Lecture 08: Some more review

February 25, 2015

Start | so lets say we give you a graph and tell you to do the following things |

1:24 | whole bunch of stuff The limit of x approaches 4 from the minus side of f of x is 2 |

1:32 | the limit as x approaches 4 from the plus side is 3 and f of 4 is 3 the limit when x approaches 5 from the minus side is positive infinity the limit as x approaches 5 from the plus side is minus infinity f of 0 is 0 f of negative 1 is 1 the limit as x approaches -1 is -1 the limit as x approaches infinity is 0 and the limit a x approaches -infinity is -infinity. so draw me a picture |

2:01 | there are many many right answers but they all have to be from the same basic ffamily alright is that long enough i think thats long enough so lets do this graph so whats a good strategy well first of all we give you some actual points we tell you that when f of 4 is 3 so when x is 4 y=3 |

2:38 | we know the graph goes to 4,3 we also know its goes 0,0 also known as the origin and it goes through negative 1,1 so that takes care, ill cross them off so i know ive taken care of something and now i start to look at the limits |

3:02 | so lets take them in order so when x approaches 4 from the minus side f of x is equal to 2 so from the minus side i have to get to 2 somehow i dont know if i go up or go down it doesnt matter but at 2 i have a circle cause its not equal to 2 there its equal to 3 there but it will be very close to 2 when x gets close to 4 now on the other side |

3:31 | so cross that one off on the other side when x approaches 4 from the positive side i go to 3, oh thats where the dot is im doing something like that so far so good and you go up and down it doesnt matter at the moment now other stuff when i approach 5 from he minus side i get infinity that means im an asymptote i go up to positive infinity something like that |

4:02 | you can squiggle on the way up it doesnt really matter no reason to get fancy just know the curve goes up like this because as we approach 5 from the left side you go up positive infinity when we approach 5 from the minus side from the positive side youre minus infinity now were down here so there we go we took care of the whole left side now that were there since were going to 5 now beyond 5 as x goes to infinity you go to 0 |

4:36 | so something like that you can go above and come down you can jiggle and come down but like i said theres no reason to show off no reason to screw it up just just like that is fine now youve taken care of almost everything |

5:00 | and notice f of -1 is positive 1 and as we approach negative 1 the limit is -1 so we have to have a circle then and we go through the origin something like that doesnt have to be perfect but some variation on that. again we know f at 0=0 and negative 1 it equals to positive 1 but when we get very close to negative 1 from either sode |

5:30 | we have to get to negative 1 cause theres a whole there because thats just the limit thats not what exactly happens at negative 1 and the last thing is when x approaches negative infinity y goes to negative infinity down like that howd we do im gonna do another one of these do we loce this one yes love yes question |

6:03 | how do we know it goes from here rather than here because
when x gets very close to negative 1
youre at negative 1
youre only at x is positive 1 when x is exactly negative 1
then you have to get aback here again
thats how we know it comes from the whole and not the point
okay that makes sense howd you do?
love this lets make up another 1 |

6:30 | ill write the rules over here so that can stay there for a while |

8:12 | alright that was fun you know f of 0 is 3, f of 1 is 4 the limit as x approaches 1 from the minus side is -5 the limit as you approach 1 from the plus side is 2 the limit as x goes to infinity is infinity the limit as x approaches negative 1 from the plus side is infinity |

8:31 | when it approaches -1 from the minus side it is infinity when it approaches -3 you get -5 f of -3 is -6 and the limit as x approaches -infinity is 0 alright first thing you do is put on all the points |

9:00 | so f of 0 is 3 f of 1 is 4 and f of -3 is -6 we know we go through there alright so we can cross these off lets deal with what happens when we approach 1 |

9:30 | well when you approach 1 from the plus side were at 2 there and when we approach 1 from the minus side we are at 5 something like that so dont know much else yet so that takes care of these 2 anything going on to the right of 1 no it just goes to infinity |

10:01 | something like that you can also go straight dont have the curve you could have a concave down curve dosent matter how do i come to which conclusion well the limit as x approaches infinity of f of x is infinity as you get and x distance that will be big y distance will be big f of x is y because this is the graph of f of x |

10:34 | now something is going on at negative 1 well at netaive 1 we have to go from the plus to infinity so we have an asymptote i dont know we can do something like that you can make it turn in there and when x approaches -1 from the minus side we also are infinity |

11:05 | so that takes care these two, i already did that well i have to go through this point and remember as i get very close to 1 at point 99999999 i have to be at 5 and .011111 i have to be at 2 so thats why i have those circles there but the graph goes to those points |

11:31 | it just doesnt actually touch those points but it get infinitily close now negative 1 from the minus side i did negative infinity and negative 3 we go to negative 5 something like that and from the other side were at negative 5 and finally we have to go to negative infinity as x goes to negative infity we go to 0 |

12:03 | so some variation on that you guys see that something like that |

12:35 | we go the idea how do we know we go through the circle at negative 5 because you know that the curve the limit as x approaches -3 is minus 5 menas as you get closer and closer to 3 from te minus value |

13:01 | you have to get closer and closer -5 from the y value so as you get closer to negative 3 you have to approach negative 5 but at -3 you leave the curve go to negative 6 and go right back again thats why its not continuous there well this is 0,3 |

13:31 | how do i know not to go through 1,4 well at 1,4 i now its equal to 4but when i get to 1 its not equal to 4 very close to 4 its either 5 or 2 depending what side you are on but at 1 we equal to 4 see theres no limits going on |

14:00 | alright im going to do one more of these because i think you could use a third |

15:44 | f of is 0 f of 1 is 2 f of 3 is 6 the limit as x approaches 3 is 4 the limit as x approaches 5 from the minus side of f of x is positive infinity the limit as x approaches 5 from the plus side is positive infinity |

16:00 | the limit as x approaches positive infinity is 2 the limit as x approaches 0 from the positive side is 0 the limit as x approaches 0 from the minus side is -1 and last the limit as x approaches -infinity of f of x is negative infinity alright lets do this 1 |

16:37 | alright what do we know about this graph grading these are going to be entertaining, maybe we wont maybe im just torturing you guys alright lets put on the points that we know we know 3 points we know it goes through the origin because f of 0 is 0 |

17:04 | we know f of 1 is 2 so 1,2 and f of 3 is 6 you know the curve is at those points you get rid of that one you get rid of that one okay |

17:31 | as x approaches 3 were at 4 so 4 i will put that halfway between 2 and 6 so when you get close to 3 from either side you get close to 4, by the way it can go the other way we havent narrowed down the graph yet do we have any other information in that area no now what happens if we approach 5 from either side of 5 we go up |

18:01 | to positive infinity so we have avertical asymptote and were gonna go up to positive infinity from both sides and as x approahes infinity i have to go to 2 something like that you dont have to draw the dotted line but it is flattening out at 2 |

18:31 | so lets convert this to color what happens between 1 and 3 i have no other information happening between 1 and 3 i can just connect those now as i approach 0 from the positive side i get to 0 |

19:02 | and im actually equal to 0 at the origin but when i leave 0 and im on the negative side of 0 i have to be down here at negative 1 and as x approaches infinity i go to infinity thats all your things we like that one were getting the hang of it? good |

19:30 | lets do a different type of graph |

20:00 | you have a spherical shape bowl and youre going to pour water into it this is kind of the homework question so water is going in the bowl start filling it up draw a graph of the hieght of water in the bowl verses time height verses time so you pour water into the bowl at a constant rate the bowl is shaped like a sphere |

20:30 | what is the height of the water going to look like |

21:44 | your graphing the height of the water verses time if you pour in water at first its empty so its going to go up at a certain rate then its going to change |

22:09 | okay so what happens, you pour wtaer in so the beginning this is very shallow so the height goes up very quickly as the bowl gets wider everytime you have a certain amount of water the height goes up lesss because it has to go out so it doesnt go ut very much |

22:32 | so it starts to flatten out till you get to the middle of the bowl then as you get closer and closer to the top the curve gets stepper again until you are full and thats whatever time you are full okay how did we do on that thats not so hard right |

23:00 | what if the bowl was shaped like a cylinder if the bowl was shaped like a cylinder it will get me a straight line because at anytime it goes up the same amount what if it was shaped like a cone |

23:34 | do you guys understand why when it gets wider the height and then curve flatten because in the beginning you add lets say a gallon of water and you fill it up this high then your next gallon of water will fill it up this high and your next gallon of water may only fill it up this high you kind of flatten like that so far so good |

24:04 | trying to think of some other shapes i dont know thats enough shapes you got the idea lets do some more graph stuff you guys love graphs |

24:51 | this is the graph of f verses the derivative draw the graph as f verses x |

25:00 | and te graph of x verses the second derivative so i want two graphs first you draw the graph x verses f then you draw the graph x verses the second derivative so youre going to go both directions so we have the derivative graph so what are the keys to the derivative graph you just have to pay attention to the things that are positive where things are negative |

25:31 | when you look at a derivative graph you dont really want to focus on the slope you want to focus on positivity and negativity so below the x axis means negative above the x axis menas positive just like the other raphs so we are below the x axis here so our slopes are negative numbers were above the x axis here so are slope are positive numbers so remmeber negative number means youre going down positive numbers are going up so were going down at this spot |

26:04 | the derivative is 0 so somewhere around here the derivative is 0, the graoh is going down till it gets there and then it gets to 0 and then it turns around and goes up by the grap could be down here we dont know and we dont care at the origin the slopes are all positive till we get to the origin then it switches to negative and heres it 0 so thats must be a maximum here on our grap[h |

26:35 | so when you get to the y axis we have to do something like that okay cause the graph is going up 0 and then its going down now all the rest fo these are negative numbers the graph is going to go down for a while its going to keep going down but its gonna kind of flatten out because it has to get to 0 something like that |

27:03 | now so bad so thats the grpah of x f of x given the derivative graph now we go the other way we want the second derivative so the second derivative this is the original and if youre graphing the derivative the second derivative is just the derivative fo the first derivative so now we dont pay attention to positive and negatives we pay attention to slope so we look at this graph and say the slope is positive |

27:32 | of all that side till right here so somewhere around here the graph of the derivative is 0 because its flat right here so we have positive numbers from here till we get to here here the slope would be again 0 cause negative numbers all the way to there then we have 0 again |

28:04 | and thats oh there it bottoms out right about there thats called point of inflection then the graph goes up again positive but it sort of turns around and gets closer to 0 so first it goes up kind of positive fast and then it slows down till it gets to 0 again so that |

28:34 | all these slopes are positive so they hard to start at big positive numbers and come down to there okay alright lets do a different type of thing this is sort of involved with graphing |

29:39 | okay so we have the function f of x is the absolute value of x minus 2 divided by x minus 2 find the limit from the left as you approach 2 the limit from the right as you approach 2 and then graph the function so whats the absolute value of 10 10 whats the absolute value of -10 |

30:00 | 10 so when you take a positive number and the absolute value of it you dont have to do anything to the number when we have a negative number and we take the absolute value of it we make it positive so how do you take a negative number and make it positive you multiply it by negative 1 so if you want to take negative 10 and turn it into a positive number multiply it by negative 1 and now it becomes positive 10 so absolute value function you can think of as two ways |

30:30 | as long as x is a positive number you do nothing cause the absolute value of any positive number is just the number what if its 0 well the absolute value of 0 is 0 what about the absolute value of a negative number well we want to turn it positive so we multiply it by negative 1 another way to think of the absolute value function Is to think of it as a peace wise function |

31:02 | as long as x is greater than or equal to 0 its just the function if its less than 0 its the negative of the function now lets think of f of x the absolute value of x-/x-2 we could say well when is x-2 positive when x is bigger than 2 so as long as x as long as x is greater than or equal to 2 we dont have to do anything |

31:33 | so far so good what about when x is less than 2 well when x is less than 2 this is a negative number in the absolute value bar so just multiply it by a negative sign sorry thats the absolute value of that part not the hole function so we could rewrite this |

32:02 | as follows you can take the absolute value of x minus/x-2 and say that it is x minus 2 over x-2 x is greater than 2 at 2 things get interesting and its negative of x minus 2 over x-2 when x is less than 2 but wait what is x-2/x-2 |

32:30 | just 1 as long as x is not 2 so we can say this is 1 as long as x is greater than 2 and negative 1 as long as x is less than 2 so what is the limit as x approaches 2 from the minus side - 1 what is the limit as x approaches 2 from the plus side positive 1 and what is the graph of this look like |

33:05 | very simple graph something like tha thats it and at 2 the limit does not exist and theres no value fo the function cause you plug in 2 you get 0/0 howd you guys do on that one |

33:45 | youre asking is it a bad sign to coming from the right side to make x 3 well to plug in 3 you get 1/1 you plug in 2.1 you get 1/1 or .1/.1 if you plug in .01 so you can certainly tst numbers |

34:01 | and you should pretty quickly come up with the idea of getting 1 everytime you plug in a number bigger than 2 and -1 everytime you plug in a number less than 1 but if you actually want to think about the algebra, thats the algebra of whats going on alright lets do some conjinuity when we still have time |

34:46 | find the value of k that makes f of x continuos if f of x is x squared minus kx+1 when x is greater than 2 |

35:00 | and 3kx+4 when x is less than or equal to 2 so find k that makes the function continuos everywhere so rememebr in order to be continuous the pencil can not leave the paper when you are graphing that means youre graphing this equation and you come along and get to x's and you get closer and closer to 2 and you get some number and then when you switch to the other piece you have to stay on the graph so another words when you approach 2 from the left side |

35:31 | you have to get to the same spot as you approached by the left side so that means the limit as x approaches 2 from the left side as to equal the limit as x approaches 2 from the right side so the limit when x approaches 2 from the minus side of f of x equals 3k 2 plus 4 which is 6k+4 |

36:03 | and the limit when x approaches 2 from the plus side of f of x well now you go to the other graph and you plug in 2 and get 2 squared mins k times 2 plus 1 which is 5 minus 2k so five minus 2k has to equal 6k plus 4 thats a five |

36:32 | so 6k plus 4 has to equal 5 minus 2k so 8k has to equal 1 k has to equal 1/8 lets do one other its just 1/8 k equals 1/8 thats all you have to do you solved the problem |

37:11 | all we are asking for is the value of k that makes it continuous 1 last quick one |

37:50 | what if we asked where is this function continuous so we have x squared -3x+1 as long as x is less than or equal to 1 |

38:03 | x+1 when x is between 1 and 2 and 4x minus 5 when x is greater than or equal to 2 where is the function continuous generally the function is continuos everywhere if theyre polynomial your only problem are going to occur at 1 and at 2 so at x equals 1 from the left side |

38:31 | the limit as x approaches 1 from the left side is 1 minus 3 plus 1 is negative 1 the limit when x approaches 1 from the plus side is 2 so its not continuous at x=1 i plugged 1 into the top branch and i plugged one into the middle branch and i dont get the same value so its not continuous at x equals 1 |

39:03 | now what happens at x equals 2 well lets do the limit as x approaches 2 from the minus side of f of x and i get 3 and i do the limit when x approaches 2 from the plus side of f of x and i get 8-5 is 3 so it is continuous at x=2 so if i said wheres this function continuous i would say all reals |

39:36 | except x equals 1 you can write that in fancier notation if you want to alright study ahrd |