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: when we represent information , such as an image , such as an image , digitally , it means we must slice it up into tiny chunks . this allows us to send an image as a sequence of color symbols , and these colors can be represented as unique numbers , using some code . consider the following challenge . alice and bob... | so for this to work , you would need to introduce letter spaces , which cancel out any savings during transmission . now , how far does this compress the message compared to the original 2,000 bits ? well , we just need to calculate the number of bits per letter on average . | how is the message decompressed ? |
: when we represent information , such as an image , such as an image , digitally , it means we must slice it up into tiny chunks . this allows us to send an image as a sequence of color symbols , and these colors can be represented as unique numbers , using some code . consider the following challenge . alice and bob... | then we repeat with the next two least likely nodes , and continue merging until you have a single node at the top . finally , we label the edges in this tree with 0 or 1 in any order . now the code for each letter is just the path from the top of the tree to the given letter . | is this tree passed along with the rest of the message to allow decoding ? |
- let 's talk about microtubules in more detail . so , first we 'll discuss the structure . so , microtubules are made up of two proteins . the first is called alphatubulin , and the second similar protein is called betatubulin , and the alphatubulin and betatubulin will join together to form a dimer . a dimer 's simpl... | and here we have a microtubule , and if you recall , the diameter of a microtubule is approximately 25 nanometers . and at one end of the microtubule , it 's going to be anchored to this place called the microtubule organizing center , or in short , we could call it the mtoc , and at the other end , it 's actually real... | you mention that microtubules shorten or lengthen via removing or adding dimers at the opposite end of the mtoc ... in anaphase , if the opposite end of the mtoc is connected to kineticore fibers also connected to centromere , how can dimers be removed , shortening the interpolar microtubules if that would ( i presume ... |
- let 's talk about microtubules in more detail . so , first we 'll discuss the structure . so , microtubules are made up of two proteins . the first is called alphatubulin , and the second similar protein is called betatubulin , and the alphatubulin and betatubulin will join together to form a dimer . a dimer 's simpl... | so , one , two , three , four , five , six , seven , eight , nine pairs , and in the center , there is one lone pair , and that 's what that two refers to . so , that 's why this arrangement is called the `` 9 + 2 '' arrangement , and between the pairs of microtubules we have this protein called nexin . it helps to kee... | should n't the microtubule arrangement in the cilia/flagella be called 9x2 + 2 arrangement ( counting the number of microtubules ) or 9 + 1 arrangement ( counting the number of microtubule pairs ) , rather than 9 + 2 arrangement ? |
- let 's talk about microtubules in more detail . so , first we 'll discuss the structure . so , microtubules are made up of two proteins . the first is called alphatubulin , and the second similar protein is called betatubulin , and the alphatubulin and betatubulin will join together to form a dimer . a dimer 's simpl... | so , let 's pick one right over here . so these are the microtubules , and if i wanted to be more specific i 'd say that these are the interpolar microtubules . so i 'm just going to point out a couple more to make it clear . | did you say that interpolar microtubules radiate from kinetochore ? |
- let 's talk about microtubules in more detail . so , first we 'll discuss the structure . so , microtubules are made up of two proteins . the first is called alphatubulin , and the second similar protein is called betatubulin , and the alphatubulin and betatubulin will join together to form a dimer . a dimer 's simpl... | for example , in cells in the respiratory tract have cilia , and they beat in an upward direction , and they help push mucus up our respiratory tract and that 's why you 'll sometimes cough up mucus , and a flagellum is a tail-like projection that comes out of a cell , and it helps the cell move , and in fact , the onl... | centrosom is characteristic for eucariotic ( animal and plant cells ) and procariotic cells or not ? |
- let 's talk about microtubules in more detail . so , first we 'll discuss the structure . so , microtubules are made up of two proteins . the first is called alphatubulin , and the second similar protein is called betatubulin , and the alphatubulin and betatubulin will join together to form a dimer . a dimer 's simpl... | let 's just go through a couple of other structures . so , these microtubules that are kind of coming out of the centrioles , those are called astral microtubules , and they 're called astral microtubules because this part , the centrioles plus those fibers that are kinda coming out of it . so , each one of these is ca... | this might be very trivial , but 8 , is n't it more accurate to say that those microtubules are nucleated from the `` centrosomes '' , not `` centrioles '' ? |
- let 's talk about microtubules in more detail . so , first we 'll discuss the structure . so , microtubules are made up of two proteins . the first is called alphatubulin , and the second similar protein is called betatubulin , and the alphatubulin and betatubulin will join together to form a dimer . a dimer 's simpl... | so , if we were to cut the flagellum , or cilia for that matter , just like that , and look at a cross section of it , it would look something like this . so , it 's made up of microtubules that are in a very specific arrangement , and we call this the `` 9 + 2 '' arrangement . let 's see why it 's called this way . | how is the basal body having a 9+2 arrangement of mircotuubules ? |
- let 's talk about microtubules in more detail . so , first we 'll discuss the structure . so , microtubules are made up of two proteins . the first is called alphatubulin , and the second similar protein is called betatubulin , and the alphatubulin and betatubulin will join together to form a dimer . a dimer 's simpl... | so , coming out of the kinetichore , those blue little fibers , those are the kinetichore fibers , and then the kinetichore fibers turn into the microtubules . so , let 's pick one right over here . so these are the microtubules , and if i wanted to be more specific i 'd say that these are the interpolar microtubules . | arent the centrosomes on right side somewhat placed opposite ? |
- let 's talk about microtubules in more detail . so , first we 'll discuss the structure . so , microtubules are made up of two proteins . the first is called alphatubulin , and the second similar protein is called betatubulin , and the alphatubulin and betatubulin will join together to form a dimer . a dimer 's simpl... | different lipids that might be necessary , and even organelles , such as mitochondria . and , kinesin and dynein are able to transport substances in this direction from the soma to the synaptic terminal , but also in the other direction , going from the synaptic terminal to the soma , and this process is called axonal ... | i may have missed it but in terms of axonal transport do kinesin and dynein operate uni-directional ( ie kinesis to , dynein from soma ) or are they interchangeable ? |
- let 's talk about microtubules in more detail . so , first we 'll discuss the structure . so , microtubules are made up of two proteins . the first is called alphatubulin , and the second similar protein is called betatubulin , and the alphatubulin and betatubulin will join together to form a dimer . a dimer 's simpl... | for example , in cells in the respiratory tract have cilia , and they beat in an upward direction , and they help push mucus up our respiratory tract and that 's why you 'll sometimes cough up mucus , and a flagellum is a tail-like projection that comes out of a cell , and it helps the cell move , and in fact , the onl... | centrosom is characteristic for eucariotic ( animal and plant cells ) and procariotic cells or not ? |
- let 's talk about microtubules in more detail . so , first we 'll discuss the structure . so , microtubules are made up of two proteins . the first is called alphatubulin , and the second similar protein is called betatubulin , and the alphatubulin and betatubulin will join together to form a dimer . a dimer 's simpl... | for example , in cells in the respiratory tract have cilia , and they beat in an upward direction , and they help push mucus up our respiratory tract and that 's why you 'll sometimes cough up mucus , and a flagellum is a tail-like projection that comes out of a cell , and it helps the cell move , and in fact , the onl... | centrosom is characteristic for eucariotic ( animal and plant cells ) and procariotic cells or not ? |
- let 's talk about microtubules in more detail . so , first we 'll discuss the structure . so , microtubules are made up of two proteins . the first is called alphatubulin , and the second similar protein is called betatubulin , and the alphatubulin and betatubulin will join together to form a dimer . a dimer 's simpl... | we 'll see in a minute what that is . but anyway , they 're called interpolar microtubules because they are between the two poles of the cell , this being one pole , and this being another pole , and you can see , the interpolar microtubules are attached to our centrioles . i 'm just gon na highlight them to make it mo... | had tought us that there are 3 types of microtubules that are activating during the cell division which they are 1-interpolar mt that attach to another microtubules from the other side 2- kinetochore mt that are attached to the chromosome and finally 3- astral mt the one that is not connected to anything ? |
i want to talk about the difference between the two words -- molarity , m-o-l-a-r-i-t-y , molarity -- and a word that 's very similar , osmolarity . and i 'm going to do it with a little example , because i think examples will help make this very clear . so i 'm going to draw a box here . and any time i draw a box , ju... | and here 's our urea . so we have n't lost our urea and glucose . it 's still there . | i understand why nacl spilt into na and cl , but why did n't glucose split into o , c and h when put into water ? |
i want to talk about the difference between the two words -- molarity , m-o-l-a-r-i-t-y , molarity -- and a word that 's very similar , osmolarity . and i 'm going to do it with a little example , because i think examples will help make this very clear . so i 'm going to draw a box here . and any time i draw a box , ju... | so now , if that 's the case , let 's go back to our question of molarity . and i 'll write up here osmolarity now , osmolarity . and let 's see if we can figure out the osmolarity of each of these things . | what is the unit for osmolarity ? |
i want to talk about the difference between the two words -- molarity , m-o-l-a-r-i-t-y , molarity -- and a word that 's very similar , osmolarity . and i 'm going to do it with a little example , because i think examples will help make this very clear . so i 'm going to draw a box here . and any time i draw a box , ju... | and here 's our urea . so we have n't lost our urea and glucose . it 's still there . | why would n't glucose or urea split up ? |
i want to talk about the difference between the two words -- molarity , m-o-l-a-r-i-t-y , molarity -- and a word that 's very similar , osmolarity . and i 'm going to do it with a little example , because i think examples will help make this very clear . so i 'm going to draw a box here . and any time i draw a box , ju... | well , you 'd say , well , i have one mole of it . and i have one liter , so one mole per liter equals one molarity . and a big m represents molarity . | how does one calculate the molarity and then osmolarity of a 1.2 % nacl solution ? |
i want to talk about the difference between the two words -- molarity , m-o-l-a-r-i-t-y , molarity -- and a word that 's very similar , osmolarity . and i 'm going to do it with a little example , because i think examples will help make this very clear . so i 'm going to draw a box here . and any time i draw a box , ju... | so now , if that 's the case , let 's go back to our question of molarity . and i 'll write up here osmolarity now , osmolarity . and let 's see if we can figure out the osmolarity of each of these things . | does osmolarity have to do with osmosis ? |
- `` ana played five rounds of golf `` and her lowest score was an 80. `` the scores of the first four rounds and the lowest round `` are shown in the following dot plot . '' and we see it right over here . the lowest round she scores an 80 , she also scores a 90 once , a 92 once , a 94 once , and a 96 once . `` it was... | so hopefully that gives you some intuition . if you removed a number that 's larger than the mean your mean is , your mean is going to go down cause you do n't have that large number anymore . if you remove a number that 's lower than the mean , well you take that out , you do n't have that small number bringing the av... | here is my question : how do we know if the mean or median is going to change if we do n't have a specific number ? |
- `` ana played five rounds of golf `` and her lowest score was an 80. `` the scores of the first four rounds and the lowest round `` are shown in the following dot plot . '' and we see it right over here . the lowest round she scores an 80 , she also scores a 90 once , a 92 once , a 94 once , and a 96 once . `` it was... | not nine and 2/5 , 90 and 2/5 . so the mean is right around here . so that 's the mean of these data points right over there . and if you remove it what is the mean going to be ? | when we add a new data point which is greater than the mean , will the mean always increase or only when the data point is really huge compared to the mean ? |
- `` ana played five rounds of golf `` and her lowest score was an 80. `` the scores of the first four rounds and the lowest round `` are shown in the following dot plot . '' and we see it right over here . the lowest round she scores an 80 , she also scores a 90 once , a 92 once , a 94 once , and a 96 once . `` it was... | median is 93 . so removing the lowest data point in this case increased the median . so the median , let me write it down here . | wo n't removing an outlier be manipulating the data set ? |
- `` ana played five rounds of golf `` and her lowest score was an 80. `` the scores of the first four rounds and the lowest round `` are shown in the following dot plot . '' and we see it right over here . the lowest round she scores an 80 , she also scores a 90 once , a 92 once , a 94 once , and a 96 once . `` it was... | so the median increased by a little bit . the median increases . now what 's going to happen to the mean ? | why `` mean '' increases ? |
- `` ana played five rounds of golf `` and her lowest score was an 80. `` the scores of the first four rounds and the lowest round `` are shown in the following dot plot . '' and we see it right over here . the lowest round she scores an 80 , she also scores a 90 once , a 92 once , a 94 once , and a 96 once . `` it was... | the median over here is going to be halfway between 92 and 94 which is 93 . so the median , the median is 93 . median is 93 . | how can you find the median with out using the sides like when you cross out the max and the min ? |
- `` ana played five rounds of golf `` and her lowest score was an 80. `` the scores of the first four rounds and the lowest round `` are shown in the following dot plot . '' and we see it right over here . the lowest round she scores an 80 , she also scores a 90 once , a 92 once , a 94 once , and a 96 once . `` it was... | - `` ana played five rounds of golf `` and her lowest score was an 80. `` the scores of the first four rounds and the lowest round `` are shown in the following dot plot . '' | is 80 a good score or a bad score ? |
- `` ana played five rounds of golf `` and her lowest score was an 80. `` the scores of the first four rounds and the lowest round `` are shown in the following dot plot . '' and we see it right over here . the lowest round she scores an 80 , she also scores a 90 once , a 92 once , a 94 once , and a 96 once . `` it was... | the lowest round she scores an 80 , she also scores a 90 once , a 92 once , a 94 once , and a 96 once . `` it was discovered that ana broke some rules when she scored `` 80 , so that score '' , so i guess cheating did n't help her , `` so that score will be removed from the data set . '' so they removed that 80 right o... | i think you should change examples.. because even if ana got her score removed , by rule that score now is 0.. so in this particular case the mean should decrease because it would be ( 0+90+92+94+96 ) /5 in the final total score.. is n't it ? |
- `` ana played five rounds of golf `` and her lowest score was an 80. `` the scores of the first four rounds and the lowest round `` are shown in the following dot plot . '' and we see it right over here . the lowest round she scores an 80 , she also scores a 90 once , a 92 once , a 94 once , and a 96 once . `` it was... | `` both the mean and the median will decrease '' , nope . `` both the mean and the median will increase , `` but the mean will increase by more than the median . '' that 's exactly , that 's exactly , what happened . | so when does the median stay the same , and when does it increase ? |
- `` ana played five rounds of golf `` and her lowest score was an 80. `` the scores of the first four rounds and the lowest round `` are shown in the following dot plot . '' and we see it right over here . the lowest round she scores an 80 , she also scores a 90 once , a 92 once , a 94 once , and a 96 once . `` it was... | so hopefully that gives you some intuition . if you removed a number that 's larger than the mean your mean is , your mean is going to go down cause you do n't have that large number anymore . if you remove a number that 's lower than the mean , well you take that out , you do n't have that small number bringing the av... | who coined the term mean ? |
- `` ana played five rounds of golf `` and her lowest score was an 80. `` the scores of the first four rounds and the lowest round `` are shown in the following dot plot . '' and we see it right over here . the lowest round she scores an 80 , she also scores a 90 once , a 92 once , a 94 once , and a 96 once . `` it was... | so we 're gon na add 80 , plus 90 , plus 92 , plus 94 , plus 96 . those are our data points . and that gets us : two plus four is six , plus six is 12 . | and the median should stay in the same place , why the data ( zero points for cheating ) disappears ? |
- `` ana played five rounds of golf `` and her lowest score was an 80. `` the scores of the first four rounds and the lowest round `` are shown in the following dot plot . '' and we see it right over here . the lowest round she scores an 80 , she also scores a 90 once , a 92 once , a 94 once , and a 96 once . `` it was... | the median over here is going to be halfway between 92 and 94 which is 93 . so the median , the median is 93 . median is 93 . | how does taking a data plot ( like in the video when they took out 80 ) make the median larger ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | 8 times the preceding term . so let 's define that as a recursive function . so first define our base case . | in what circumstances would the the recursive function be more useful than the explicit ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . | why does x have to be a positive integer ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | so it 's equal to 9 times 8 -- that 's g of 2 -- times 8 again . and so you see we get the exact same results . so this is the recursive definition of this function . | is there any formula to get sum of terms in geometric sequences ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and on . | is the explicit formula of a geometric sequence a logarithmic function , which could be represented graphically ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | so let 's try a couple of them . 1 , 2 , 3 , 4 . and then see what the corresponding g of x is . | do you have to make from the 0.2 ^n-1 a notation with a numarator and dominator , like x/y ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . | where can i find the formal syntax/grammar rules for defining these functions/sets/sequences ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | so let 's see ... when x is equal to 1 , so g of 1 -- well if x equals 1 , it 's equal to 9 . it 's equal to 9 . | i 'm pretty sure this is beyond the scope of this lesson , but if i have the following recursion : $ c_n=6n*c_ { n-1 } $ , is there any way to solve the recursion and solve for c_n explicitly ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | and then see what the corresponding g of x is . g of x . so when x is equal to 1 , g of x is 9 times 8 to the 1 minus 1 power , 9 times 8 to the 0 power , or 9 times 1 . | so then how do you convert a recursive formula ( with the information of the first g ( x ) number ) into an explicit formula ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | so once again , we 're going to have x and we 're going to have g of x . but this time we 're going to use this recursive definition for g of x . and the reason why it 's recursive is it 's referring to itself . | why would one use the recursive ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | 8 times the preceding term . so let 's define that as a recursive function . so first define our base case . | is there a way of converting a recursive function to a explicit function ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | so let 's see ... when x is equal to 1 , so g of 1 -- well if x equals 1 , it 's equal to 9 . it 's equal to 9 . | since it 's a geometric sequence the x-1 should be an exponent , no ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | 8 times the preceding term . so let 's define that as a recursive function . so first define our base case . | how do you find the constant difference and recursive equation to an arithmetic sequence ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | so let 's first just try to understand the inputs and outputs here . so let 's make a little table . let 's make a table here . and let 's think about what happens when we put in various x 's into this function definition . | how do i find an explicit formula for a function with only a table and graph provided ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | so we could write that right here . times 8 . so for any other x other than 1 , g of x is equal to the previous entry -- so it 's g of -- i 'll do that in a blue color -- g of x minus 1 times 8 . | ( 8 to the second power ) ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | so let 's try a couple of them . 1 , 2 , 3 , 4 . and then see what the corresponding g of x is . | what would the third term be if a1 = 12 and an= 5an-1 - 14 , if n is greater than or is equal to 2 ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | and then see what the corresponding g of x is . g of x . so when x is equal to 1 , g of x is 9 times 8 to the 1 minus 1 power , 9 times 8 to the 0 power , or 9 times 1 . so g of x is going to be just 9 . | should the 8 or the x-1 be in parenthesis when x-1 is used as an exponent of 8 ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | so let 's try a couple of them . 1 , 2 , 3 , 4 . and then see what the corresponding g of x is . | i understand that 4+3n is a ( 1 ) =4 ; a ( n-1 ) + 4 , but why would n't a ( 1 ) = 3 ; a ( n-1 ) -6 be 3-6n ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | well g of 1 is right over here . g of 1 is 9 . so this is going to be equal to 9 times 8 . | the same as g ( n ) =n-1*ratio : is this the equation for any of these types of sequences ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | the previous entry times 8 . so we could write that right here . times 8 . | this could n't be geometrical as there is no common ratio , nor addition as there is no constant between terms , so how would it be named , and/or expressed ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | let me write it as just 9 times 8 . 9 times 8 . then when x is equal to 3 , what 's going on here ? | 9*8*8 can be simplified as 9 ( 8^2 ) , correct ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | so let 's see ... when x is equal to 1 , so g of 1 -- well if x equals 1 , it 's equal to 9 . it 's equal to 9 . | when x is one how is g ( x ) 9 because x-1 where x -1 will be = 0 hence 9^0 = 1 ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | 8 times the preceding term . so let 's define that as a recursive function . so first define our base case . | i understand how to convert an explicit function to a recursive function , but how would you do the opposite ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | and so you see we get the exact same results . so this is the recursive definition of this function . | is there a more generic term for converting an explicit function to a recursive definition ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | 8 times the preceding term . so let 's define that as a recursive function . so first define our base case . | are recursive functions only associated with arithmetic and/or geometric sequences ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | and then see what the corresponding g of x is . g of x . so when x is equal to 1 , g of x is 9 times 8 to the 1 minus 1 power , 9 times 8 to the 0 power , or 9 times 1 . | do you always have to use the letters f or g when writing g ( x ) ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | and then see what the corresponding g of x is . g of x . so when x is equal to 1 , g of x is 9 times 8 to the 1 minus 1 power , 9 times 8 to the 0 power , or 9 times 1 . | can x be negative ? |
so i have the function g of x is equal to 9 times 8 to the x minus 1 power . and it 's defined for x being a positive -- or if x is a positive -- integer . if x is a positive integer . so we could say the domain of this function , or all the valid inputs here are positive integers . so 1 , 2 , 3 , 4 , 5 , on and on and... | and then see what the corresponding g of x is . g of x . so when x is equal to 1 , g of x is 9 times 8 to the 1 minus 1 power , 9 times 8 to the 0 power , or 9 times 1 . | what is the exact definition of g ( x ) in simple terms ? |
in previous videos , we 've already seen that benzene is aromatic because it fits the following criteria . benzene contains a ring of continuously overlapping p orbitals . so each of the six carbons in benzene has a double bond to it . so each of those six carbons is sp2 hybridized , which means that each of those carb... | they 're actually delocalized . and that lone pair of electrons can now participate in resonance . but we still do n't have the exact pyrimidine structure here . | why is it that only one of the electron pairs can participate in resonance ? |
in previous videos , we 've already seen that benzene is aromatic because it fits the following criteria . benzene contains a ring of continuously overlapping p orbitals . so each of the six carbons in benzene has a double bond to it . so each of those six carbons is sp2 hybridized , which means that each of those carb... | we have a lone pair of electrons on that nitrogen . and then we also have some electrons already participating in resonance . and so this nitrogen is also sp2 hybridized . | also how are you supposed to know if electrons are localized to the atom or participate in the hybrid resonance ? |
in previous videos , we 've already seen that benzene is aromatic because it fits the following criteria . benzene contains a ring of continuously overlapping p orbitals . so each of the six carbons in benzene has a double bond to it . so each of those six carbons is sp2 hybridized , which means that each of those carb... | each one of them has a p orbital . and we now can see a little bit better that there actually are six pi electrons that can be delocalized throughout this ring . and now maybe it 's a little bit more obvious that the thymine molecule contains the pyrimidine ring . and therefore , it is aromatic and has some extra stabi... | so if we conclude that thymine is aromatic as demonstrated then is it correct to conclude that the `` native form '' or most commonly encountered form of thymine has 2 positively charged n and 2 negatively charged o ? |
in previous videos , we 've already seen that benzene is aromatic because it fits the following criteria . benzene contains a ring of continuously overlapping p orbitals . so each of the six carbons in benzene has a double bond to it . so each of those six carbons is sp2 hybridized , which means that each of those carb... | and we can go ahead and draw in the rest of the molecule as well . so there is our resonance structure for thymine . now let 's go ahead and analyze the nitrogen after we drew the resonance structure here . | is n't the first resonance structure of thymine more likely than the one that was drawn in the video ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | and then you 're done . you can graph your hyperbola . see you in the next video . | how do you know where the foci are on the graph ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | i 'm working on this hyperbola right here , not this one , and then i 'm going to just shift it later . and then let 's say multiply both sides by minus 4 and you get y squared is equal to -- see the minus cancels out with that and then 4 over 16 is x squared over 4 minus 4 and so y is equal to plus or minus square roo... | why does that make a difference when x approaches infinity we forget about the +or- 4 for us now that x is basically in the area of + or - 10 for arguments sake +or - 4 should make a huge difference ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | let 's do that . let 's draw our asymptotes . and remember , these are the asymptotes for this situation . | can somebody say what the equation to find the asymptotes is ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | let me do it over here . these get messy . x minus 1 squared is equal to 16 . | i dont get how that is the product ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | this is more important than just memorizing the formula , because it gives you an intuition of where those equations for the lines of the asymptote actually come from . because these are what this graph or this equation or this function approaches as x approaches positive or negative infinity . as x approaches positive... | why not y approaches +- infinity ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | this is more important than just memorizing the formula , because it gives you an intuition of where those equations for the lines of the asymptote actually come from . because these are what this graph or this equation or this function approaches as x approaches positive or negative infinity . as x approaches positive... | what does `` approaches '' means ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | especially if it has the same asymptotes just shifted , but centered at 0 it would look like this : x squared over 16 minus y squared over 4 is equal to 1 . and the difference between this hyperbola and this hyperbola the center of this hyperbola is at the point x is equal to 1 y is equal to minus 1 . and the way to th... | is it possible for a hyperbola to have no x and y intercepts whatsoever ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | let 's plot those 2 . so 5 , 1 2 3 4 5 , negative 1 and 3 , negative 1 . is that right ? | do the final coordinates ; ( 5 , -1 ) & ( -3 , -1 ) counted from the shifted origin , or the ( 0,0 ) origin ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | so that 's the first one . let me draw that asymptote . looks something like that , and then we draw it from this point to that point . | did n't sal forget to shift the asymptote ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | especially if it has the same asymptotes just shifted , but centered at 0 it would look like this : x squared over 16 minus y squared over 4 is equal to 1 . and the difference between this hyperbola and this hyperbola the center of this hyperbola is at the point x is equal to 1 y is equal to minus 1 . and the way to th... | what happens to the hyperbola as the numbers underneath the x and y get larger , does it just make the middle of the hyperbola further away from the place where the asymptotes meet ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems where the first point is just to identify what type of conic section we have and then the second step is actuall... | why are conic section equations always set = to one ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | take the square root of both sides . x minus 1 is equal to positive or negative 4 . and so if x is equal to positive 4 , if you add 1 to that x would be equal to 5 . and then if x minus 1 would be minus 4 and you add 1 to that you will have x is equal to 3 . | at 0 why does n't `` x-1 = +-4 '' become `` x = +- 5 '' ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | because these are what this graph or this equation or this function approaches as x approaches positive or negative infinity . as x approaches positive or negative infinity , what is y approximately equal to , in this case ? well once again , this term is going to dominate . | what does `` as x approaches positive or negative infinity ... '' means ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | and does that ever happen ? can x be equal to 1 ? if x is equal to 1 here this term is 0 . | why ca n't x = 1 ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | it 's going to look something like that . so we 've drawn our asymptotes for this function , and now we have to figure out if it 's going to be a vertical hyperbola or a horizontal hyperbola . and the easy way to think about it is to try and make -- and we can do it two ways . | what is the domain of a horizontal hyperbola ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | and then the other asymptote is going to have a minus 1/2 slope . remember this is our center 1 minus 1 , so if i go down 1 and over . so when i go over 2 , i go down 1 , so that will be right there , let me draw that asymptote . | why the point ( 1 , -1 ) is the center ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | it 's going to look something like that . so we 've drawn our asymptotes for this function , and now we have to figure out if it 's going to be a vertical hyperbola or a horizontal hyperbola . and the easy way to think about it is to try and make -- and we can do it two ways . | can the asymptotes not be vertical and horizontal , making the hyperbola tilted ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | and to figure out the asymptotes you just have to think about well what happens as x approaches positive or negative infinity . as x gets really positive or x gets really negative . and we 've done this a bunch of times already . | why sal says that x could not be negative one ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | so our 2 points or our 2 points closest to our center are the points 5 comma negative 1 and 3 comma negative 1 . let 's plot those 2 . so 5 , 1 2 3 4 5 , negative 1 and 3 , negative 1 . | is n't y equal to plus or minus sqrt of ( a^2/b^2 ) x^2 -b ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | and to figure out the asymptotes you just have to think about well what happens as x approaches positive or negative infinity . as x gets really positive or x gets really negative . and we 've done this a bunch of times already . | how do i know the difference between a and b in respect to x and y ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | and then let 's say multiply both sides by minus 4 and you get y squared is equal to -- see the minus cancels out with that and then 4 over 16 is x squared over 4 minus 4 and so y is equal to plus or minus square root of x squared over 4 minus 4 . and to figure out the asymptotes you just have to think about well what ... | is it easiest to think of a correlating to x , and b correlating to y ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | so we get minus y squared over 4 . subtracting x squared over 16 from both sides minus x squared over 16 plus 1 . i 'm working on this hyperbola right here , not this one , and then i 'm going to just shift it later . | because when you have y first ( ex : ( y^2/16 ) - ( x^2/4 ) =1 ) is 16 the distance between the focus and the vertex ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | let me draw that asymptote . looks something like that , and then we draw it from this point to that point . got to have a steady hand . | which equations includes a point on its graph that is 10 units away from one focus and 6 units away from another focus ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | let 's try that out . if y is equal to negative 1 , this term right here disappears . so when y is equal to negative 1 , you 're just left with -- x minus 1 squared over 16 is equal to 1 . | so , is it true to say that if the constant term under the radical is negative , the hyperbola will open horizontally , and if negative , it will open vertically ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | i just got rid of the shift just so i could figure out the asymptotes but of course this is the real thing that we 're trying to graph , so let me do that . this is my y-axis this is my x-axis and the center of this is at 1 negative 1 . so x is equal to 1 y is equal to minus 1 . | how is the equation modified to reflect a cutting plane that is not parallel to that axis but the angle is known ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | especially if it has the same asymptotes just shifted , but centered at 0 it would look like this : x squared over 16 minus y squared over 4 is equal to 1 . and the difference between this hyperbola and this hyperbola the center of this hyperbola is at the point x is equal to 1 y is equal to minus 1 . and the way to th... | why does sal use the equation for a hyperbola with centre at origin to get the equations of asymptotes of the hyperbola centred at ( 1 , -1 ) ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | and then you 're done . you can graph your hyperbola . see you in the next video . | why does the graph of the hyperbola never touch the asymptotes ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | that 's what i always like to do whenever i 'm graphing a hyperbola . so we get minus y squared over 4 . subtracting x squared over 16 from both sides minus x squared over 16 plus 1 . i 'm working on this hyperbola right here , not this one , and then i 'm going to just shift it later . | ( x squared over y square ) did you make x and y zero to get that ... where exactly did you do to get them ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | and y equal to minus 1 makes this whole term 0 . and on here , of course , the center is the origin . center is 0 , 0 . | why do you need to make the center at the origin to calculate the asymptotes ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | so our 2 points or our 2 points closest to our center are the points 5 comma negative 1 and 3 comma negative 1 . let 's plot those 2 . so 5 , 1 2 3 4 5 , negative 1 and 3 , negative 1 . | in most other situations when i have seen ( a-b ) ^2 or ( a+b ) ^2 , you would solve them using foil , why is this not the case here ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | this is my y-axis this is my x-axis and the center of this is at 1 negative 1 . so x is equal to 1 y is equal to minus 1 . and then the slopes of the asymptotes were positive and negative 1/2 . | why do hyperbolas and ellipses equal one ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | let me draw that asymptote . looks something like that , and then we draw it from this point to that point . got to have a steady hand . | why is it that hyperbolas are the difference between two foci and a point and ellipses are the positive distance between two foci and a point ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | and to figure out the asymptotes you just have to think about well what happens as x approaches positive or negative infinity . as x gets really positive or x gets really negative . and we 've done this a bunch of times already . | how do you know where does the asymptote cuts the x axis ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | and the easy way to think about it is to try and make -- and we can do it two ways . i mean if you just look at this equation right here . when you 're taking the positive square root , we 're always going to be slightly below the asymptote . | what does b represent in the equation ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | and then you 're done . you can graph your hyperbola . see you in the next video . | what i mean is , how can i identify b if i have the graph and the value of a but not the value of c ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | so our 2 points or our 2 points closest to our center are the points 5 comma negative 1 and 3 comma negative 1 . let 's plot those 2 . so 5 , 1 2 3 4 5 , negative 1 and 3 , negative 1 . is that right ? | in why did n't sal factor x^2/4-4 in ( x/2+2 ) ( x/2-2 ) ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | so x is equal to 1 y is equal to minus 1 . and then the slopes of the asymptotes were positive and negative 1/2 . so let 's do the positive 1/2 . | would n't the 2 asymptotes have to be perpendicular ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | i just got rid of the shift just so i could figure out the asymptotes but of course this is the real thing that we 're trying to graph , so let me do that . this is my y-axis this is my x-axis and the center of this is at 1 negative 1 . so x is equal to 1 y is equal to minus 1 . | what is a transverse axis ? |
let 's see if we can tackle a slightly more difficult hyperbola graphing problem . let 's add the hyperbola . make this up on the fly x minus 1 squared over 16 minus y plus 1 squared over 4 is equal to 1 . so the first thing to recognize is that this is a hyperbola and we 'll in a few videos , do a bunch of problems wh... | and the easy way to think about it is to try and make -- and we can do it two ways . i mean if you just look at this equation right here . when you 're taking the positive square root , we 're always going to be slightly below the asymptote . | what does a and b variables stand for in the equation of hyperbolas ? |
all right . what we 've got here are 12 pirates . they 're going to divide out a treasure chest of gold . and here 's how they 're going to do it . first pirate 's going to come along , take 1/12 of the gold that 's in the chest . second pirate 's going to come along , take 2/12 of whatever 's left after the first pira... | leaves 2/3 of this amount , takes away a factor of 3 , throws in another factor of 2 . next one 's going to come along , take 5/12 , leave 7/12 . so that 's 7 times 55 . | how come amc test is very challenging ? |
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