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what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . | is the equation of ak=1+a ( k-1 ) is the same thing to say tn=a ( n-1 ) d ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | i want to make it clear -- i have essentially defined a function here . if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restr... | what does k stand for ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . | when sal said that the top sequence was explicit and the bottom one is recursive , is this due to the finite and infinite ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . | sal writes { a sub k } k = 1 , is it correct for him to write that k = 1 because if you look at the sequence then you can see that ( a sub 1 ) is 3 ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so let 's see . when k is 3 , we added 3 twice . let me make it clear . | why is ( k-1 ) multiplied by 3 ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so the same exact sequence , i could define it as a sub k from k equals 1 to 4 , with -- instead of explicitly writing the numbers here , i could say a sub k is equal to some function of k. so let 's see what happens . when k is 1 , we get 1 . when k is 2 , we get 4 . | what if a1=.5 and a2=2 an+3=an + 4an+1/ n+1 ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . | how would you go about finding the first 5 terms of this recursively defined sequence ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . | what is the practical value of math sequencing ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and i want to be clear -- not every sequence can be defined as either an explicit function like this , or as a recursive function . but many can , including this , which is an arithmetic sequence , where we keep adding the same quantity over and over again . so how would we do that ? | if a sequence is not geometric , is it arithmetic ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so it 's going to be plus 4 times k minus 1 . so this is another way of defining this infinite sequence . now , in both of these cases , i defined it as an explicit function . | so , recursively defining a sequence means adding the needed amount to a number that has already happened in the sequence ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so this is a sub k minus 1 . so a given term is equal to the previous term . let me make it clear -- this is the previous term , plus -- in this case , we 're adding 3 every time . | and explicit is basing it off of the term 's placement in the sequence ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | so how would we do that ? well , we could also -- another way of defining this first sequence , we could say a sub k , starting at k equals 1 and going to 4 with . and when you define a sequence recursively , you want to define what your first term is , with a sub 1 equaling 1 . you can define every other term in terms... | is there any way to calculate a term in a sequence purely recursively without any knowledge of terms that go before it except the first ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | if i wanted a more traditional function notation , i could have written a of k , where k is the term that i care about . a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . | when sal uses the expression a ( k ) =1+3 ( k-1 ) , would the positive one and the negative one cancel each other out ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . | what is the exact definition of a sequence ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and you do n't always have to add the same thing . we 'll explore fancier sequences . the sequences where you keep adding the same amount , we call these arithmetic sequences , which we will also explore in more detail . | how will we use arithemetic sequences in real life ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | and let 's say i only have these four terms right over here . so this one we would call a finite sequence . i could also have an infinite sequence . so an example of an infinite sequence -- let 's say we start at 3 , and we keep adding 4 . | what is the difference between a a sequence and a series ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . | what 's the difference between the two ways ? |
what i want to do in this video is familiarize ourselves with the notion of a sequence . and all a sequence is is an ordered list of numbers . so for example , i could have a finite sequence -- that means i do n't have an infinite number of numbers in it -- where , let 's say , i start at 1 and i keep adding 3 . so 1 p... | a of k is equal to 1 plus 3 times k minus 1 . this is essentially a function , where an allowable input , the domain , is restricted to positive integers . now , how would i denote this business right over here ? | why is the domain restricted to two positives integers instead of 2 positive ranges ? |
sal : this is sal , and i 'm still at stanford medical school with dr. connolly . doctor connolly : sal , we 're going to continue going . you going to get one unit of credit for today . sal : excellent . my mother would be proud . doctor connolly : so what we 're going to look at today is a colon polyp . we had said b... | sal : how would you know that ? doctor connolly : and so -- sal : right , because it never crossed a boundary line . doctor connolly : and there 's a reason why the boundary is important , is that if you look here , these are all the surface cells . | do all polyps have the potential to become cancerous and are not because they have n't crossed the boundary or are there different types of polyps ? |
sal : this is sal , and i 'm still at stanford medical school with dr. connolly . doctor connolly : sal , we 're going to continue going . you going to get one unit of credit for today . sal : excellent . my mother would be proud . doctor connolly : so what we 're going to look at today is a colon polyp . we had said b... | sal : excellent . my mother would be proud . doctor connolly : so what we 're going to look at today is a colon polyp . | how would you cure a polyp ? |
sal : this is sal , and i 'm still at stanford medical school with dr. connolly . doctor connolly : sal , we 're going to continue going . you going to get one unit of credit for today . sal : excellent . my mother would be proud . doctor connolly : so what we 're going to look at today is a colon polyp . we had said b... | sal : how would you know that ? doctor connolly : and so -- sal : right , because it never crossed a boundary line . doctor connolly : and there 's a reason why the boundary is important , is that if you look here , these are all the surface cells . | are cancers always determined by a boundary in organs and are diagnosed when they cross that boundary ? |
sal : this is sal , and i 'm still at stanford medical school with dr. connolly . doctor connolly : sal , we 're going to continue going . you going to get one unit of credit for today . sal : excellent . my mother would be proud . doctor connolly : so what we 're going to look at today is a colon polyp . we had said b... | sal : i see . doctor connolly : so carcinoma in situ means it 's not an invasive cancer . and therefore , most people would not really call it a cancer . | `` carcinoma in situ '' means the boundary has n't been crossed ? |
sal : this is sal , and i 'm still at stanford medical school with dr. connolly . doctor connolly : sal , we 're going to continue going . you going to get one unit of credit for today . sal : excellent . my mother would be proud . doctor connolly : so what we 're going to look at today is a colon polyp . we had said b... | doctor connolly : yeah . we find that people , if they have colon cancer , often they have lots of other pre-cancer lesions . and you can see -- sal : i see . | then what does `` benign cancer '' mean ? |
sal : this is sal , and i 'm still at stanford medical school with dr. connolly . doctor connolly : sal , we 're going to continue going . you going to get one unit of credit for today . sal : excellent . my mother would be proud . doctor connolly : so what we 're going to look at today is a colon polyp . we had said b... | it would continue to grow . cells divide . cells divide too much , and then one of them made a mistake . | what are the test tube cells supposed to do ? |
sal : this is sal , and i 'm still at stanford medical school with dr. connolly . doctor connolly : sal , we 're going to continue going . you going to get one unit of credit for today . sal : excellent . my mother would be proud . doctor connolly : so what we 're going to look at today is a colon polyp . we had said b... | doctor connolly : and there 's a reason why the boundary is important , is that if you look here , these are all the surface cells . down here , these are the vessels that go to the rest of the body . these are blood vessels . | since the carcinoma in situ does n't reach the vessels that go to the rest of the body , can it be harmful in any way to the patient before it crosses that boundary ? |
sal : this is sal , and i 'm still at stanford medical school with dr. connolly . doctor connolly : sal , we 're going to continue going . you going to get one unit of credit for today . sal : excellent . my mother would be proud . doctor connolly : so what we 're going to look at today is a colon polyp . we had said b... | you going to get one unit of credit for today . sal : excellent . my mother would be proud . | how do you find the boundary sal mentions ? |
sal : this is sal , and i 'm still at stanford medical school with dr. connolly . doctor connolly : sal , we 're going to continue going . you going to get one unit of credit for today . sal : excellent . my mother would be proud . doctor connolly : so what we 're going to look at today is a colon polyp . we had said b... | and so what this , this is now called dysplasia . sal : dysplasia . doctor connolly : d-y-s. sal : right , plasia in the wrong thing . | what are foods to avoid to help avoid this dysplasia ? |
sal : this is sal , and i 'm still at stanford medical school with dr. connolly . doctor connolly : sal , we 're going to continue going . you going to get one unit of credit for today . sal : excellent . my mother would be proud . doctor connolly : so what we 're going to look at today is a colon polyp . we had said b... | doctor connolly : yeah . we find that people , if they have colon cancer , often they have lots of other pre-cancer lesions . and you can see -- sal : i see . | how does cancer spread like a virus or bacteria ? |
sal : this is sal , and i 'm still at stanford medical school with dr. connolly . doctor connolly : sal , we 're going to continue going . you going to get one unit of credit for today . sal : excellent . my mother would be proud . doctor connolly : so what we 're going to look at today is a colon polyp . we had said b... | doctor connolly : yeah . we find that people , if they have colon cancer , often they have lots of other pre-cancer lesions . and you can see -- sal : i see . | do polyps break off very often ? |
so , managing schizophrenia is a tricky business . typically , the main problem that we want to try to manage are psychotic symptoms , like hallucinations and delusions using medications that are called anti-psychotics . now , anti-psychotics can be broken up into older drugs , sometimes called first generation anti-p... | and there are a couple of factors that seems to be associated with a worse prognosis , and simply being male is one of them , but we 're not quite sure why . an early onset seems to also be associated with worse prognosis . and , finally , a strong family history of schizophrenia , meaning the more family members that ... | given that males are more likely to have early onset ( higher numbers of males at younger age over females that gradually equilibrate over time ) , are n't those two factors that worsen prognosis connected ? |
so , managing schizophrenia is a tricky business . typically , the main problem that we want to try to manage are psychotic symptoms , like hallucinations and delusions using medications that are called anti-psychotics . now , anti-psychotics can be broken up into older drugs , sometimes called first generation anti-p... | now , sgas are actually usually preferred over fgas because they usually have less side effects that induce movement disorders , like , parkinsonism . these movement type side effects are also called extrapyramidal side effects . and the main difference between fgas and sgas is thought to be that sgas have this tendenc... | could extrapyamidal be any type of movement including being unable to sit in one place for very long ? |
so , managing schizophrenia is a tricky business . typically , the main problem that we want to try to manage are psychotic symptoms , like hallucinations and delusions using medications that are called anti-psychotics . now , anti-psychotics can be broken up into older drugs , sometimes called first generation anti-p... | now , sgas are actually usually preferred over fgas because they usually have less side effects that induce movement disorders , like , parkinsonism . these movement type side effects are also called extrapyramidal side effects . and the main difference between fgas and sgas is thought to be that sgas have this tendenc... | 7 what exactly are the endocrine side effects and what is prolactine ? |
jada is ordering mother 's day gifts online . the shipping costs are based on the dollar amount of the order . for orders less than $ 20 , shipping costs $ 4 . for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dol... | so if we have the shipping costs as an input , will that map to exactly -- for a given input , will we get exactly one output for the amount of order ? in order for this to be represented as a function , we have to input a shipping cost , a shipping cost where this relationship is defined . we need to input a shipping ... | why is shipping cost the input ? |
jada is ordering mother 's day gifts online . the shipping costs are based on the dollar amount of the order . for orders less than $ 20 , shipping costs $ 4 . for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dol... | for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dollar amount of the order -- so can the amount of order be represented as a function of shipping -- let me do that in that blue color costs . | between pots 2 & 3 , the amount of rupees stays the same ( 6 ) , however this can still be represented as a function ? |
jada is ordering mother 's day gifts online . the shipping costs are based on the dollar amount of the order . for orders less than $ 20 , shipping costs $ 4 . for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dol... | so they 're saying , can the dollar amount of the order -- so can the amount of order be represented as a function of shipping -- let me do that in that blue color costs . so if we have the shipping costs as an input , will that map to exactly -- for a given input , will we get exactly one output for the amount of orde... | how to know what x input goes into the domain ? |
jada is ordering mother 's day gifts online . the shipping costs are based on the dollar amount of the order . for orders less than $ 20 , shipping costs $ 4 . for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dol... | so they 're saying , can the dollar amount of the order -- so can the amount of order be represented as a function of shipping -- let me do that in that blue color costs . so if we have the shipping costs as an input , will that map to exactly -- for a given input , will we get exactly one output for the amount of orde... | should n't the input be the shipping order ? |
jada is ordering mother 's day gifts online . the shipping costs are based on the dollar amount of the order . for orders less than $ 20 , shipping costs $ 4 . for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dol... | so if we have the shipping costs as an input , will that map to exactly -- for a given input , will we get exactly one output for the amount of order ? in order for this to be represented as a function , we have to input a shipping cost , a shipping cost where this relationship is defined . we need to input a shipping ... | why is the shipping cost the input ? |
jada is ordering mother 's day gifts online . the shipping costs are based on the dollar amount of the order . for orders less than $ 20 , shipping costs $ 4 . for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dol... | so if we have the shipping costs as an input , will that map to exactly -- for a given input , will we get exactly one output for the amount of order ? in order for this to be represented as a function , we have to input a shipping cost , a shipping cost where this relationship is defined . we need to input a shipping ... | how can you tell for sure which cost is the dependent and the independent ? |
jada is ordering mother 's day gifts online . the shipping costs are based on the dollar amount of the order . for orders less than $ 20 , shipping costs $ 4 . for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dol... | so they 're saying , can the dollar amount of the order -- so can the amount of order be represented as a function of shipping -- let me do that in that blue color costs . so if we have the shipping costs as an input , will that map to exactly -- for a given input , will we get exactly one output for the amount of orde... | in the above video what exactly are the dependent and independent variables and why ? |
jada is ordering mother 's day gifts online . the shipping costs are based on the dollar amount of the order . for orders less than $ 20 , shipping costs $ 4 . for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dol... | we could do a similar thing if we input 7 into this relationship . if 7 was -- and i could put literally an infinite number of numbers . it could be a million dollars . | can a function return a range instead of a specific number ? |
jada is ordering mother 's day gifts online . the shipping costs are based on the dollar amount of the order . for orders less than $ 20 , shipping costs $ 4 . for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dol... | can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dollar amount of the order -- so can the amount of order be represented as a function of shipping -- let me do that in that blue color costs . so if we have the shipping costs as an input , will that map to ... | why is n't the amount of the order the input , and the shipping cost the output in the function ? |
jada is ordering mother 's day gifts online . the shipping costs are based on the dollar amount of the order . for orders less than $ 20 , shipping costs $ 4 . for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dol... | if someone tells you the shipping costs and you do n't necessarily know what the order size was , this is not a function . you can not represent the amount of order as a function of the shipping cost . so , no . | the amount of order is n't a function of shipping cost but shipping cost is a function of amount of order , right ? |
jada is ordering mother 's day gifts online . the shipping costs are based on the dollar amount of the order . for orders less than $ 20 , shipping costs $ 4 . for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dol... | for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dollar amount of the order -- so can the amount of order be represented as a function of shipping -- let me do that in that blue color costs . | is it accurate to say the following : for a given function y=f ( x ) , then x would be an independent variable , and therefore the domain , and y would be the dependent variable , and therefore the range ? |
jada is ordering mother 's day gifts online . the shipping costs are based on the dollar amount of the order . for orders less than $ 20 , shipping costs $ 4 . for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dol... | what are the possible inputs here ? well , there 's only two possible shipping costs . shipping costs are either going to be $ 4 , or they 're going to be $ 7 . so let 's think about what happens when we input $ 4 in as a shipping cost . so if we input $ 4 into our relationship , so we input $ 4 into our little potenti... | overall , sal is saying that just because you have to pay the shipping cost of $ 4 , it doesnt mean the price of the item is exactly one price ? |
jada is ordering mother 's day gifts online . the shipping costs are based on the dollar amount of the order . for orders less than $ 20 , shipping costs $ 4 . for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dol... | if someone tells you the shipping costs and you do n't necessarily know what the order size was , this is not a function . you can not represent the amount of order as a function of the shipping cost . so , no . | is shipping cost as a function of amount of order a function ? |
jada is ordering mother 's day gifts online . the shipping costs are based on the dollar amount of the order . for orders less than $ 20 , shipping costs $ 4 . for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dol... | in order for this to be represented as a function , we have to input a shipping cost , a shipping cost where this relationship is defined . we need to input a shipping cost , put it into our relationship , and get exactly one dollar amount of the order in order for this to be a function . if we get multiple dollar amou... | of order be the input ? |
jada is ordering mother 's day gifts online . the shipping costs are based on the dollar amount of the order . for orders less than $ 20 , shipping costs $ 4 . for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dol... | so they 're saying , can the dollar amount of the order -- so can the amount of order be represented as a function of shipping -- let me do that in that blue color costs . so if we have the shipping costs as an input , will that map to exactly -- for a given input , will we get exactly one output for the amount of orde... | so why would n't you input the amount of order and output the shipping cost ? |
jada is ordering mother 's day gifts online . the shipping costs are based on the dollar amount of the order . for orders less than $ 20 , shipping costs $ 4 . for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dol... | for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dollar amount of the order -- so can the amount of order be represented as a function of shipping -- let me do that in that blue color costs . | so is the shipping cost a function of the dollar amount of dollar ? |
jada is ordering mother 's day gifts online . the shipping costs are based on the dollar amount of the order . for orders less than $ 20 , shipping costs $ 4 . for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dol... | so because for a valid shipping cost , for each of these valid shipping costs , i can get many , many , many potential outputs , i do n't know which output it necessarily will output to . if someone tells you the shipping costs and you do n't necessarily know what the order size was , this is not a function . you can n... | why is n't the example a representation of a function ? |
jada is ordering mother 's day gifts online . the shipping costs are based on the dollar amount of the order . for orders less than $ 20 , shipping costs $ 4 . for orders $ 20 or more , shipping is $ 7 . can the dollar amount of the order be represented as a function of shipping costs ? so they 're saying , can the dol... | so let 's think about it . what are the possible inputs here ? well , there 's only two possible shipping costs . | for what jay had said it makes more sense to me now , but how can we identify inputs and outputs in general for any word problem ? |
so i 've got four cartesian coordinates here . this first one is minus 1 , 0 . i tried to draw them ahead of time . so minus 1 , 0 is this point right there . doing this in these new colors . the next point is a 0 , 1 , which is that point right there . then the next point is 1 , 2 , which is that point right up there ... | you could find a line that maybe goes through these points , but it 's not going to go through this point over here . if you try to make a line to goes through these two points , it 's not going to go through those points there . so you 're not going to be able to find a solution that goes through those points . | so , if i am correct , this is how a calculator takes a set of data points and creates a linear regression ? |
so i 've got four cartesian coordinates here . this first one is minus 1 , 0 . i tried to draw them ahead of time . so minus 1 , 0 is this point right there . doing this in these new colors . the next point is a 0 , 1 , which is that point right there . then the next point is 1 , 2 , which is that point right up there ... | so a transpose will look like this . b minus 1 , 1 , 0 , 1 , 1 , 1 , and then 2 , 1 . this first column becomes this first row ; this second column becomes this second row . | why cant we take the line equation as `` ax+by=1 '' instead of `` y=mx+b '' and find the values of a and b using least square method ? |
so i 've got four cartesian coordinates here . this first one is minus 1 , 0 . i tried to draw them ahead of time . so minus 1 , 0 is this point right there . doing this in these new colors . the next point is a 0 , 1 , which is that point right there . then the next point is 1 , 2 , which is that point right up there ... | the second equation tells us that f of 0 , which is equal to 0 times m , which is just 0 plus b is equal to 1. f of 0 is 1 . this is f of x . the next one -- let me do it in this yellow color -- tells us that f of 1 , which is equal to 1 times m , or just m , plus b , is going to be equal to 2 . | i want to fit a `` 2nd degree curve '' having equation as y=f ( x ) =a0+a1* ( x ) +a2* ( x^2 ) , then how should i proceed ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a generalized formula for the observed frequency . so that 's how fast he 's moving to the righ... | but , is n't the velocity of the wave for moving particle the sum vw and vs , because the speed of the wave would also include the speed of the object emitting the light ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | what happens if the wave source is moving faster than the waves themselves ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | so let 's just say it has some period of -- let me write it this way . the source period , which is the period of the wave from the perspective of the source , so the source period , we 'll call it t sub source , and the source frequency , which would just be -- we 've learned , hopefully it 's intuitive now -- would b... | what would happen if the source travels faster than the wave ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | what are superposition of waves ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | well , you multiply the velocity of that first pulse times the time . velocity times time is going to give you a distance . if you do n't believe me , i 'll show you an example . | and if the time is different how is the calculation going to differ ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | so that 's how fast he 's moving to the right , and he 's emitting a wave . let 's say the wave that he 's emitting -- so the velocity of wave -- let 's call that v sub w radially outward . we 've got to give a magnitude and a direction . | wavelength of a wave depends on what factors ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | when the soure of the sound is moving away form someone in a speed faster than sound , then that person will never hear the sound , right ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | this thing has also moved . how far ? well , we do the same thing . | according to this formula , would n't the observed frequency be the same in this scenario regardless of how far away the observer is from the source ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | and i 'll rewrite them , just because the observed period of the observer is going to be the period from the point of view of the source times the velocity of the wave minus the velocity of the source -- that 's the velocity of the source -- divided by the velocity of the wave itself . the frequency , from the point of... | so when vs approaches vw , does that mean the frequency observed becomes infinitely large ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | how can the doppler effect be relative , how can that make sense ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | this is all classical mechanics . we 're not going to be talking about relativistic speed , so we do n't have to worry about all of the strange things that happen as things approach the speed of light . so let 's just say it has some period of -- let me write it this way . | if two ambulances travel a the same speed together , one in front of the other , would the front ambulance and the cars in front who are driving at a slower speed hear the same noise before the ambulance passes them ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | i 'm saying it 's v sub w radially outward . this is n't a vector quantity . this is just a number you can imagine . | could n't really understand what doppler effect is ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | and i 'll rewrite them , just because the observed period of the observer is going to be the period from the point of view of the source times the velocity of the wave minus the velocity of the source -- that 's the velocity of the source -- divided by the velocity of the wave itself . the frequency , from the point of... | then , how would we get observed frequency if the observer was not at that distance ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | and this distance right here , this radius right there , that is equal to velocity times time . the velocity of that first pulse , v sub w , that 's actually the speed . i 'm saying it 's v sub w radially outward . this is n't a vector quantity . | why v sub w is not a vector here ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | we 're not going to be talking about relativistic speed , so we do n't have to worry about all of the strange things that happen as things approach the speed of light . so let 's just say it has some period of -- let me write it this way . the source period , which is the period of the wave from the perspective of the ... | why does sal divide vw by vw*ts-vs*ts to find out the period from the perspective of the observer ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | let 's say it 's moved right over here . in this video , we 're assuming that the velocity of our source is strictly less than the velocity of the wave . some pretty interesting things happen right when they 're equal , and , obviously , when it goes the other way . | what happens if the velocity of the source equals the velocity of the wave ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | so period of the wave t sub s. so after one period of the wave , if that 's 5 seconds , then we 'll say , after 5 seconds , the source has moved this far , v sub s times t sub s , and that first crest of our wave has moved that far , v sub w times t sub s. now , the time that we 're talking about , that 's the period o... | what exactly is a pitch ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | what should i do when the observer is at an angle instead of being right on the path of the source ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | let 's say it 's moved right over here . in this video , we 're assuming that the velocity of our source is strictly less than the velocity of the wave . some pretty interesting things happen right when they 're equal , and , obviously , when it goes the other way . | why does velocity of wave is higher than velocity of source ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | so although this is radially outward from the point at which it was emitted , this thing is n't standing still . we saw this in the last video . this thing has also moved . | in the above video the wave is not reflected at any place , why ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | this is all classical mechanics . we 're not going to be talking about relativistic speed , so we do n't have to worry about all of the strange things that happen as things approach the speed of light . so let 's just say it has some period of -- let me write it this way . | how would this formula be connected with the speed of sound ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | and i 'll rewrite them , just because the observed period of the observer is going to be the period from the point of view of the source times the velocity of the wave minus the velocity of the source -- that 's the velocity of the source -- divided by the velocity of the wave itself . the frequency , from the point of... | is there any other possible expression for the frequency with respect to the speed of sound ? |
i 've got this source of a wave right here that 's moving to the right at some velocity . so let 's just say that the velocity of the source -- let 's call it v sub s to the right -- so we 're really going to do what we do in the last video , but we 're going to do it in more abstract terms so we can come up with a gen... | v sub w times the period , times t of s. i know it 's abstract , but just think , this is just the distance times the time . if this was moving at 10 meters per second and if the period is 2 seconds , this is how far . it will have gone 10 meters after 2 seconds . now , this thing we said at the beginning of the video ... | i dont understand , if period time is 2 seconds , that means 1 period - which is 1 wavelength - needs 2 seconds , if speed is 5 m/s , the wavelenght should be 10 meters or not ? |
hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? my fellow grammarian , i am so glad you asked . modality is when we use these verbs to express conditions on st... | so let 's take a look at some examples . so the word must , right ? we can use the word must in a couple different ways . | require an apostrophe in the word 'its ' ? |
hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? my fellow grammarian , i am so glad you asked . modality is when we use these verbs to express conditions on st... | you can also use modal verbs to give advice to someone , as in `` you should n't do that . '' and something that is special to will , and , to a lesser extent , shall , is that they can form the future . shall used to be a lot more popular , and it still is in british english , but less so in standard american . | what is about `` will '' in future simple tense ? |
hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? my fellow grammarian , i am so glad you asked . modality is when we use these verbs to express conditions on st... | you take a word like talk , for instance , and you would say i talk , diane talks . right , you add the s. but any subject in any sentence is going to take the modal verb the same way . let me show you . | is it just me , or can `` will '' be used as a non-modal verb as well ? |
hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? my fellow grammarian , i am so glad you asked . modality is when we use these verbs to express conditions on st... | those are the modals . so here 's what modal verbs can do . thing number one , they agree with everything . | let me ask you guys , is there a limit to how many modal verbs are in a sentence ? |
hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? my fellow grammarian , i am so glad you asked . modality is when we use these verbs to express conditions on st... | you take a word like talk , for instance , and you would say i talk , diane talks . right , you add the s. but any subject in any sentence is going to take the modal verb the same way . let me show you . | what is the difference of an auxiliary verb vs a modal verb ? |
hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? my fellow grammarian , i am so glad you asked . modality is when we use these verbs to express conditions on st... | roll call , here we go . may , might , must , can , could , shall , should , will , would . those are the modals . | is might and could is used as may and can ? |
hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? my fellow grammarian , i am so glad you asked . modality is when we use these verbs to express conditions on st... | it 's all the same . they agree with everything . something else that modals do is actually something that they do n't do . | are the modal verb really agree with everything ? |
hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? my fellow grammarian , i am so glad you asked . modality is when we use these verbs to express conditions on st... | you should learn how to use them and in fact if you can complete the exercises , then you will master these tricky verbs . you can learn anything . david out . | speaking of `` you can learn anything '' , is `` you might learn anything '' or `` you will learn anything '' both correct ? |
hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? my fellow grammarian , i am so glad you asked . modality is when we use these verbs to express conditions on st... | those are the modals . so here 's what modal verbs can do . thing number one , they agree with everything . | did there use to be more modal verbs in the old times , such as in the 1700 's or anytime else ? |
hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? my fellow grammarian , i am so glad you asked . modality is when we use these verbs to express conditions on st... | hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? | is `` modal verbs '' equal to `` helping verbs '' ? |
hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? my fellow grammarian , i am so glad you asked . modality is when we use these verbs to express conditions on st... | roll call , here we go . may , might , must , can , could , shall , should , will , would . those are the modals . | is `` could '' the past tense of `` can '' ? |
hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? my fellow grammarian , i am so glad you asked . modality is when we use these verbs to express conditions on st... | so , if you imagine a detective looking up from some tracks , some muddy bootprints on the floor , with her magnifying glass . she looks up and she says `` he must have gone that way ! '' that detective is using must to express a likelihood , a high likelihood . | in 5 he says i will have been eating cookies , does that make sense ? |
hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? my fellow grammarian , i am so glad you asked . modality is when we use these verbs to express conditions on st... | by a similar token , if you enter someone 's house and they demand that you take off your shoes , they would say `` you must remove your shoes . '' similarly with may , we can use may to express a possibility , like saying `` it may rain . '' that 's something that could possibly happen . | i know i 'm probably wrong , but for some reason i feel like the words `` may '' and `` might '' pretty much do the same thing in a sentence.for example `` it may rain '' and `` it might rain '' is this true ? |
hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? my fellow grammarian , i am so glad you asked . modality is when we use these verbs to express conditions on st... | you ca n't say i want to should . that is an illegal operation in english . it is ungrammatical . | if 0 the use of to ( modal word ) is an illegal operation in sentence structure , does that mean that shakespearian authors , which use to shall in every other publication , are committing an illegal action in sentence structure ? |
hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? my fellow grammarian , i am so glad you asked . modality is when we use these verbs to express conditions on st... | those are the modals . so here 's what modal verbs can do . thing number one , they agree with everything . | what is the hardest type of sentence you people can use with modal verbs ? |
hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? my fellow grammarian , i am so glad you asked . modality is when we use these verbs to express conditions on st... | so let 's take a look at some examples . so the word must , right ? we can use the word must in a couple different ways . | require an apostrophe in the word 'its ' ? |
hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? my fellow grammarian , i am so glad you asked . modality is when we use these verbs to express conditions on st... | you take a word like talk , for instance , and you would say i talk , diane talks . right , you add the s. but any subject in any sentence is going to take the modal verb the same way . let me show you . | in the sentence , `` i do not understand , '' is the word `` do '' a helping verb or a main verb ? |
hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? my fellow grammarian , i am so glad you asked . modality is when we use these verbs to express conditions on st... | those are the modals . so here 's what modal verbs can do . thing number one , they agree with everything . | at when david is giving a list of modal verbs , are those all of the modal verbs in our english language or are there more modal verbs in addition to what david listed ? |
hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? my fellow grammarian , i am so glad you asked . modality is when we use these verbs to express conditions on st... | or to give someone permission , like `` you may enter . '' you can use a word like can to express ability . you could say `` i can eat ten pounds of broccoli ! '' | why the word verb look like uerb a bit ? |
hello grammarians ! today we 're gon na talk about a class of auxiliary , or helper verbs , called the modal verbs . these are verbs that have special properties and help other verbs . but what is modality ? my fellow grammarian , i am so glad you asked . modality is when we use these verbs to express conditions on st... | this must have happened . but if you 're at an amusement park and the roller coaster you want to ride has a height requirement , the sign probably says `` you must be this tall to ride the doom-coaster . '' this is like a necessary condition . | should i ask who wants to ride the doom-coaster ? |
let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actuall... | so then i have my x to the fourth term . so it is 2x to the fourth . and we are done . | how did the first term of 2x^5 go to 2x^4 ? |
let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actuall... | it 's going to be -- and let me work backwards . so i 'll start with our remainder . so our remainder is 487 . and it 's going to be 487 over x minus 3 . | my teachers taught me to draw a line between the remainder and the rest before we solve so we know its the remainder does this make a difference ? |
let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actuall... | and then i have a positive 3 times x squared . negative 2 times x . and then i have a constant term , or zero degree term , of 7 . | in one question , there was a polynomial ; 9a^2 - 30ab + 25b^2 divided by 3a - 5b , what do i do then ? |
let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . but here it 's going to be another just , let 's go through the process of it just so that you get comfortable with it . and now is a good chance to give it a shot , to actuall... | let 's do another synthetic division example . and in another video , we actually have the why this works relative to algebraic long division . | does anyone know how/who synthetic divison was thought of/who though of it ? |
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