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any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | and these are line segments because they have two end points . they each have two end points . and they continue forever . | are lines made up of points ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | and these are line segments because they have two end points . they each have two end points . and they continue forever . | how can a ray have two end points , but only continue in 1 direction ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | let me make these two points parallel . and these are line segments because they have two end points . they each have two end points . | do line segments actually go on forever ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | any pair of points can be connected by a line segment . that 's right . | what are the exact definitions of a ray , line segment , and line ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | so i could create one segment that connects this point to this point and then another one that connects this point to this point . and they look pretty parallel . in fact , i think this is the right answer . | how is perpendicular or parallel is involve in this suituation ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | and these are line segments because they have two end points . they each have two end points . and they continue forever . | how is it a ray without a arrow on the other end ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . | find the equation of a line parallel to the given line and through the given points ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | drag the ray so it has an endpoint at a , so we want to make its endpoint at a where the ray terminates and goes through one of the other black points . the ray should also be parallel to the pink line . so i have two options . | can a ray and a line be parallel ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | so i could create one segment that connects this point to this point and then another one that connects this point to this point . and they look pretty parallel . in fact , i think this is the right answer . | can lines ever be parallel and perpendicular ? |
( piano music ) - [ steven ] we 're in the metropolitan museum of art looking at a gigantic clay pot . - [ elizabeth ] this is form ancient greece . - [ steven ] long before the classical period . the shape of this vase makes it a crater and it was found at the dipylon cemetery in athens . - [ elizabeth ] normally when... | you see diamonds and triangles and circles and meanders . - [ elizabeth ] we also see broad areas of black paint and stripped areas that form the base . - [ steven ] and this particular pot has pictorial bands which we call friezes and in them , and this is a little bit unusual for the geometric period , we see human f... | why is the slip red on one side , but black on other parts ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | i really did n't have to pick up my pen as i drew this right over here . and so you can see at least the way this continuous function that i 've drawn , it 's clear that there 's an absolute maximum and absolute minimum point over this interval . the absolute minimum point , well it seems like we hit it right over here... | what is the absolute max of a constant function ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | and i encourage you , actually pause this video and try to construct that function on your own . try to construct a non-continuous function over a closed interval where it would be very difficult or you ca n't really pick out an absolute minimum or an absolute maximum value over that interval . well let 's see , let me... | so if the intervals is not closed meaning the endpoints are not included , there are no absolute max or min ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have some function that is continuous over a closed interval , let 's say the closed interval from a to b . | 0 , is it a bit convenient for us when the discontinuity points ( holes ) coincidentally overlap the extreme points ( maxima & minima ) ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | and i encourage you , actually pause this video and try to construct that function on your own . try to construct a non-continuous function over a closed interval where it would be very difficult or you ca n't really pick out an absolute minimum or an absolute maximum value over that interval . well let 's see , let me... | are n't there some curves that are continuous but do n't have absolute maxima and minima ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . | sal , are you addressing the extreme value theorem or the theorem that addresses the existence of extreme values ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | i really did n't have to pick up my pen as i drew this right over here . and so you can see at least the way this continuous function that i 've drawn , it 's clear that there 's an absolute maximum and absolute minimum point over this interval . the absolute minimum point , well it seems like we hit it right over here... | because it seems to me that sal does not explicit what is `absolute maximum or minimum point ' means , does it mean just a complete way to get extrema , without the criteria you must n't ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | and our minimum point happens at a . for a flat function we could put any point as a maximum or the minimum point . and we 'll see that this would actually be true . | for a constant function this does not occur- f prime at a given point is 0 ... ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . | what about there exist a undefined point that is neither a maximum , nor a minimum , will extreme value theorem still hold ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | so you could say , well let 's a little closer here . maybe this number right over here is 5 . so you could say , maybe the maximum is 4.9 . | what is the meaning of the symbol that sal writes 3 right after d ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | and if we wanted to do an open interval right over here , that 's a and that 's b . and let 's just pick very simple function , let 's say a function like this . so right over here , if a were in our interval , it looks like we hit our minimum value at a. f of a would have been our minimum value . | what does mean that a function have some undefined points ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | so they 're members of the set that are in the interval such that -- and i 'm just using the logical notation here . such that f c is less than or equal to f of x , which is less than or equal to f of d for all x in the interval . just like that . | could n't you just define the maximum value as the limit f ( x ) as x approaches 5 and include the caveat that x does not equal 5 ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | and you could draw a bunch of functions here that are continuous over this closed interval . here our maximum point happens right when we hit b . and our minimum point happens at a . for a flat function we could put any point as a maximum or the minimum point . and we 'll see that this would actually be true . | if there was more than one x point that results in an absolute maximum , minimum point , how would i represent that in the `` mathy '' notation provided in ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . | this extreme value theorem wo n't apply to a function that we draw as a horizontal line , right ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | let 's say our function did something like this . let 's say our function did something right where you would have expected to have a maximum value let 's say the function is not defined . and right where you would have expected to have a minimum value , the function is not defined . and so right over here you could sa... | would the min/max of a function with a vertical asymptote be -inf and inf or just undefined ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | and our minimum point happens at a . for a flat function we could put any point as a maximum or the minimum point . and we 'll see that this would actually be true . | on the graph what if the discontinuity occurs in the middle of the function - meaning that the function is discontinuous just at a point between the interval a and b ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | so they 're members of the set that are in the interval such that -- and i 'm just using the logical notation here . such that f c is less than or equal to f of x , which is less than or equal to f of d for all x in the interval . just like that . | so what are the extreme points of f ( x ) =1/x for [ -2,2 ] ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | and i encourage you , actually pause this video and try to construct that function on your own . try to construct a non-continuous function over a closed interval where it would be very difficult or you ca n't really pick out an absolute minimum or an absolute maximum value over that interval . well let 's see , let me... | if the max and min of a non continuous line are difficult to construct , does it even exist ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . | what are the white lines for ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | then you could get your x even closer to this value and make your y be 4.99 , or 4.999 . you could keep adding another 9 . so there is no maximum value . | can someone give another explanation of continuity ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | here our maximum point happens right when we hit b . and our minimum point happens at a . for a flat function we could put any point as a maximum or the minimum point . | if the absolute minimum is at `` c '' , how is the minimum point at `` a '' as stated ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | and if we wanted to do an open interval right over here , that 's a and that 's b . and let 's just pick very simple function , let 's say a function like this . so right over here , if a were in our interval , it looks like we hit our minimum value at a. f of a would have been our minimum value . | does the function count these maximums as having the same x coordinates , or just the same y coordinates ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | and that might give us a little bit more intuition about it . so the extreme value theorem says if we have some function that is continuous over a closed interval , let 's say the closed interval from a to b . and when we say a closed interval , that means we include the end points a and b . | how to find a example of a function on a bounder closed interval which achieves a maximum but not a minimum on that interval ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . | what if the holes were of jump discontinuity ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | and you could draw a bunch of functions here that are continuous over this closed interval . here our maximum point happens right when we hit b . and our minimum point happens at a . | so the point below the hole must be the absolute minima right ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | and so right over here you could say , well look , the function is clearly approaching , as x approaches this value right over here , the function is clearly approaching this limit . but that limit ca n't be the maxima because the function never gets to that . so you could say , well let 's a little closer here . | are the absolute and relative maxima/minima same with global and local maxima/minima ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | and i encourage you , actually pause this video and try to construct that function on your own . try to construct a non-continuous function over a closed interval where it would be very difficult or you ca n't really pick out an absolute minimum or an absolute maximum value over that interval . well let 's see , let me... | so what if a function on a closed interval has a hole that does n't take place of the absolute max , absolute min , or the intervals ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . | would the extreme value theorem still be true even though it 's not continuous at that point ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . | what if the graph was a curve and the maximum and minimum was not at the point of intervals.. would extreme value theorem be valid then ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | you could keep adding another 9 . so there is no maximum value . similarly here , on the minimum . | what about if we had removable discontinuities at the maximum and minimum value of the function ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | let 's say our function did something like this . let 's say our function did something right where you would have expected to have a maximum value let 's say the function is not defined . and right where you would have expected to have a minimum value , the function is not defined . and so right over here you could sa... | what would happen in the case of jump discontinuities ( cases where the function is defined but discontinuous at the point ) ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | and that might give us a little bit more intuition about it . so the extreme value theorem says if we have some function that is continuous over a closed interval , let 's say the closed interval from a to b . and when we say a closed interval , that means we include the end points a and b . | would the rule still apply that no local extreme values exists within the closed interval ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | and that might give us a little bit more intuition about it . so the extreme value theorem says if we have some function that is continuous over a closed interval , let 's say the closed interval from a to b . and when we say a closed interval , that means we include the end points a and b . | the graph that sal drew , if it was continuous but the interval was not closed [ thus ( a , b ) ] would the function had any maxima or minima ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | well let 's imagine that it was an open interval . let 's imagine open interval . and sometimes , if we want to be particular , we could make this is the closed interval right of here in brackets . | what is the center of the interval is the vertex of a parabola ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | for a flat function we could put any point as a maximum or the minimum point . and we 'll see that this would actually be true . but let 's dig a little bit deeper as to why f needs to be continuous , and why this needs to be a closed interval . | would that mean you have two maxima/minima ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | so they 're members of the set that are in the interval such that -- and i 'm just using the logical notation here . such that f c is less than or equal to f of x , which is less than or equal to f of d for all x in the interval . just like that . | can there be f ( x ) = x^2 over closed interval ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | that is we have these brackets here instead of parentheses . then there will be an absolute maximum value for f and an absolute minimum value for f. so then that means there exists -- this is the logical symbol for there exists -- there exists an absolute maximum value of f over interval and absolute minimum value of f... | can there be 2 absolute maximum , or for my example would there just be 2 relative maximum and 1 relative minimum ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . | how do horizontal lines obey the extreme value theorem ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | so they 're members of the set that are in the interval such that -- and i 'm just using the logical notation here . such that f c is less than or equal to f of x , which is less than or equal to f of d for all x in the interval . just like that . | now what happens if the given function f ( x ) is a horizontal line ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | i really did n't have to pick up my pen as i drew this right over here . and so you can see at least the way this continuous function that i 've drawn , it 's clear that there 's an absolute maximum and absolute minimum point over this interval . the absolute minimum point , well it seems like we hit it right over here... | what is the difference between the range of a function and the absolute maximum and minimum points ? |
so we 'll now think about the extreme value theorem . which we 'll see is a bit of common sense . but in all of these theorems it 's always fun to think about the edge cases . why is it laid out the way it is ? and that might give us a little bit more intuition about it . so the extreme value theorem says if we have so... | and we 'll see in a second why the continuity actually matters . so this is my x-axis , that 's my y-axis . and let 's draw the interval . | will the maxima and the minima be the same for the graph of a line parallel to x-axis ? |
: we 've already seen how to form acetals . if we start with an aldehyde or a ketone , and we add an excess of alcohol in an acidic environment we can form our acetal . we talked about ways to increase the formation of your acetal by removing , in this case , water from your reaction to drive the equilibrium to the ri... | and so , now we can add our lithium aluminum hydride , so we add our lithium aluminum hydride here in the first step . and that is going to reduce our carboxylic acid . and so in the second step we can add a source of protons , and we can add some excess water . | 0 why does n't the carboxylic acid get turned into an acetal ? |
: we 've already seen how to form acetals . if we start with an aldehyde or a ketone , and we add an excess of alcohol in an acidic environment we can form our acetal . we talked about ways to increase the formation of your acetal by removing , in this case , water from your reaction to drive the equilibrium to the ri... | so , we 're going to use , we 're going to protect our ketone using an acetal , of course . and we 're going to react it with ethylene glycol . so if we react our starting compound here with ethylene glycol , and we use an acid catalyst , we 're going to form an acetal . | does n't the 'ethylene ' imply the presence of a double bond when in fact only single bonds ( associated with -anes ) exist ? |
: we 've already seen how to form acetals . if we start with an aldehyde or a ketone , and we add an excess of alcohol in an acidic environment we can form our acetal . we talked about ways to increase the formation of your acetal by removing , in this case , water from your reaction to drive the equilibrium to the ri... | so we can use an acidic environment here . and we 're going to form a cyclic thioacetal . so very similar to what we did before . | should n't it be called cyclic ketal ? |
: we 've already seen how to form acetals . if we start with an aldehyde or a ketone , and we add an excess of alcohol in an acidic environment we can form our acetal . we talked about ways to increase the formation of your acetal by removing , in this case , water from your reaction to drive the equilibrium to the ri... | so , we 're going to use , we 're going to protect our ketone using an acetal , of course . and we 're going to react it with ethylene glycol . so if we react our starting compound here with ethylene glycol , and we use an acid catalyst , we 're going to form an acetal . | why isnt an ester formed when ethylene glycol reacts with -cooh ? |
: we 've already seen how to form acetals . if we start with an aldehyde or a ketone , and we add an excess of alcohol in an acidic environment we can form our acetal . we talked about ways to increase the formation of your acetal by removing , in this case , water from your reaction to drive the equilibrium to the ri... | so , we 're going to use , we 're going to protect our ketone using an acetal , of course . and we 're going to react it with ethylene glycol . so if we react our starting compound here with ethylene glycol , and we use an acid catalyst , we 're going to form an acetal . | 6 , why does n't the ethylene glycol react with the carboxylic acid as well as the ketone ? |
: we 've already seen how to form acetals . if we start with an aldehyde or a ketone , and we add an excess of alcohol in an acidic environment we can form our acetal . we talked about ways to increase the formation of your acetal by removing , in this case , water from your reaction to drive the equilibrium to the ri... | and so , now we can add our lithium aluminum hydride , so we add our lithium aluminum hydride here in the first step . and that is going to reduce our carboxylic acid . and so in the second step we can add a source of protons , and we can add some excess water . | around 0 , can the acid protonate the hydroxyl portion of the carboxylic group , thereby forming a water molecule which may become a leaving group ? |
: we 've already seen how to form acetals . if we start with an aldehyde or a ketone , and we add an excess of alcohol in an acidic environment we can form our acetal . we talked about ways to increase the formation of your acetal by removing , in this case , water from your reaction to drive the equilibrium to the ri... | so if we react our starting compound here with ethylene glycol , and we use an acid catalyst , we 're going to form an acetal . alright , so the ethylene glycol is going to react with the ketone portion of the molecule to form an acetal , specifically a cyclic acetal . alright , so over here on the left we still have o... | could we have used a simpler acetal such as ch3oh ? |
: we 've already seen how to form acetals . if we start with an aldehyde or a ketone , and we add an excess of alcohol in an acidic environment we can form our acetal . we talked about ways to increase the formation of your acetal by removing , in this case , water from your reaction to drive the equilibrium to the ri... | so this is completely analogous to the formation of an acetal . so you start off with an aldehyde or a ketone . and this time , instead of using an alcohol you 're going to use a thiol , so instead of having an oxygen here , you have a sulfur . | if i have a ketone and a aldehyde in the same molecule reacting with `` hoch2ch2oh '' forming a protecting group , would this protector be added to the aldehyde since is more reactive than a ketone ? |
: we 've already seen how to form acetals . if we start with an aldehyde or a ketone , and we add an excess of alcohol in an acidic environment we can form our acetal . we talked about ways to increase the formation of your acetal by removing , in this case , water from your reaction to drive the equilibrium to the ri... | and we have two equivalents of it . so that 'd be one , two , three , four carbons . so , by looking at our acetal and thinking about hydrolyzing it and thinking about where those portions came from , we can easily come up with the products of this hydrolysis of this acetal . | to hydrolyze the acetyl into a ketone , would a water molecule attack one of the two carbons forming the ring ? |
: we 've already seen how to form acetals . if we start with an aldehyde or a ketone , and we add an excess of alcohol in an acidic environment we can form our acetal . we talked about ways to increase the formation of your acetal by removing , in this case , water from your reaction to drive the equilibrium to the ri... | so here 's your sulfur , two carbons , two carbons and then another sulfur , so a cyclic thioacetal and then there 's actually a reaction for converting this compound into our target compound . and this involves a special kind of nickel , so if we use raney nickel here . it 's a special kind of finely divided nickel th... | when talking about reducing the cyclo-dithiol acetal and adding the two hydrogen groups , is this all done by raney nickel ? |
: we 've already seen how to form acetals . if we start with an aldehyde or a ketone , and we add an excess of alcohol in an acidic environment we can form our acetal . we talked about ways to increase the formation of your acetal by removing , in this case , water from your reaction to drive the equilibrium to the ri... | alright , let 's look at another type of reaction here . so this one 's formation of a thioacetal . so this is completely analogous to the formation of an acetal . | after the formation of the thioacetal , is it possible to reform the ketone/aldehyde ? |
: we 've already seen how to form acetals . if we start with an aldehyde or a ketone , and we add an excess of alcohol in an acidic environment we can form our acetal . we talked about ways to increase the formation of your acetal by removing , in this case , water from your reaction to drive the equilibrium to the ri... | so once again you have your r double prime group coming from your thiol , and then you have a sulfur here instead of an oxygen . and then you have two of these to form your thioacetal . and so , one reason you might want to form a thioacetal instead of an acetal , is thioacetals have an additional reaction that they un... | would the rate of forming thioacetal be faster than that of plain acetal because of the molecular size or weight or do these factors not affect the rate of the reaction ? |
: we 've already seen how to form acetals . if we start with an aldehyde or a ketone , and we add an excess of alcohol in an acidic environment we can form our acetal . we talked about ways to increase the formation of your acetal by removing , in this case , water from your reaction to drive the equilibrium to the ri... | and so we 've now been able to make our desired product . and so that 's one way to use an acetal as a protecting group . alright , let 's look at another type of reaction here . | if you have two ketones , which one does the protecting group protect ? |
: we 've already seen how to form acetals . if we start with an aldehyde or a ketone , and we add an excess of alcohol in an acidic environment we can form our acetal . we talked about ways to increase the formation of your acetal by removing , in this case , water from your reaction to drive the equilibrium to the ri... | you would push the equilibrium to the left , and you would hydrolyze your acetal and convert it back into your original aldehyde or ketone and your alcohol . so this can actually be a useful reaction if you want to use an acetal as a protecting group . and so let 's look at an example of hydrolyzing an acetal . | for the last reaction , ca n't we use clemenson 's reagent ( zn hg & h cl ) to reduce kentone ? |
: we 've already seen how to form acetals . if we start with an aldehyde or a ketone , and we add an excess of alcohol in an acidic environment we can form our acetal . we talked about ways to increase the formation of your acetal by removing , in this case , water from your reaction to drive the equilibrium to the ri... | and so you can see that we have reduced our ketone here to form this cyclohexene portion . so reducing your ketone . so the first thing we could do , to do this transformation , is to form a thioacetal . | in the first step of the protection of the ketone , how does the carbonyl oxygen `` disappear '' in solution ? |
: we 've already seen how to form acetals . if we start with an aldehyde or a ketone , and we add an excess of alcohol in an acidic environment we can form our acetal . we talked about ways to increase the formation of your acetal by removing , in this case , water from your reaction to drive the equilibrium to the ri... | you would push the equilibrium to the left , and you would hydrolyze your acetal and convert it back into your original aldehyde or ketone and your alcohol . so this can actually be a useful reaction if you want to use an acetal as a protecting group . and so let 's look at an example of hydrolyzing an acetal . | for the last reaction , ca n't we use clemenson 's reagent ( zn hg & h cl ) to reduce kentone ? |
: we 've already seen how to form acetals . if we start with an aldehyde or a ketone , and we add an excess of alcohol in an acidic environment we can form our acetal . we talked about ways to increase the formation of your acetal by removing , in this case , water from your reaction to drive the equilibrium to the ri... | you would push the equilibrium to the left , and you would hydrolyze your acetal and convert it back into your original aldehyde or ketone and your alcohol . so this can actually be a useful reaction if you want to use an acetal as a protecting group . and so let 's look at an example of hydrolyzing an acetal . | for the last reaction , ca n't we use clemenson 's reagent ( zn hg & h cl ) to reduce kentone ? |
: we 've already seen how to form acetals . if we start with an aldehyde or a ketone , and we add an excess of alcohol in an acidic environment we can form our acetal . we talked about ways to increase the formation of your acetal by removing , in this case , water from your reaction to drive the equilibrium to the ri... | you would push the equilibrium to the left , and you would hydrolyze your acetal and convert it back into your original aldehyde or ketone and your alcohol . so this can actually be a useful reaction if you want to use an acetal as a protecting group . and so let 's look at an example of hydrolyzing an acetal . | for the last reaction , ca n't we use clemenson 's reagent ( zn hg & h cl ) to reduce kentone ? |
: we 've already seen how to form acetals . if we start with an aldehyde or a ketone , and we add an excess of alcohol in an acidic environment we can form our acetal . we talked about ways to increase the formation of your acetal by removing , in this case , water from your reaction to drive the equilibrium to the ri... | you would push the equilibrium to the left , and you would hydrolyze your acetal and convert it back into your original aldehyde or ketone and your alcohol . so this can actually be a useful reaction if you want to use an acetal as a protecting group . and so let 's look at an example of hydrolyzing an acetal . | for the last reaction , ca n't we use clemenson 's reagent ( zn hg & h cl ) to reduce kentone ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | and i 've divided it into 5 equal sections . and so we already know that the total amount of his weekend spent studying is 1/5 . so that 's the total amount studying for the weekend is 1/5 . | but how do you know what number goes first in the equation ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | so he 's going to divide this by 4 . now , we 've already seen that dividing by a number is the same thing as multiplying by its reciprocal . you might say , hey , well , what 's the reciprocal of 4 ? you just have to remind yourself that 4 is the same thing as 4/1 . | so basically an reciprocal is the opposite ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | so he 's going to divide this by 4 . now , we 've already seen that dividing by a number is the same thing as multiplying by its reciprocal . you might say , hey , well , what 's the reciprocal of 4 ? | why does multiplying by a reciprocal is the same than divining fractions ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | you just have to remind yourself that 4 is the same thing as 4/1 . so 1/5 divided by 4/1 is the same thing as 1/5 times 1/4 . and you could also view this as 1/4 of 1/5 or 1/5 of 1/4 , either way . | 1/5 / 4/1 = 1/5 = 5/1 *4 = 20 why not fliping 1/5 to 5/1 ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | and what is that ? well , that 's 1 over -- how many equal sections are there of that size in the weekend ? well , i 've just drawn out the grid . | how many different types of fractions are there ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | so he 's going to divide this by 4 . now , we 've already seen that dividing by a number is the same thing as multiplying by its reciprocal . you might say , hey , well , what 's the reciprocal of 4 ? | how would you pictorially represent dividing to fractions where the first is less than the second ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | you just have to remind yourself that 4 is the same thing as 4/1 . so 1/5 divided by 4/1 is the same thing as 1/5 times 1/4 . and you could also view this as 1/4 of 1/5 or 1/5 of 1/4 , either way . | if you had 1/8 divided by 1/2 , how would you get 1/4 if you have to cut it into 1/2 pieces ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | and he 's going to spend that much time on each subject . so he 's going to divide this by 4 . now , we 've already seen that dividing by a number is the same thing as multiplying by its reciprocal . | is it possible to divide negative fractions by negative numbers ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . | and why do we use negative numbers in math ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . | what does reciprical mean ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | and he 's going to spend that much time on each subject . so he 's going to divide this by 4 . now , we 've already seen that dividing by a number is the same thing as multiplying by its reciprocal . | in word problems , how can you tell whether you should multiply or divide ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | you just have to remind yourself that 4 is the same thing as 4/1 . so 1/5 divided by 4/1 is the same thing as 1/5 times 1/4 . and you could also view this as 1/4 of 1/5 or 1/5 of 1/4 , either way . | why you write 4 after 1/5 why you dont write 4/1 divide by 1/5 ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | and he 's going to spend that much time on each subject . so he 's going to divide this by 4 . now , we 've already seen that dividing by a number is the same thing as multiplying by its reciprocal . | when should i put the number after or before the divide sign ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of the weekend . | single numbers in fraction problems are improper fractions ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | so he 's going to divide this by 4 . now , we 've already seen that dividing by a number is the same thing as multiplying by its reciprocal . you might say , hey , well , what 's the reciprocal of 4 ? | when dividing fractions by whole numbers , does the fraction always come first or does it not matter ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . | write and solve a division equation to determine the speed limit of the interstate highway ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | and he 's going to spend that much time on each subject . so he 's going to divide this by 4 . now , we 've already seen that dividing by a number is the same thing as multiplying by its reciprocal . | i still do n't get how to divide fractions , can you help me more ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | so he 's going to divide this by 4 . now , we 've already seen that dividing by a number is the same thing as multiplying by its reciprocal . you might say , hey , well , what 's the reciprocal of 4 ? | when dividing fractions in word problems , how would i know which number goes first ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | you just have to remind yourself that 4 is the same thing as 4/1 . so 1/5 divided by 4/1 is the same thing as 1/5 times 1/4 . and you could also view this as 1/4 of 1/5 or 1/5 of 1/4 , either way . | in the video the problem is 1/5 divided by 4 , why does the four go second and why does the 1/5 go first ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | so he 's going to divide this by 4 . now , we 've already seen that dividing by a number is the same thing as multiplying by its reciprocal . you might say , hey , well , what 's the reciprocal of 4 ? | why is it that dividing is the same as multiplying the fractions ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | you just have to remind yourself that 4 is the same thing as 4/1 . so 1/5 divided by 4/1 is the same thing as 1/5 times 1/4 . and you could also view this as 1/4 of 1/5 or 1/5 of 1/4 , either way . | when you divide a fraction less than 1 by a whole number greater than 1 how does the quotient compare to the dividend ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | and i 've divided it into 5 equal sections . and so we already know that the total amount of his weekend spent studying is 1/5 . so that 's the total amount studying for the weekend is 1/5 . now , he has to divide this into 4 equals section . | does using reciprocals in your equation lessen the amount of work that you have to show ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | and he 's going to spend that much time on each subject . so he 's going to divide this by 4 . now , we 've already seen that dividing by a number is the same thing as multiplying by its reciprocal . | is there i easier way to divide fractions to whole numbers ? |
tommy is studying for final exams this weekend . he will spend 1/5 of the weekend studying . what fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject ? so the total amount of time he 's going to spend studying this weekend is 1/5 of th... | so he 's going to divide this by 4 . now , we 've already seen that dividing by a number is the same thing as multiplying by its reciprocal . you might say , hey , well , what 's the reciprocal of 4 ? you just have to remind yourself that 4 is the same thing as 4/1 . | what 's the difference between an inverse and reciprocal ? |
so quadrilateral abcd , they 're telling us it is a rhombus , and what we need to do , we need to prove that the area of this rhombus is equal to 1/2 times ac times bd . so we 're essentially proving that the area of a rhombus is 1/2 times the product of the lengths of its diagonals . so let 's see what we can do over ... | we also know , because this is a rhombus , and we proved this in the last video , that the diagonals , not only do they bisect each other , but they are also perpendicular . so we know that this is a right angle , this is a right angle , that is a right angle , and then this is a right angle . so the easiest way to thi... | in a rhombus , are the angles alwyas right angles ? |
so quadrilateral abcd , they 're telling us it is a rhombus , and what we need to do , we need to prove that the area of this rhombus is equal to 1/2 times ac times bd . so we 're essentially proving that the area of a rhombus is 1/2 times the product of the lengths of its diagonals . so let 's see what we can do over ... | what is the height here ? well we know that this diagonal right over here , that it 's a perpendicular bisector . so the height is just the distance from be . | whats the difference between perpendicular and parelel ? |
so quadrilateral abcd , they 're telling us it is a rhombus , and what we need to do , we need to prove that the area of this rhombus is equal to 1/2 times ac times bd . so we 're essentially proving that the area of a rhombus is 1/2 times the product of the lengths of its diagonals . so let 's see what we can do over ... | and so we can say that the area -- so because of that , we know that the area of abcd is just going to be equal to 2 times the area of , we could pick either one of these . we could say 2 times the area of abc . because area of abcd -- actually let me write it this way . | the plural of rhombus is rhombi , but how does one know which words to say `` uses '' and which words to say `` i '' ? |
so quadrilateral abcd , they 're telling us it is a rhombus , and what we need to do , we need to prove that the area of this rhombus is equal to 1/2 times ac times bd . so we 're essentially proving that the area of a rhombus is 1/2 times the product of the lengths of its diagonals . so let 's see what we can do over ... | this side is congruent to that side , and they both share a c right over here . so this is by side-side-side . and so we can say that the area -- so because of that , we know that the area of abcd is just going to be equal to 2 times the area of , we could pick either one of these . | but what if it asked a question like `` find the area of rhombus when a side and diagonal are given '' like when a diagonal is 12 and side is 10 ? |
so quadrilateral abcd , they 're telling us it is a rhombus , and what we need to do , we need to prove that the area of this rhombus is equal to 1/2 times ac times bd . so we 're essentially proving that the area of a rhombus is 1/2 times the product of the lengths of its diagonals . so let 's see what we can do over ... | because it 's a parallelogram , we know the diagonals bisect each other . so we know that this length -- let me call this point over here b , let 's call this e. we know that be is going to be equal to ed . so that 's be , we know that 's going to be equal to ed . | what does a , b , c , d , and e mean ? |
so quadrilateral abcd , they 're telling us it is a rhombus , and what we need to do , we need to prove that the area of this rhombus is equal to 1/2 times ac times bd . so we 're essentially proving that the area of a rhombus is 1/2 times the product of the lengths of its diagonals . so let 's see what we can do over ... | so it 's 1/2 -- i 'll color code it . the base is ac . and then what is the height ? | ( dot ) while writing ac.bd instead of ac*bd why should n't it be a multiplication sign ? |
so quadrilateral abcd , they 're telling us it is a rhombus , and what we need to do , we need to prove that the area of this rhombus is equal to 1/2 times ac times bd . so we 're essentially proving that the area of a rhombus is 1/2 times the product of the lengths of its diagonals . so let 's see what we can do over ... | this side is congruent to that side , and they both share a c right over here . so this is by side-side-side . and so we can say that the area -- so because of that , we know that the area of abcd is just going to be equal to 2 times the area of , we could pick either one of these . | what does sss mean or 'side side side ' ? |
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