t need them."},{"Start":"01:35.030 ","End":"01:38.990","Text":"Now we\u0027re going to make our equivalent of the table,"},{"Start":"01:38.990 ","End":"01:40.190","Text":"which is not a table,"},{"Start":"01:40.190 ","End":"01:44.270","Text":"just draw the number line and on this number line,"},{"Start":"01:44.270 ","End":"01:46.820","Text":"I\u0027m going to write the extremum."},{"Start":"01:46.820 ","End":"01:51.380","Text":"Let\u0027s mark this point here, 3 and really,"},{"Start":"01:51.380 ","End":"01:53.165","Text":"I have the point 3,"},{"Start":"01:53.165 ","End":"01:55.970","Text":"what\u0027s less than 3, and what\u0027s larger than 3."},{"Start":"01:55.970 ","End":"02:00.050","Text":"Now, I want to relate to both the derivative,"},{"Start":"02:00.050 ","End":"02:03.334","Text":"which is the red function f prime,"},{"Start":"02:03.334 ","End":"02:09.875","Text":"and to correlate that with stuff in the original function f. For the derivative,"},{"Start":"02:09.875 ","End":"02:15.260","Text":"what we said was that we have increasing here,"},{"Start":"02:15.260 ","End":"02:21.320","Text":"maximum here, and decreasing here."},{"Start":"02:21.320 ","End":"02:23.705","Text":"As for the original function,"},{"Start":"02:23.705 ","End":"02:27.800","Text":"this corresponds to inflection."},{"Start":"02:27.800 ","End":"02:32.390","Text":"The increasing is the concave"},{"Start":"02:32.390 ","End":"02:39.180","Text":"up and the decreasing is concave down."},{"Start":"02:39.180 ","End":"02:41.895","Text":"Other than summarizing it,"},{"Start":"02:41.895 ","End":"02:44.670","Text":"we\u0027re done, let\u0027s do the quick summary."},{"Start":"02:44.670 ","End":"02:52.550","Text":"What we have is the inflection at the point where x equals 3,"},{"Start":"02:52.550 ","End":"03:02.760","Text":"concave up when x is less than 3 and concave down where x is bigger than 3 and that\u0027s it."}],"ID":6246},{"Watched":false,"Name":"Exercise 3","Duration":"4m 6s","ChapterTopicVideoID":6234,"CourseChapterTopicPlaylistID":4030,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"In this exercise, we\u0027re given the graph"},{"Start":"00:03.000 ","End":"00:06.120","Text":"of the derivative of a function that\u0027s f prime of x."},{"Start":"00:06.120 ","End":"00:09.870","Text":"But we don\u0027t have the graph of the original function f of x."},{"Start":"00:09.870 ","End":"00:14.280","Text":"We\u0027re going to make some deductions about it based on the derivative."},{"Start":"00:14.280 ","End":"00:18.555","Text":"We basically have to find 3 things about the original function."},{"Start":"00:18.555 ","End":"00:21.520","Text":"We have to find its extrema."},{"Start":"00:21.520 ","End":"00:26.420","Text":"We have to find the intervals where it\u0027s increasing"},{"Start":"00:26.420 ","End":"00:29.630","Text":"and the intervals where it\u0027s decreasing."},{"Start":"00:29.630 ","End":"00:33.950","Text":"Now, we said that each property of the original function"},{"Start":"00:33.950 ","End":"00:36.920","Text":"has a corresponding property in a derived function."},{"Start":"00:36.920 ","End":"00:44.645","Text":"Extrema correspond to places where the derivative crosses the axis."},{"Start":"00:44.645 ","End":"00:50.305","Text":"So that would be here, here, and here."},{"Start":"00:50.305 ","End":"00:52.240","Text":"That\u0027s about the extrema."},{"Start":"00:52.240 ","End":"00:58.370","Text":"The increasing corresponds to where the function is positive"},{"Start":"00:58.370 ","End":"01:01.280","Text":"and decreasing as to where it\u0027s negative."},{"Start":"01:01.280 ","End":"01:07.295","Text":"So this is where it\u0027s positive and this is where it\u0027s positive,"},{"Start":"01:07.295 ","End":"01:13.445","Text":"negative in here, and negative is here."},{"Start":"01:13.445 ","End":"01:17.390","Text":"Now I want to summarize that not exactly in a table,"},{"Start":"01:17.390 ","End":"01:20.075","Text":"something that is equivalent to a table,"},{"Start":"01:20.075 ","End":"01:23.585","Text":"draw the number line and on this number line,"},{"Start":"01:23.585 ","End":"01:28.700","Text":"I\u0027m going to mark the important points which is where this thing crosses the axis."},{"Start":"01:28.700 ","End":"01:32.170","Text":"So we have the point below here on,"},{"Start":"01:32.170 ","End":"01:35.220","Text":"and the next one would be 5,"},{"Start":"01:35.220 ","End":"01:41.025","Text":"so say here\u0027s 5 and the next one is 7."},{"Start":"01:41.025 ","End":"01:44.510","Text":"As far as the function that we have,"},{"Start":"01:44.510 ","End":"01:46.310","Text":"which is f prime,"},{"Start":"01:46.310 ","End":"01:50.435","Text":"and we\u0027re going to write some things about f prime."},{"Start":"01:50.435 ","End":"01:54.395","Text":"That is, that here it is positive."},{"Start":"01:54.395 ","End":"02:02.855","Text":"Here it is 0 and then we have the area between here and here where the graph is negative."},{"Start":"02:02.855 ","End":"02:08.825","Text":"Then here it\u0027s 0 again and it\u0027s the kind of 0 which crosses the axis."},{"Start":"02:08.825 ","End":"02:15.215","Text":"This crossing the axis is where the original function has its extremum."},{"Start":"02:15.215 ","End":"02:19.550","Text":"Then here we have again that the function is positive"},{"Start":"02:19.550 ","End":"02:26.570","Text":"and then here again we have 0 of the kind that crosses from positive to negative here."},{"Start":"02:26.570 ","End":"02:28.354","Text":"For each of these,"},{"Start":"02:28.354 ","End":"02:32.810","Text":"we have the corresponding property for the original function, which we don\u0027t have."},{"Start":"02:32.810 ","End":"02:37.980","Text":"The plus here, means that this is going to be increasing,"},{"Start":"02:37.980 ","End":"02:43.640","Text":"this place is decreasing and between increasing and decreasing, it\u0027s an extremum."},{"Start":"02:43.640 ","End":"02:48.140","Text":"But I even know specifically that this will be of type maximum,"},{"Start":"02:48.140 ","End":"02:50.005","Text":"up and then down is maximum."},{"Start":"02:50.005 ","End":"02:53.470","Text":"Here we have down and then up and in the middle"},{"Start":"02:53.470 ","End":"02:58.180","Text":"we have a minimum of the original and then here again,"},{"Start":"02:58.180 ","End":"03:05.555","Text":"we have transition from increasing to decreasing so it\u0027s going to be a maximum."},{"Start":"03:05.555 ","End":"03:10.570","Text":"What we\u0027ll have to do is now summarize this in a table."},{"Start":"03:10.570 ","End":"03:13.469","Text":"We asked for extrema and I\u0027m going to be more specific"},{"Start":"03:13.469 ","End":"03:16.780","Text":"and separate the minimum from the maximum."},{"Start":"03:16.780 ","End":"03:24.640","Text":"We have maximum, 2 of those at x equals 1 and x equals 7."},{"Start":"03:24.640 ","End":"03:30.640","Text":"Let\u0027s see, minimum only got 1 of those at x equals 5."},{"Start":"03:30.640 ","End":"03:34.985","Text":"Then we talking about increasing and decreasing,"},{"Start":"03:34.985 ","End":"03:38.780","Text":"which for increasing, that\u0027s where the arrows go up."},{"Start":"03:38.780 ","End":"03:44.735","Text":"This would correspond to x less than 1 or x between 5 and 7,"},{"Start":"03:44.735 ","End":"03:48.750","Text":"so 5 less than x less than 7."},{"Start":"03:48.750 ","End":"03:53.550","Text":"Finally, decreasing, look for the down arrows"},{"Start":"03:53.550 ","End":"03:59.490","Text":"so that would be between 0 and 5 or from 7 onwards,"},{"Start":"03:59.490 ","End":"04:02.685","Text":"which means x bigger than 7."},{"Start":"04:02.685 ","End":"04:07.720","Text":"That\u0027s all there is to it and we\u0027re done."}],"ID":6247},{"Watched":false,"Name":"Exercise 4","Duration":"3m 4s","ChapterTopicVideoID":6235,"CourseChapterTopicPlaylistID":4030,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.230","Text":"In this exercise, we\u0027re given the graph of the derivative of a"},{"Start":"00:04.230 ","End":"00:08.730","Text":"function f. So f prime is not the function, it\u0027s derivative."},{"Start":"00:08.730 ","End":"00:10.500","Text":"But using this graph,"},{"Start":"00:10.500 ","End":"00:14.505","Text":"we have to find out various things about the original function."},{"Start":"00:14.505 ","End":"00:17.009","Text":"We need the inflection points."},{"Start":"00:17.009 ","End":"00:20.460","Text":"We need the places where the intervals where the function is"},{"Start":"00:20.460 ","End":"00:24.329","Text":"concave up and where it\u0027s concave down."},{"Start":"00:24.329 ","End":"00:27.420","Text":"Now, each of these properties on the function f has"},{"Start":"00:27.420 ","End":"00:30.509","Text":"a corresponding property on the function f prime."},{"Start":"00:30.509 ","End":"00:33.554","Text":"The inflection points correspond to the extrema,"},{"Start":"00:33.554 ","End":"00:35.130","Text":"and extrema, we have 2 of them."},{"Start":"00:35.130 ","End":"00:37.484","Text":"We have 1 here which is a minimum,"},{"Start":"00:37.484 ","End":"00:40.665","Text":"and 1 here which is a maximum,"},{"Start":"00:40.665 ","End":"00:44.720","Text":"and the property of being concave up has"},{"Start":"00:44.720 ","End":"00:49.910","Text":"the corresponding property of the derivative be increasing."},{"Start":"00:49.910 ","End":"00:54.800","Text":"Let\u0027s say use a color green for the increasing here, it\u0027s increasing,"},{"Start":"00:54.800 ","End":"01:00.155","Text":"and the concave down corresponds to decreasing."},{"Start":"01:00.155 ","End":"01:03.610","Text":"Let\u0027s say I\u0027ll use this color for decreasing."},{"Start":"01:03.610 ","End":"01:08.690","Text":"Decreasing, minimum, increasing, maximum, decreasing."},{"Start":"01:08.690 ","End":"01:13.880","Text":"What I have to do is just write this information in a table,"},{"Start":"01:13.880 ","End":"01:15.440","Text":"not exactly a table."},{"Start":"01:15.440 ","End":"01:18.050","Text":"I\u0027ll just use the number line,"},{"Start":"01:18.050 ","End":"01:21.335","Text":"and I\u0027m marking the important points which are here,"},{"Start":"01:21.335 ","End":"01:23.210","Text":"the extrema of the derivative,"},{"Start":"01:23.210 ","End":"01:26.960","Text":"which is 3 and 6."},{"Start":"01:26.960 ","End":"01:32.260","Text":"I\u0027m going to write the properties of the function f prime,"},{"Start":"01:32.260 ","End":"01:36.050","Text":"and then I\u0027m going to write after that the corresponding properties"},{"Start":"01:36.050 ","End":"01:39.920","Text":"of the function f. For f prime,"},{"Start":"01:39.920 ","End":"01:41.975","Text":"what I have is that up to 3,"},{"Start":"01:41.975 ","End":"01:46.095","Text":"it is decreasing, so I\u0027ll indicate with an arrow."},{"Start":"01:46.095 ","End":"01:48.290","Text":"At 3, and so if I have an extremum,"},{"Start":"01:48.290 ","End":"01:51.835","Text":"but I\u0027ll write that specifically that extremum is a minimum."},{"Start":"01:51.835 ","End":"01:56.220","Text":"Then between 3 and 6, we are increasing."},{"Start":"01:56.220 ","End":"01:58.445","Text":"At 6, we have another extremum,"},{"Start":"01:58.445 ","End":"02:00.890","Text":"which I\u0027ll write as max,"},{"Start":"02:00.890 ","End":"02:03.200","Text":"and from 6 onwards again,"},{"Start":"02:03.200 ","End":"02:06.400","Text":"we are decreasing, so a downward arrow."},{"Start":"02:06.400 ","End":"02:13.550","Text":"The corresponding property or feature for f is decreasing is concave down,"},{"Start":"02:13.550 ","End":"02:15.950","Text":"which I\u0027ll just symbolically write this way."},{"Start":"02:15.950 ","End":"02:22.505","Text":"Increasing is concave up and decreasing is again concave down,"},{"Start":"02:22.505 ","End":"02:27.470","Text":"and the extrema, minimum and maximum are inflection points."},{"Start":"02:27.470 ","End":"02:29.600","Text":"That basically answers the question,"},{"Start":"02:29.600 ","End":"02:36.375","Text":"but I just like to write it in a nicer way that the question asked for inflection points."},{"Start":"02:36.375 ","End":"02:37.590","Text":"I\u0027ll write it in full,"},{"Start":"02:37.590 ","End":"02:44.620","Text":"inflection I have at x equals 3 and x equals 6,"},{"Start":"02:44.620 ","End":"02:50.365","Text":"and then the concave up is between 3 and 6,"},{"Start":"02:50.365 ","End":"02:54.120","Text":"and the concave down is 2 places."},{"Start":"02:54.120 ","End":"02:57.210","Text":"It\u0027s below 3 and above 6."},{"Start":"02:57.210 ","End":"03:02.805","Text":"So x less than 3 or x bigger than 6,"},{"Start":"03:02.805 ","End":"03:04.960","Text":"and we are done."}],"ID":6248},{"Watched":false,"Name":"Exercise 5","Duration":"3m 16s","ChapterTopicVideoID":6236,"CourseChapterTopicPlaylistID":4030,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.310","Text":"In this exercise, we\u0027re given the graph of the derivative of a function f,"},{"Start":"00:05.310 ","End":"00:07.410","Text":"and we have to find, essentially,"},{"Start":"00:07.410 ","End":"00:09.930","Text":"3 things about the original function f."},{"Start":"00:09.930 ","End":"00:13.080","Text":"We need to find the local extrema,"},{"Start":"00:13.080 ","End":"00:15.749","Text":"which are the minimum and maximum points,"},{"Start":"00:15.749 ","End":"00:18.330","Text":"and then we want the intervals where the original function"},{"Start":"00:18.330 ","End":"00:21.525","Text":"is increasing and where it\u0027s decreasing."},{"Start":"00:21.525 ","End":"00:25.455","Text":"We can tell all this by looking at the graph of the derivative."},{"Start":"00:25.455 ","End":"00:28.290","Text":"Now, it turns out that each of these criteria"},{"Start":"00:28.290 ","End":"00:31.545","Text":"has a corresponding thing for the derivative."},{"Start":"00:31.545 ","End":"00:36.254","Text":"Local extrema are where the graph of the derivative crosses the axis,"},{"Start":"00:36.254 ","End":"00:38.610","Text":"and I have 2 such places I see,"},{"Start":"00:38.610 ","End":"00:43.485","Text":"here, it crosses the axis and here, it crosses the axis."},{"Start":"00:43.485 ","End":"00:45.905","Text":"The next thing is the increasing."},{"Start":"00:45.905 ","End":"00:49.250","Text":"Increasing is where the function is positive,"},{"Start":"00:49.250 ","End":"00:50.570","Text":"meaning above the axis."},{"Start":"00:50.570 ","End":"00:56.450","Text":"Let\u0027s say, the derivative is positive here and it\u0027s positive over here."},{"Start":"00:56.450 ","End":"01:04.960","Text":"For decreasing, what we need is where it is negative or below the axis, there it is."},{"Start":"01:04.960 ","End":"01:09.515","Text":"I\u0027m going to put all this information into the equivalent of a table,"},{"Start":"01:09.515 ","End":"01:10.760","Text":"not exactly a table,"},{"Start":"01:10.760 ","End":"01:12.440","Text":"this is a number line."},{"Start":"01:12.440 ","End":"01:19.535","Text":"The important point or where it crosses the axis are the point 0 and the point 4."},{"Start":"01:19.535 ","End":"01:24.665","Text":"What I want to do is write some properties of derivative."},{"Start":"01:24.665 ","End":"01:29.300","Text":"Then from here, I\u0027ll conclude some stuff about the original function."},{"Start":"01:29.300 ","End":"01:33.020","Text":"Here, the function f prime is positive."},{"Start":"01:33.020 ","End":"01:37.100","Text":"Here, it crosses the axis and it\u0027s 0."},{"Start":"01:37.100 ","End":"01:40.955","Text":"Here, all the way up to here it is negative."},{"Start":"01:40.955 ","End":"01:44.275","Text":"At 4 itself it\u0027s also 0,"},{"Start":"01:44.275 ","End":"01:47.180","Text":"and beyond 4 it\u0027s positive."},{"Start":"01:47.180 ","End":"01:51.710","Text":"What I\u0027ve written here in green is just to reflect the state of f prime,"},{"Start":"01:51.710 ","End":"01:55.130","Text":"positive 0, negative 0, positive."},{"Start":"01:55.130 ","End":"02:00.170","Text":"Correspondingly, each of these as an interpretation for the original function,"},{"Start":"02:00.170 ","End":"02:03.185","Text":"this plus means as an arrow here,"},{"Start":"02:03.185 ","End":"02:07.075","Text":"when the derivative is positive the function is increasing,"},{"Start":"02:07.075 ","End":"02:10.140","Text":"when it\u0027s negative it\u0027s decreasing."},{"Start":"02:10.140 ","End":"02:13.500","Text":"When it\u0027s positive the original function is increasing."},{"Start":"02:13.500 ","End":"02:16.700","Text":"Between them, this value 0 means it\u0027s"},{"Start":"02:16.700 ","End":"02:20.360","Text":"an extra number because it\u0027s between increasing and decreasing."},{"Start":"02:20.360 ","End":"02:23.615","Text":"I know specifically that it\u0027s of type maximum."},{"Start":"02:23.615 ","End":"02:25.910","Text":"Similarly here, the point 4,"},{"Start":"02:25.910 ","End":"02:29.135","Text":"the function goes from negative to positive."},{"Start":"02:29.135 ","End":"02:33.020","Text":"The original function is from decreasing to increasing,"},{"Start":"02:33.020 ","End":"02:35.945","Text":"so this here is a minimum."},{"Start":"02:35.945 ","End":"02:38.930","Text":"Let me just write the answer to the question properly."},{"Start":"02:38.930 ","End":"02:43.009","Text":"They asked for extrema and I can be more specific."},{"Start":"02:43.009 ","End":"02:49.055","Text":"I can say that I have a minimum at the point x equals 4."},{"Start":"02:49.055 ","End":"02:54.350","Text":"I can say that I have a maximum at the point x equals 0."},{"Start":"02:54.350 ","End":"03:00.615","Text":"I can say that I have the interval of increasing at 2 places,"},{"Start":"03:00.615 ","End":"03:05.760","Text":"x less than 0 or x bigger than 4."},{"Start":"03:05.760 ","End":"03:09.705","Text":"The decreasing part is between 0 and 4,"},{"Start":"03:09.705 ","End":"03:12.540","Text":"so 0 less than x, less than 4."},{"Start":"03:12.540 ","End":"03:17.400","Text":"This essentially summarizes the question and we are done."}],"ID":6249},{"Watched":false,"Name":"Exercise 6","Duration":"3m 43s","ChapterTopicVideoID":6237,"CourseChapterTopicPlaylistID":4030,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.063","Text":"In this exercise, we\u0027re given the"},{"Start":"00:02.063 ","End":"00:04.800","Text":"graph of the derivative of a function."},{"Start":"00:04.800 ","End":"00:06.285","Text":"There was a function f of x."},{"Start":"00:06.285 ","End":"00:08.790","Text":"It has a derivative f prime, and it\u0027s a function"},{"Start":"00:08.790 ","End":"00:11.535","Text":"and this is the red one that\u0027s been sketched here."},{"Start":"00:11.535 ","End":"00:14.190","Text":"Now, from this function f prime,"},{"Start":"00:14.190 ","End":"00:16.590","Text":"we actually can find out a lot of things about"},{"Start":"00:16.590 ","End":"00:19.674","Text":"the original function f of x before it"},{"Start":"00:19.674 ","End":"00:22.095","Text":"was differentiated to get f prime."},{"Start":"00:22.095 ","End":"00:25.320","Text":"We have to find out 3 things, and we can,"},{"Start":"00:25.320 ","End":"00:26.921","Text":"and these things are the"},{"Start":"00:26.921 ","End":"00:30.575","Text":"inflection points of the function f and"},{"Start":"00:30.575 ","End":"00:33.033","Text":"the intervals where the function f is"},{"Start":"00:33.033 ","End":"00:37.255","Text":"concave up and where it\u0027s concave down."},{"Start":"00:37.255 ","End":"00:41.280","Text":"Let\u0027s start with the inflection points."},{"Start":"00:41.280 ","End":"00:44.585","Text":"The thing is this about inflection points that"},{"Start":"00:44.585 ","End":"00:48.155","Text":"when the original function has an inflection point,"},{"Start":"00:48.155 ","End":"00:51.206","Text":"then the derivative function has an"},{"Start":"00:51.206 ","End":"00:54.305","Text":"extremum that\u0027s a maximum or minimum."},{"Start":"00:54.305 ","End":"00:55.568","Text":"Let\u0027s look here and see if we"},{"Start":"00:55.568 ","End":"00:57.335","Text":"can see a maximum or minimum."},{"Start":"00:57.335 ","End":"01:01.400","Text":"Oh, yeah, I can only see one, and that is this."},{"Start":"01:01.400 ","End":"01:02.780","Text":"This is a minimum, and that\u0027s"},{"Start":"01:02.780 ","End":"01:04.150","Text":"the only one that we have."},{"Start":"01:04.150 ","End":"01:05.910","Text":"There are no other special points."},{"Start":"01:05.910 ","End":"01:07.625","Text":"Now that we have that"},{"Start":"01:07.625 ","End":"01:09.800","Text":"let\u0027s go on to the second thing, concave up."},{"Start":"01:09.800 ","End":"01:10.703","Text":"Now, in the case of the"},{"Start":"01:10.703 ","End":"01:12.230","Text":"original function, concave up,"},{"Start":"01:12.230 ","End":"01:15.344","Text":"what it translates to is increasing as"},{"Start":"01:15.344 ","End":"01:17.370","Text":"far as the derivative function goes."},{"Start":"01:17.370 ","End":"01:20.210","Text":"We have to look for areas where it\u0027s increasing."},{"Start":"01:20.210 ","End":"01:23.006","Text":"I can see it\u0027s from the minimum and"},{"Start":"01:23.006 ","End":"01:25.340","Text":"to the right, so I\u0027ll just mark that,"},{"Start":"01:25.340 ","End":"01:28.375","Text":"that this is the increasing path; I\u0027ll use green."},{"Start":"01:28.375 ","End":"01:31.549","Text":"You guessed it, the analog for concave down,"},{"Start":"01:31.549 ","End":"01:32.930","Text":"for the original function is"},{"Start":"01:32.930 ","End":"01:35.989","Text":"decreasing for the derived function."},{"Start":"01:35.989 ","End":"01:37.925","Text":"Let\u0027s just choose a color,"},{"Start":"01:37.925 ","End":"01:39.290","Text":"oh, I\u0027ll go for this,"},{"Start":"01:39.290 ","End":"01:42.190","Text":"and that will be for the decreasing part."},{"Start":"01:42.190 ","End":"01:45.750","Text":"Decreasing, minimum, increasing."},{"Start":"01:45.750 ","End":"01:49.670","Text":"Now we\u0027re going to use our technique of this axis."},{"Start":"01:49.670 ","End":"01:51.263","Text":"It\u0027s going to get all the information"},{"Start":"01:51.263 ","End":"01:54.170","Text":"that we need on it that will help us."},{"Start":"01:54.170 ","End":"01:56.260","Text":"This time the only point that"},{"Start":"01:56.260 ","End":"01:57.780","Text":"we can draw is the point 3,"},{"Start":"01:57.780 ","End":"02:01.165","Text":"and I\u0027ll write the number 3 over here."},{"Start":"02:01.165 ","End":"02:04.895","Text":"We can say some things about the red function,"},{"Start":"02:04.895 ","End":"02:08.465","Text":"f prime of x. I want to write a few things about this,"},{"Start":"02:08.465 ","End":"02:10.240","Text":"and then I\u0027ll be able to conclude"},{"Start":"02:10.240 ","End":"02:13.550","Text":"some things about the original function f of x,"},{"Start":"02:13.550 ","End":"02:15.110","Text":"which I\u0027ll write in blue."},{"Start":"02:15.110 ","End":"02:16.352","Text":"Meanwhile, let\u0027s see what we"},{"Start":"02:16.352 ","End":"02:18.190","Text":"can say to summarize what we said."},{"Start":"02:18.190 ","End":"02:21.845","Text":"Up to the point 3 or x equals 3,"},{"Start":"02:21.845 ","End":"02:24.907","Text":"the red function was decreasing,"},{"Start":"02:24.907 ","End":"02:28.175","Text":"so I\u0027ll write that as an arrow down."},{"Start":"02:28.175 ","End":"02:30.515","Text":"At point 3 itself,"},{"Start":"02:30.515 ","End":"02:34.225","Text":"it was an extremum, specifically a minimum."},{"Start":"02:34.225 ","End":"02:36.432","Text":"From 3 onwards, the function was"},{"Start":"02:36.432 ","End":"02:39.555","Text":"increasing, so it\u0027s like an upward arrow."},{"Start":"02:39.555 ","End":"02:42.240","Text":"Translating these things from derivative"},{"Start":"02:42.240 ","End":"02:45.320","Text":"to the original function gives us the following."},{"Start":"02:45.320 ","End":"02:47.991","Text":"The decreasing for the derivative"},{"Start":"02:47.991 ","End":"02:51.319","Text":"corresponds to concave down,"},{"Start":"02:51.319 ","End":"02:53.360","Text":"which I\u0027ll just draw like this."},{"Start":"02:53.360 ","End":"02:57.155","Text":"The increasing for the derivative interprets"},{"Start":"02:57.155 ","End":"03:00.100","Text":"as concave up for the original function,"},{"Start":"03:00.100 ","End":"03:03.560","Text":"and the extremum for the derivative"},{"Start":"03:03.560 ","End":"03:07.490","Text":"translates as an inflection point."},{"Start":"03:07.490 ","End":"03:10.160","Text":"Basically, we\u0027ve answered the questions; it\u0027s just"},{"Start":"03:10.160 ","End":"03:12.910","Text":"that I want to write it a little bit more formally."},{"Start":"03:12.910 ","End":"03:15.215","Text":"To summarize, they asked about,"},{"Start":"03:15.215 ","End":"03:17.140","Text":"first of all, about inflections."},{"Start":"03:17.140 ","End":"03:20.713","Text":"I have an inflection point at exactly"},{"Start":"03:20.713 ","End":"03:22.955","Text":"x equals 3 and nowhere else."},{"Start":"03:22.955 ","End":"03:28.010","Text":"The concave up is this, and that is from 3 onwards,"},{"Start":"03:28.010 ","End":"03:30.685","Text":"that means x bigger than 3."},{"Start":"03:30.685 ","End":"03:35.375","Text":"The concave down is this thing"},{"Start":"03:35.375 ","End":"03:37.040","Text":"which is to the left of 3,"},{"Start":"03:37.040 ","End":"03:39.365","Text":"so x is less than 3."},{"Start":"03:39.365 ","End":"03:41.330","Text":"This summarizes the questions we"},{"Start":"03:41.330 ","End":"03:44.160","Text":"were asked, and we are done."}],"ID":6250},{"Watched":false,"Name":"Exercise 7","Duration":"3m 41s","ChapterTopicVideoID":6238,"CourseChapterTopicPlaylistID":4030,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.190","Text":"In this exercise, once again,"},{"Start":"00:02.190 ","End":"00:06.255","Text":"we\u0027re given the graph of the derivative function of another function."},{"Start":"00:06.255 ","End":"00:09.160","Text":"From certain properties of the derivative function,"},{"Start":"00:09.160 ","End":"00:14.340","Text":"we\u0027ll be able to choose all kinds of properties or features of the original function."},{"Start":"00:14.340 ","End":"00:17.790","Text":"In this case, 3 categories we are going to,"},{"Start":"00:17.790 ","End":"00:18.930","Text":"for the original function,"},{"Start":"00:18.930 ","End":"00:21.630","Text":"find its local extrema."},{"Start":"00:21.630 ","End":"00:27.475","Text":"We\u0027re going to determine the intervals where it is increasing and where it is decreasing."},{"Start":"00:27.475 ","End":"00:31.160","Text":"Normally, the local extrema of a function are"},{"Start":"00:31.160 ","End":"00:35.240","Text":"found by looking for places where the derivative is 0."},{"Start":"00:35.240 ","End":"00:40.010","Text":"Now, there certainly is a place where the derivative is 0 and that is this point."},{"Start":"00:40.010 ","End":"00:42.110","Text":"But it turns out, and I\u0027ve said it before,"},{"Start":"00:42.110 ","End":"00:45.890","Text":"that just being 0 does not guarantee the extrema."},{"Start":"00:45.890 ","End":"00:47.150","Text":"In fact, in this case,"},{"Start":"00:47.150 ","End":"00:54.485","Text":"there is no extremum of the original function because touching the x-axis is not enough,"},{"Start":"00:54.485 ","End":"00:57.650","Text":"the function would have to actually cross the x-axis."},{"Start":"00:57.650 ","End":"01:01.955","Text":"We\u0027ll see this more when we get into the table,"},{"Start":"01:01.955 ","End":"01:05.360","Text":"and I discuss increasing and decreasing intervals."},{"Start":"01:05.360 ","End":"01:09.230","Text":"You\u0027ll see this clearly that something doesn\u0027t happen here that you would expect."},{"Start":"01:09.230 ","End":"01:14.360","Text":"Now, normally, function is increasing when derivative function is positive."},{"Start":"01:14.360 ","End":"01:17.690","Text":"Now, it seems to be positive pretty much everywhere with"},{"Start":"01:17.690 ","End":"01:21.995","Text":"a slight exception that here it\u0027s not positive, it\u0027s 0."},{"Start":"01:21.995 ","End":"01:24.260","Text":"Let\u0027s see what this means."},{"Start":"01:24.260 ","End":"01:26.365","Text":"We\u0027re going to draw this on an axis."},{"Start":"01:26.365 ","End":"01:32.910","Text":"What we\u0027re going to do is take this axis and we\u0027re going to put any unusual points."},{"Start":"01:32.910 ","End":"01:38.200","Text":"Normally, I\u0027d be putting the points where the derivative crosses the x-axis,"},{"Start":"01:38.200 ","End":"01:40.670","Text":"but really we should also include where it just touches"},{"Start":"01:40.670 ","End":"01:43.480","Text":"the x-axis and we\u0027ll sort it all out later."},{"Start":"01:43.480 ","End":"01:46.540","Text":"Here we have the value 4."},{"Start":"01:46.540 ","End":"01:49.250","Text":"Let me just label that here above"},{"Start":"01:49.250 ","End":"01:52.549","Text":"the line I\u0027m going to be talking about properties of f prime,"},{"Start":"01:52.549 ","End":"01:54.350","Text":"and below the line,"},{"Start":"01:54.350 ","End":"01:58.850","Text":"I\u0027m going to be talking about properties of just f of x."},{"Start":"01:58.850 ","End":"02:04.320","Text":"At this point, what I can say is that something happens around 4,"},{"Start":"02:04.320 ","End":"02:06.855","Text":"it\u0027s positive after 4,"},{"Start":"02:06.855 ","End":"02:10.115","Text":"it\u0027s positive before we get to 4."},{"Start":"02:10.115 ","End":"02:11.730","Text":"At that 1 single point,"},{"Start":"02:11.730 ","End":"02:14.345","Text":"4, it happens to be 0."},{"Start":"02:14.345 ","End":"02:16.355","Text":"That\u0027s as far as f prime goes."},{"Start":"02:16.355 ","End":"02:19.100","Text":"Now, as far as f of x goes,"},{"Start":"02:19.100 ","End":"02:22.325","Text":"what we can say is that where the derivative is positive,"},{"Start":"02:22.325 ","End":"02:25.300","Text":"this thing is increasing."},{"Start":"02:25.300 ","End":"02:28.280","Text":"At this particular point,"},{"Start":"02:28.280 ","End":"02:33.740","Text":"it happens to be 0, but 0 not of a kind of crossing the axis, but plus the plus."},{"Start":"02:33.740 ","End":"02:36.800","Text":"It\u0027s increasing before and increasing after."},{"Start":"02:36.800 ","End":"02:40.070","Text":"What happens is that in these cases,"},{"Start":"02:40.070 ","End":"02:42.975","Text":"when there\u0027s an odd point that\u0027s temporarily,"},{"Start":"02:42.975 ","End":"02:45.240","Text":"the derivative is not positive,"},{"Start":"02:45.240 ","End":"02:48.950","Text":"that 1 point doesn\u0027t really affect the general flow of increasing."},{"Start":"02:48.950 ","End":"02:53.160","Text":"I\u0027ll just give you a small example that you\u0027ve already encountered,"},{"Start":"02:53.160 ","End":"02:55.399","Text":"and this is just to illustrate this concept"},{"Start":"02:55.399 ","End":"02:58.925","Text":"that positive derivative is a sign of increase,"},{"Start":"02:58.925 ","End":"03:04.520","Text":"but we can allow an exception of 0 in a range of positive derivatives,"},{"Start":"03:04.520 ","End":"03:06.755","Text":"and that\u0027s how it works."},{"Start":"03:06.755 ","End":"03:13.400","Text":"My answer would be simply that there are no extrema."},{"Start":"03:13.400 ","End":"03:17.050","Text":"We also have no decreasing intervals,"},{"Start":"03:17.050 ","End":"03:20.075","Text":"and increasing is everywhere."},{"Start":"03:20.075 ","End":"03:22.970","Text":"You can either say it as all x,"},{"Start":"03:22.970 ","End":"03:26.375","Text":"you could say it as minus infinity,"},{"Start":"03:26.375 ","End":"03:29.220","Text":"less than x, less than infinity."},{"Start":"03:29.220 ","End":"03:32.180","Text":"If you know the interval notation,"},{"Start":"03:32.180 ","End":"03:39.305","Text":"you could say that x belongs to the interval that goes from minus infinity to infinity,"},{"Start":"03:39.305 ","End":"03:41.940","Text":"and we are done."}],"ID":6251},{"Watched":false,"Name":"Exercise 8","Duration":"3m 7s","ChapterTopicVideoID":6239,"CourseChapterTopicPlaylistID":4030,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.385","Text":"In this exercise, we\u0027re given the graph of a function f prime of x,"},{"Start":"00:05.385 ","End":"00:09.585","Text":"which implies that there is an original function f of x."},{"Start":"00:09.585 ","End":"00:14.040","Text":"There are 3 things we have to find out about f of x"},{"Start":"00:14.040 ","End":"00:17.370","Text":"by using this graph information on f prime."},{"Start":"00:17.370 ","End":"00:19.230","Text":"Those 3 things are as follows,"},{"Start":"00:19.230 ","End":"00:23.790","Text":"we want to find the inflection points of the function f."},{"Start":"00:23.790 ","End":"00:29.130","Text":"We want to know where it\u0027s concave up and we want to know where it\u0027s concave down,"},{"Start":"00:29.130 ","End":"00:34.545","Text":"and of course we don\u0027t have the graph of the original f only the graph of f prime."},{"Start":"00:34.545 ","End":"00:36.593","Text":"What I\u0027d like to say is that each of these"},{"Start":"00:36.593 ","End":"00:41.440","Text":"has its correlation in terms of the graph of the derivative."},{"Start":"00:41.440 ","End":"00:45.835","Text":"Inflection is where the derivative has an extremum,"},{"Start":"00:45.835 ","End":"00:51.140","Text":"concave up is where the derivative is increasing,"},{"Start":"00:51.140 ","End":"00:54.290","Text":"and concave down is where the derivative is decreasing."},{"Start":"00:54.290 ","End":"00:56.675","Text":"Let\u0027s see if we can determine these things."},{"Start":"00:56.675 ","End":"00:59.375","Text":"Let\u0027s start off with the extrema."},{"Start":"00:59.375 ","End":"01:02.150","Text":"There\u0027s only 1 extremum of the red function,"},{"Start":"01:02.150 ","End":"01:05.410","Text":"and that\u0027s here where x is equal to 4,"},{"Start":"01:05.410 ","End":"01:11.915","Text":"and then the concave up corresponds to where the derivative is increasing."},{"Start":"01:11.915 ","End":"01:13.805","Text":"Let\u0027s mark that part."},{"Start":"01:13.805 ","End":"01:19.975","Text":"The derivative is increasing here and here we are decreasing."},{"Start":"01:19.975 ","End":"01:21.080","Text":"That\u0027s basically it."},{"Start":"01:21.080 ","End":"01:24.660","Text":"Decreasing, extremum, and increasing."},{"Start":"01:24.660 ","End":"01:28.490","Text":"Now I\u0027m going to write all that in or alternative to a table."},{"Start":"01:28.490 ","End":"01:34.060","Text":"What we do is we take the x-axis and we write on it first of all, the important points,"},{"Start":"01:34.060 ","End":"01:35.720","Text":"and as I said, in this case,"},{"Start":"01:35.720 ","End":"01:38.540","Text":"just the extremum of the derivative,"},{"Start":"01:38.540 ","End":"01:40.955","Text":"which is where x is equal to 4."},{"Start":"01:40.955 ","End":"01:44.270","Text":"Then we also look at the intervals before and after."},{"Start":"01:44.270 ","End":"01:47.240","Text":"Above the line I\u0027m going to use f prime"},{"Start":"01:47.240 ","End":"01:50.870","Text":"and below the line I\u0027m going to use f of x"},{"Start":"01:50.870 ","End":"01:53.030","Text":"and we\u0027re going to write some things about f prime"},{"Start":"01:53.030 ","End":"01:54.530","Text":"and then infer about f."},{"Start":"01:54.530 ","End":"01:58.370","Text":"Like we said, up to here, it\u0027s decreasing."},{"Start":"01:58.370 ","End":"02:01.820","Text":"I\u0027ll just, instead writing the word decreasing, we write a down arrow."},{"Start":"02:01.820 ","End":"02:04.055","Text":"From 4 onwards it\u0027s increasing,"},{"Start":"02:04.055 ","End":"02:05.975","Text":"so I write an up arrow."},{"Start":"02:05.975 ","End":"02:08.300","Text":"At 4 itself, It\u0027s an extremum,"},{"Start":"02:08.300 ","End":"02:10.490","Text":"in this case, specifically a minimum,"},{"Start":"02:10.490 ","End":"02:12.890","Text":"and I will write that as a minimum."},{"Start":"02:12.890 ","End":"02:15.460","Text":"But really all I care about is that it\u0027s an extremum."},{"Start":"02:15.460 ","End":"02:21.920","Text":"So for f of x, what it implies is that derivative decreasing means concave down,"},{"Start":"02:21.920 ","End":"02:23.470","Text":"so something like this."},{"Start":"02:23.470 ","End":"02:26.825","Text":"For increasing intervals of f prime,"},{"Start":"02:26.825 ","End":"02:30.305","Text":"we have intervals of concave up,"},{"Start":"02:30.305 ","End":"02:33.470","Text":"and between these 2 we have inflection."},{"Start":"02:33.470 ","End":"02:35.780","Text":"This essentially answers the question,"},{"Start":"02:35.780 ","End":"02:37.820","Text":"but I would like to just put all our results"},{"Start":"02:37.820 ","End":"02:41.480","Text":"in a more organized fashion and answer the question fully."},{"Start":"02:41.480 ","End":"02:45.034","Text":"We were asked about inflection points,"},{"Start":"02:45.034 ","End":"02:47.210","Text":"and that is only one such place,"},{"Start":"02:47.210 ","End":"02:49.505","Text":"and that is where x equals 4."},{"Start":"02:49.505 ","End":"02:53.035","Text":"Then we were asked about concave up."},{"Start":"02:53.035 ","End":"02:57.900","Text":"Concave up we saw when x is bigger than 4."},{"Start":"02:57.900 ","End":"03:01.425","Text":"Concave down, concave down,"},{"Start":"03:01.425 ","End":"03:03.045","Text":"what we have is this,"},{"Start":"03:03.045 ","End":"03:05.970","Text":"which is x is less than 4."},{"Start":"03:05.970 ","End":"03:08.080","Text":"We are done."}],"ID":6252}],"Thumbnail":null,"ID":4030}]

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1.1

1