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so i know we talked about different pacemakers in the body , but i thought it 'd be fun to revisit that and show you an interesting example . so let 's start out by laying out the table we 'd set up before . we talked about the heart rate in beats per minute , and we talked about the heartbeat itself -- the length of t... | this very first one will get to the av node somewhere around here , because that 's 4 and 2/3 of a second . so at 4 and 2/3 of a second , this one -- let me somehow show you without making this too messy -- will make it to the av node right here . of course at that time , the sa node itself is letting out its seventh a... | are the thresholds somehow raised ? |
so i know we talked about different pacemakers in the body , but i thought it 'd be fun to revisit that and show you an interesting example . so let 's start out by laying out the table we 'd set up before . we talked about the heart rate in beats per minute , and we talked about the heartbeat itself -- the length of t... | so that 's the length of a heartbeat there . and then we have the av node . i 'm just going to quickly go through this . | are more calcium ions being added by the av node ? |
so i know we talked about different pacemakers in the body , but i thought it 'd be fun to revisit that and show you an interesting example . so let 's start out by laying out the table we 'd set up before . we talked about the heart rate in beats per minute , and we talked about the heartbeat itself -- the length of t... | and they 're going to all just continue and basically , just what are we going to get ? a total of probably 9 , right ? we 're going to get 9 signals sent off . | 0 in thinking about heartbeats : can monitor techs detect escape beats on an ekg and know that is what is happening ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . | how was pascals triangle discovered ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . | is it just coincidence that the sum of the coefficients of ( a+b ) ^n = 2^n ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . | whats the difference between a^3 + b^3 and ( a+b ) ^3 ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | and then for the second term i start at the lowest power , at zero . and then b to first , b squared , b to the third power , and then b to the fourth , and then i just add those terms together . and there you have it . | what if you have terms with exponents inside the brackets ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem --... | and also what do you do if there are multiple variables with exponents multiplied within the brackets ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? | i encountered this one reacently and became uncertain : ( 3x^3y^2z ) ^4 how would that be solved and could you use pascals triangle for this ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . one a to the fourth b to the zero : that 's just a to the fourth . | binomial theorum and pascal 's triangle ( -p+q ) ^5 my answer was -p^5 + 5p^4q - 10p^3q^2 + 10p^2q^3 - 5pq^4 -q^5 but the answer for the question was listed with the last term +q^5 my question is why is n't it -q^5 for the last term ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . | how could you figure out numbers in larger rows of pascal 's triangle ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know about the expansion . so once again let me write down what we 're trying to calculate . we 'r... | when you are trying to solve a problem with pascal 's triangle , you would have to memorize it or write it out again , correct ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . | is there a way to do the theorem with trinomials ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . | is it true that we can obtain sierpinski triangle with the help of pascals triangle ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . i could go like that , i could go like that , i could go like that , and i can go like that . | after constructing pascals triangle to a large extent , does there not become countless ways to get to every number ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . | how would i solve ( a+b ) ( a+b ) ( a+b ) instead of ( a+b ) ^2 ( a+b ) ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . | is there an equation for pascal 's triangle , say for instance if we were to calculate ( a+b ) ^100 ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . | is there an equation for pascal 's triangle , say for instance if we were to calculate ( a+b ) ^100 ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . | is there an equation for pascal 's triangle , say for instance if we were to calculate ( a+b ) ^100 ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | and then you 're going to have plus this b times that a so that 's going to be another a times b . plus b times b which is b squared . now this is interesting right over here . | is there an equation for pascal 's triangle , say for instance if we were to calculate ( a+b ) ^100 ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | it would have been useful if we did even a higher power -- a plus b to the seventh power , a plus b to the eighth power . but what i want to do in this video is show you that there 's another way of thinking about it and this would be using `` pascal 's triangle '' . and if we have time we 'll also think about why thes... | can somone show me how to expand ( x + 1 ) 1 = x + 1 please ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . | why does the binomial formula work ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | and so i guess you see that this gave me an equivalent result . now an interesting question is 'why did this work ? ' and i encourage you to pause this video and think about it on your own . | why does the binomial formula with combinatorics work ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . | this question might be stupid but , why is it that the pascal 's triangle works for figuring out leading coefficients for the binomial theorem ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . | in pascal 's triangle , is term the same thing as position in that they are both 1 greater than r , in the combination ncr ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | i have just figured out the expansion of a plus b to the fourth power . it 's exactly what i just wrote down . this term right over here , a to the fourth , that 's what this term is . | how exactly do we know that , in the video , the numbers in the bottom layer represent the coefficients of the expanded polynomial and that the layer numbers represent the exponents ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . | what does the numbers of ways of reaching to a point have to do anything with binomial expansion ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | and then there 's only one way to get to that point right over there . and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . | can we just directly add the powers like 1+3 in the 3rd level to get 4 in the fourth level in the pascal 's triangle ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | we 're going to add these together . and then when you multiply it , you have -- so this is going to be equal to a times a . you get a squared . | why is 0 factorial equal to 1 ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . | why does pascal 's triangle work and how do you know when to stop going ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . | how is 4 choose 0 = 1 ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . | what is the quotient of the problem 36 divided by 6 ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . | where did you get x to the power of 6 and y to the power of 6 ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | and so let 's add a fifth level because this was actually what we care about when we think about something to the fourth power . this is essentially zeroth power -- binomial to zeroth power , first power , second power , third power . so let 's go to the fourth power . | 0 and 8 when he adds the a to the 0th power and b to the 0th power , what would b & a be ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . | how do you do the equation ( x-2 ) ^3 using pascal 's triangle ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . | how would you figure out the coefficients for a bigger row like the tenth row for example without having to draw or do pascal 's triangle ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | so there 's two ways to get here . one way to get there , one way to get there . the only way i get there is like that , the only way i can get there is like that . | is there a similar way to calculate for binomials such as ( 2x+3y ) ^6 ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . | was pascal 's triangle made just for the binomial theorem ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . | is it a coincidence that the 1st level of pascal 's triangle is ( 11^0 ) , the 2nd level is ( 11^1 ) and so on ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | and if we have time we 'll also think about why these two ideas are so closely related . so instead of doing a plus b to the fourth using this traditional binomial theorem -- i guess you could say -- formula right over here , i 'm going to calculate it using pascal 's triangle and some of the patterns that we know abou... | are there any other applications to using the pascal 's triangle , besides for binomial expansion ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . and one way to think about it is , it 's a triangle where if you start it u... | is it possible to start pascal 's triangle with 0 instead of 1 ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | we 're trying to calculate a plus b to the fourth power -- i 'll just do this in a different color -- to the fourth power . so what i 'm going to do is set up pascal 's triangle . so pascal 's triangle -- so we 'll start with a one at the top . | what 's the difference between pascal 's triangle and the fibonacci sequence ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | so how many ways are there to get here ? well i just have to go all the way straight down along this left side to get here , so there 's only one way . there 's four ways to get here . | is there an easy way to figure out what numbers go on the next rows ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | you can multiply this a times that b , or this b times that a . there are -- just hit the point home -- there are two ways , two ways of getting an ab term . and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . | what does it mean when sal says there are two ways to get the ab term ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | and then we could add a fourth level where -- let 's see , if i have -- there 's only one way to go there but there 's three ways to go here . one plus two . how are there three ways ? | so if in two dimensions we can find a binomial , can we find a quadratic by adding another dimension ? |
in the previous video we were able to apply the binomial theorem in order to figure out what a plus b to fourth power is in order to expand this out . and we did it . and it was a little bit tedious but hopefully you appreciated it . it would have been useful if we did even a higher power -- a plus b to the seventh pow... | and so , when you take the sum of these two you are left with a squared plus two times ab plus b squared . notice the exact same coefficients : one two one , one two one . why is that like that ? | how does one denote the set of numbers in the range in sigma notation ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done . | i can not resolve this in my head : i get the math , but if at 24 years armaan is 3x as old as diya at 8 , would that mean that arman is always 3 times as old as diya ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | you 're going to be left with 2 y 's . now we can get rid of the constant term here . so we will subtract 6 from both sides . | i mean , we all experience time at a constant rate ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? | mary age is 2/3 that of peter , s .two years ago mary , s age was 1/2 of what peter , s age will be in 5 years time .how old is peter now ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . | if i am 25 years old and my little brother jonas is 5 years old , how long would it take for me to be three times jonas ' age ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . | what are their present ages ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? | benson and jackson are brother , benson age is two years less than twice jackson 's age the sum 19 years , how old each brother ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | so in y years , arman is going to be 18 plus y . and what about diya ? how old will she be in y years ? | wo n't armaan and diya have the same age difference , no matter how much time passes ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . | how long will it take for ee to be 4 times uu 's age ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | we now know that this thing is 6 . so in six years , arman is going to be 18 plus 6 , which is 24 years old . how old is diya going to be ? | how long ago in years did ship sink ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | well , let 's think about how old arman will be in y years . how old will he be ? let me write here . | how old is umaima now ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | well , let 's think about how old arman will be in y years . how old will he be ? let me write here . | how old is william now ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | well , let 's think about how old arman will be in y years . how old will he be ? let me write here . | how old is christopher now ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . | if the perimeter of the floor of the room is 90. find the length and the width of the room ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . | wait how come the equation is 8+x = 3 ( 2+y ) ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | we want to solve for y such that 18 plus y is going to be equal to 3 times 2 plus y . notice , this is arman in y years . this is diya in y years . | should n't sal have noticed he wrote `` armaan '' when it should be arman ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . | what does ''magenta '' mean ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . | a man is 4times as old as his son and five years ago the product of their ages is 234 find their present ages ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . | mrs. zamora is 36 years old and her son is 4 when will the mother be thrice as old as her son ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . | couldnt you of just multiplied 3 times the age of diya to equal 6 years ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done . | it will be 6 years before arman is 3 times as old as diya , right ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | well , let 's think about how old arman will be in y years . how old will he be ? let me write here . | if the sum of their ages is 64 , how old is joan ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | well , let 's think about how old arman will be in y years . how old will he be ? let me write here . | if the sum of their ages is 33 , how old is each person ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | well , let 's think about how old arman will be in y years . how old will he be ? let me write here . | if the sum of their ages is 47 , how old is each person ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | that 's what we 're curious about . how many years will it take for arman to be three times as old as diya ? so let 's set some variable -- let 's say , y for years . | how many years will it take until christopher is only 3 times as old as gabriela ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | and lo and behold , 24 is , indeed , three times as old as 8 . in 6 years -- arman is 24 , diya is 8 -- arman is three times as old as diya , and we are done . | since arman is the one who will be 3 times older than diya , should n't arman 's age in y years be multiplied by 3 instead ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | and on the right hand side , i can distribute this 3 . so 3 times 2 is 6 . 3 times y is 3y . 6 plus 3y . | is n't arman already greater than 3 times diya 's age ? |
let 's say that arman today is 18 years old . and let 's say that diya today is 2 years old . and what i am curious about in this video is how many years will it take -- and let me write this down -- how many years will it take for arman to be three times as old as diya ? so that 's the question right there , and i enc... | and what about diya ? how old will she be in y years ? well , she 's 2 now , and in y years , she will just be 2 plus y . | a man is 4times as old as his son and five years ago the product of their ages is 234 find their present ages ? |
hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards ... | let 's talk about how all of these work . i 'm just going to list a couple of the most famous prepositions of time and write an example sentence . so let 's go through these . | is there a limit to the amount of prepositions you can have in a sentence ? |
hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards ... | `` since 1974 , our company has made `` nothing but toasters . '' `` until '' is also precise , but it 's a precise ending time . `` you have until midnight to rescue the ambassador , break the curse and save prince wilbur . | hey david , can we use until instead of by and vice versa as both of them indicate a precise ending ? |
hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards ... | it 's a bounded duration . `` on '' has a specific connotation . it 's something that happens on a specific day . you could say something like , `` on the 4th of july , `` many americans watch fireworks and eat encaged meats . '' | do i can use 'on ' only at specific day ? |
hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards ... | hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . | is there any difference between till and until ? |
hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards ... | there 's the , oh that 's not the sun . the sun would burn a vampire . no that 's a little clock . | how would one distinguish use of since as a conjunction versus the standard conjunction of because ? |
hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards ... | anyway , okay . `` by , '' this is a really precise end time , but not a very precise beginning time . so you could say something like , `` this place had better be clean by 3 p.m. , buddy . '' | what is the difference between 'by ' and 'until ' with reference to time ? |
hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards ... | so you could say , `` i 've been a chef for 40 years . '' i have n't , but that would be cool and difficult . but you know what , this is khan academy , you can cook anything . | and respectively , would n't those usages of `` after '' and `` before '' both be considered a conjunction instead of a preposition ? |
hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards ... | and before this occurs prior to some point in time , so it is behind an action . so you can say , `` can you take out the garbage `` before you leave the house ? '' `` at '' is very precise . | but how can you take a garbage out , if you have n't leaved the house ? |
hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards ... | `` by , '' this is a really precise end time , but not a very precise beginning time . so you could say something like , `` this place had better be clean by 3 p.m. , buddy . '' if you say something like that , you 're not especially concerned that the place might be clean before three . | i 'm not a native speaker , so it 's weird to me ... but why does the word `` had '' is in the sentence `` this place had better be clean by 3 p.m. '' ? |
hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards ... | it 's a metaphor . we 're using space to represent time . anyway , i just thought that was really cool . | if the prepositions of time were used to denote space , then what spatial relation did for denote ? |
hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards ... | it 's a metaphor . we 're using space to represent time . anyway , i just thought that was really cool . | can you use a time preposition as a space preposition as well ? |
hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards ... | it 's got a tail because it 's like a little wacky cat clock . no , bat clock . that does n't have a tail . | why in in the video the bat drawing suddenly finishes the legs ? |
hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . but , if you want to become an additional time wizard , if you want to get , i do n't know , a second hat , because everyone knows all wizards ... | hello , grammarians . we are once again learning how to master time and become time wizards , which is , of course , what you will be if you master all the tenses of english . | is `` during '' a preposition ? |
we 've already looked at a carbon-hydrogen bond , and in the last video , we actually calculated an approximate wavenumber for where we would expect the signal for a carbon-hydrogen bond stretch to appear on our ir spectrum . and we got a value of a little bit over 3000 wavenumbers . however , that wavenumber depends o... | and so it can be tricky sometimes , and just glancing at that part of your spectrum . let 's think about the double bond region too . so the double bond region right up here , so this is where -- this is 1600 , that 's for this one . | in which region does the c-c bond appear ? |
we 've already looked at a carbon-hydrogen bond , and in the last video , we actually calculated an approximate wavenumber for where we would expect the signal for a carbon-hydrogen bond stretch to appear on our ir spectrum . and we got a value of a little bit over 3000 wavenumbers . however , that wavenumber depends o... | that 's a carbon-hydrogen bond , where the carbon is sp2-hydribized , so that stretch occurs at this frequency , or this wavenumber , and so we know an sp2-hybridized carbon must be present , and obviously here with this double bond , we know we have an sp2-hybridized carbon . notice the difference between this one and... | can one figure out the number of certain functional group from just analyzing the ir diagram ? |
we 've already looked at a carbon-hydrogen bond , and in the last video , we actually calculated an approximate wavenumber for where we would expect the signal for a carbon-hydrogen bond stretch to appear on our ir spectrum . and we got a value of a little bit over 3000 wavenumbers . however , that wavenumber depends o... | let 's go down here , and let 's look at some spectra . so let me just go down here and we can look at two ir spectra . the first one is for this molecule , which is decane . so hopefully i have the right number of carbons drawn there . | for example , would two c ( sp2 ) -h 's ina molecule generate somewhat different ir diagram from a molecule that has just one c ( sp2 ) -h ? |
we 've already looked at a carbon-hydrogen bond , and in the last video , we actually calculated an approximate wavenumber for where we would expect the signal for a carbon-hydrogen bond stretch to appear on our ir spectrum . and we got a value of a little bit over 3000 wavenumbers . however , that wavenumber depends o... | so this signal must be a carbon-hydrogen bond , where the carbon is sp-hybridized . and that would be right here . so we have a carbon here and we have a hydrogen right here . | would the peak broader or deeper for the molecule having two c ( sp2 ) -h ? |
we 've already looked at a carbon-hydrogen bond , and in the last video , we actually calculated an approximate wavenumber for where we would expect the signal for a carbon-hydrogen bond stretch to appear on our ir spectrum . and we got a value of a little bit over 3000 wavenumbers . however , that wavenumber depends o... | and so it can be tricky sometimes , and just glancing at that part of your spectrum . let 's think about the double bond region too . so the double bond region right up here , so this is where -- this is 1600 , that 's for this one . | what is the double bond region range ? |
we 've already looked at a carbon-hydrogen bond , and in the last video , we actually calculated an approximate wavenumber for where we would expect the signal for a carbon-hydrogen bond stretch to appear on our ir spectrum . and we got a value of a little bit over 3000 wavenumbers . however , that wavenumber depends o... | the electron density is closest to the nucleus . so , if that 's the case , then we 're talking about this bond right here , this bond being the shortest bond , because the electron density is closest to the nucleus . the more s character you have . | is a sterically hindered oh bond that has reduced h-bonding going to always show up at ~3600 ? |
we 've already looked at a carbon-hydrogen bond , and in the last video , we actually calculated an approximate wavenumber for where we would expect the signal for a carbon-hydrogen bond stretch to appear on our ir spectrum . and we got a value of a little bit over 3000 wavenumbers . however , that wavenumber depends o... | so that 's what we 're looking at there . let 's go down and look at the dot structure , and see if we can figure out what it means on the dot structure . so this signal must be a carbon-hydrogen bond , where the carbon is sp-hybridized . | if the structure was not drawn before and we had to identify what the structure were to be , how would we know ? |
let h be the vertical distance between the graphs of f and g and region s. find the rate at which h changes with respect to x when x is equal to 1.8 . we have region s right over here . you ca n't see it that well since i drew over it . and what you see when we 're in region s , the function f is greater than the func... | we could write that as h prime of , we could say h prime of x but we want the rate when x is equal to 1.8 . so h prime of 1.8 is what we wan na figure out . we could evaluate f prime of 1.8 and g prime of 1.8 and to do that we would take the derivatives of each of these things and we know how to do that . | how do i know it is h ' ( 1.8 ) ? |
let h be the vertical distance between the graphs of f and g and region s. find the rate at which h changes with respect to x when x is equal to 1.8 . we have region s right over here . you ca n't see it that well since i drew over it . and what you see when we 're in region s , the function f is greater than the func... | let me delete that . and so notice this is h of x . we have our f of x , which was 1 plus x plus e to the x squared minus 2x , and then from that we subtract g of x. g of x was a positive x to the fourth but we 're subtracting so negative x to the fourth , let me show you g of x right over here . | it makes sense when sal says it , but on a real ap test , how do i interpret the written question correctly to assume they are asking for the derivative ( h ' ( x ) ) ? |
let h be the vertical distance between the graphs of f and g and region s. find the rate at which h changes with respect to x when x is equal to 1.8 . we have region s right over here . you ca n't see it that well since i drew over it . and what you see when we 're in region s , the function f is greater than the func... | we could write that as h prime of , we could say h prime of x but we want the rate when x is equal to 1.8 . so h prime of 1.8 is what we wan na figure out . we could evaluate f prime of 1.8 and g prime of 1.8 and to do that we would take the derivatives of each of these things and we know how to do that . | how do i know it is h ' ( 1.8 ) ? |
let h be the vertical distance between the graphs of f and g and region s. find the rate at which h changes with respect to x when x is equal to 1.8 . we have region s right over here . you ca n't see it that well since i drew over it . and what you see when we 're in region s , the function f is greater than the func... | let me delete that . and so notice this is h of x . we have our f of x , which was 1 plus x plus e to the x squared minus 2x , and then from that we subtract g of x. g of x was a positive x to the fourth but we 're subtracting so negative x to the fourth , let me show you g of x right over here . | it makes sense when sal says it , but on a real ap test , how do i interpret the written question correctly to assume they are asking for the derivative ( h ' ( x ) ) ? |
today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size . and let 's talk about how it works . so first , you need to have the gel . this can be made out o... | this shows us that the yellow sample has a band that 's 200 base pairs long . and the green sample actually was composed of two different fragments , one that was 100 base pairs and another that was 200 base pairs . so what would we do with this information ? | what does it mean for the green sample to have two fragments of different sizes ? |
today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size . and let 's talk about how it works . so first , you need to have the gel . this can be made out o... | this shows us that the yellow sample has a band that 's 200 base pairs long . and the green sample actually was composed of two different fragments , one that was 100 base pairs and another that was 200 base pairs . so what would we do with this information ? if you needed a particular size of dna , say for the next st... | why would there be one fragment larger than another ? |
today , we 'll be talking about gel electrophoresis . what is gel electrophoresis , you might ask . well , it 's a lab technique usually used in the biochemistry lab for separating out dna or proteins based on their size . and let 's talk about how it works . so first , you need to have the gel . this can be made out o... | once you have these bands , what will happen once you turn on the electrical field ? remember that dna has a negatively charged backbone because of all the phosphate groups , so what you 'll actually observe is that they 'll travel towards the positive end . so maybe after a short period of time , what you 'll see is t... | i 'm very confused , i understand that dna travels towards the positive end but should n't the positive end be the cathode ? |
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