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let 's now generalize what we did in the last video . so if this is my y-axis and this is -- that 's not that straight . this right over here is my x-axis . and let 's say i have two functions . so i 'm just going to say it in general terms . so let 's say i have a function right over here . so let 's say it looks some... | so if we rotate this thing around the x-axis , we end up with a washer . that 's why we 're going to call this the washer method . and it 's really just kind of the disk method , where you 're gutting out the inside of a disk . | how do we know when to use the washer method or the disc method ? |
let 's now generalize what we did in the last video . so if this is my y-axis and this is -- that 's not that straight . this right over here is my x-axis . and let 's say i have two functions . so i 'm just going to say it in general terms . so let 's say i have a function right over here . so let 's say it looks some... | of pi times -- what 's f of x squared ? f of x squared , square root of x squared is just x , minus g of x squared . g of x is x , that squared is x squared . and then we multiply times dx . | does using [ f ( x ) - g ( x ) ] ^2 for the squared radius give you the volume if the axis of rotation is y=g ( x ) ? |
let 's now generalize what we did in the last video . so if this is my y-axis and this is -- that 's not that straight . this right over here is my x-axis . and let 's say i have two functions . so i 'm just going to say it in general terms . so let 's say i have a function right over here . so let 's say it looks some... | f of x squared , square root of x squared is just x , minus g of x squared . g of x is x , that squared is x squared . and then we multiply times dx . | we know that sqrt ( x ) is above x in the coordinate plane but what if , instead of integral of sqrt ( x ) - x , we integrate x - sqrt ( x ) and revolve that around the x-axis ? |
let 's now generalize what we did in the last video . so if this is my y-axis and this is -- that 's not that straight . this right over here is my x-axis . and let 's say i have two functions . so i 'm just going to say it in general terms . so let 's say i have a function right over here . so let 's say it looks some... | so what 's the area of the inside ? this part right over here ? well , we 're going to subtract it out . | so what happens when you do n't know the equation of the part that creates a washer ? |
let 's now generalize what we did in the last video . so if this is my y-axis and this is -- that 's not that straight . this right over here is my x-axis . and let 's say i have two functions . so i 'm just going to say it in general terms . so let 's say i have a function right over here . so let 's say it looks some... | so let me see how well i can draw this . so depth dx . that 's the side of this washer . | is dx the same for all the washers ? |
let 's now generalize what we did in the last video . so if this is my y-axis and this is -- that 's not that straight . this right over here is my x-axis . and let 's say i have two functions . so i 'm just going to say it in general terms . so let 's say i have a function right over here . so let 's say it looks some... | so let 's say it looks something like this . so that 's one function . so this is y is equal to f of x . | to do the washer method , does one function have to be greater than the other over the entire interval for which the shape 's volume is being calculated ? |
let 's now generalize what we did in the last video . so if this is my y-axis and this is -- that 's not that straight . this right over here is my x-axis . and let 's say i have two functions . so i 'm just going to say it in general terms . so let 's say i have a function right over here . so let 's say it looks some... | so if we rotate this thing around the x-axis , we end up with a washer . that 's why we 're going to call this the washer method . and it 's really just kind of the disk method , where you 're gutting out the inside of a disk . | what 's the big difference between washer , shell and disc method ? |
let 's now generalize what we did in the last video . so if this is my y-axis and this is -- that 's not that straight . this right over here is my x-axis . and let 's say i have two functions . so i 'm just going to say it in general terms . so let 's say i have a function right over here . so let 's say it looks some... | so if this is my y-axis and this is -- that 's not that straight . this right over here is my x-axis . and let 's say i have two functions . | the equation seems to work only when rotating about the x-axis , so what about when it is rotated about the y-axis or when y=-3 ? |
let 's now generalize what we did in the last video . so if this is my y-axis and this is -- that 's not that straight . this right over here is my x-axis . and let 's say i have two functions . so i 'm just going to say it in general terms . so let 's say i have a function right over here . so let 's say it looks some... | but this is our interval we 're saying in general terms , from a to b . and this will give our volume . this right over here is the volume of each washer . | why the calculated volume is exact and not an approximation ? |
hello everyone . so i 'm talking about how to find the tangent plane to a graph . and i think the first step of that is to just figure out how we control planes in three dimensions in the first place . so what i have pictured here is a red dot representing a point in three dimensions , and the coordinates of that poin... | and the reason i 'm doing this , notice this does n't change the partial derivative information , it 's just if we take the partial derivative with respect to x , this will still be two , and when we take it with respect to y , this will still be one . but the reason i 'm putting these in here , is because we 're gon n... | grant , what programme are you using to make these 3d plots ? |
( piano music ) beth : when you walk into the gallery , this painting just stands out . it almost seems to glow . steven : the hands are unbelievable . beth : he does seem particularly interested in hands . we follow them from the figure of joseph holding out his hands , looking questioning , and then the delicate way ... | the wealth of the king 's crown sits neglected against that rock . he 's far more interested in christ and in his devotion to the divine than that worldly wealth , that worldly power , but it 's so interesting , because christ , in mary 's lap , is not actually paying attention to the king . christ stares out directly ... | why is mary so pale ? |
( piano music ) beth : when you walk into the gallery , this painting just stands out . it almost seems to glow . steven : the hands are unbelievable . beth : he does seem particularly interested in hands . we follow them from the figure of joseph holding out his hands , looking questioning , and then the delicate way ... | beth : he does seem particularly interested in hands . we follow them from the figure of joseph holding out his hands , looking questioning , and then the delicate way that mary holds up christ 's left hand , and then the figure of the first magi , whose hands are in prayer as he worships the christ child . steven : th... | why is joseph so old ? |
( piano music ) beth : when you walk into the gallery , this painting just stands out . it almost seems to glow . steven : the hands are unbelievable . beth : he does seem particularly interested in hands . we follow them from the figure of joseph holding out his hands , looking questioning , and then the delicate way ... | the wealth of the king 's crown sits neglected against that rock . he 's far more interested in christ and in his devotion to the divine than that worldly wealth , that worldly power , but it 's so interesting , because christ , in mary 's lap , is not actually paying attention to the king . christ stares out directly ... | why does mary always wear dark blue ? |
( piano music ) beth : when you walk into the gallery , this painting just stands out . it almost seems to glow . steven : the hands are unbelievable . beth : he does seem particularly interested in hands . we follow them from the figure of joseph holding out his hands , looking questioning , and then the delicate way ... | beth : yeah , that really seems like skin . steven : does n't it ? steven : look at the way we can see the light on his palm , between his fingers . | why does n't christ have a penis ? |
( piano music ) beth : when you walk into the gallery , this painting just stands out . it almost seems to glow . steven : the hands are unbelievable . beth : he does seem particularly interested in hands . we follow them from the figure of joseph holding out his hands , looking questioning , and then the delicate way ... | ( piano music ) beth : when you walk into the gallery , this painting just stands out . it almost seems to glow . | is there any symbolic significance in the choice of painting iris and columbine flowers ? |
( piano music ) beth : when you walk into the gallery , this painting just stands out . it almost seems to glow . steven : the hands are unbelievable . beth : he does seem particularly interested in hands . we follow them from the figure of joseph holding out his hands , looking questioning , and then the delicate way ... | steven : the hands are unbelievable . beth : he does seem particularly interested in hands . we follow them from the figure of joseph holding out his hands , looking questioning , and then the delicate way that mary holds up christ 's left hand , and then the figure of the first magi , whose hands are in prayer as he w... | especially the faces in the background seem to have gotten less of the artist 's attention to detail ... no ? |
let me do a little experiment . let 's say i have oxygen here , and we know that oxygen is about 21 % of the atmosphere . and i decide to take a cup , let 's say a cup like this -- simple cup of water . and i leave it out on the counter . and it 's about room temperature , about 25 degrees celsius here . and i want to ... | so this is my calculation for oxygen . and so far , so good , right ? but now i decide to challenge myself and say , let 's do this again . | the h2co3 is carbonic acid , right ? |
let me do a little experiment . let 's say i have oxygen here , and we know that oxygen is about 21 % of the atmosphere . and i decide to take a cup , let 's say a cup like this -- simple cup of water . and i leave it out on the counter . and it 's about room temperature , about 25 degrees celsius here . and i want to ... | i would actually measure the concentration there , and that 's the measure of oxygen . so i 've learned about henry 's law , and i can think well , you know , i know the partial pressure now . and i can rearrange the formula so that it looks something like this . | i know you are meant to divide partial pressure by concentration to get kh but how exactly did you get 769 and 29 ? |
let me do a little experiment . let 's say i have oxygen here , and we know that oxygen is about 21 % of the atmosphere . and i decide to take a cup , let 's say a cup like this -- simple cup of water . and i leave it out on the counter . and it 's about room temperature , about 25 degrees celsius here . and i want to ... | now , you might say , well , that 's fine for 25 degrees celsius . but what about body temperature ? what 's happening in our actual body ? | the difference in the ratios of co2 solubility to o2 solubility between room temperature and body temperature- is that more due to co2 becoming less soluble at warmer temps , or o2 becoming more soluble ? |
let me do a little experiment . let 's say i have oxygen here , and we know that oxygen is about 21 % of the atmosphere . and i decide to take a cup , let 's say a cup like this -- simple cup of water . and i leave it out on the counter . and it 's about room temperature , about 25 degrees celsius here . and i want to ... | and so these k sub h values are actually temperature dependent . and they 're going to change as you increase the temperature . so at this new temperature , it turns out that carbon dioxide is about 22 times more soluble than oxygen . | and if you have hypothermia , is the increase in co2 solubility or decrease in o2 solubility pronounced enough to lead to acidosis or hypoxia , respectively ? |
let me do a little experiment . let 's say i have oxygen here , and we know that oxygen is about 21 % of the atmosphere . and i decide to take a cup , let 's say a cup like this -- simple cup of water . and i leave it out on the counter . and it 's about room temperature , about 25 degrees celsius here . and i want to ... | let me do a little experiment . let 's say i have oxygen here , and we know that oxygen is about 21 % of the atmosphere . | does that contribute at all to death from hypothermia ? |
let me do a little experiment . let 's say i have oxygen here , and we know that oxygen is about 21 % of the atmosphere . and i decide to take a cup , let 's say a cup like this -- simple cup of water . and i leave it out on the counter . and it 's about room temperature , about 25 degrees celsius here . and i want to ... | and on this side , on the first side of our experiment , we had lots of oxygens leaving this water . they did n't like being in water . they were leaving readily . | is n't most of the o2 carried by our blood carried bound to hemoglobin ? |
let me do a little experiment . let 's say i have oxygen here , and we know that oxygen is about 21 % of the atmosphere . and i decide to take a cup , let 's say a cup like this -- simple cup of water . and i leave it out on the counter . and it 's about room temperature , about 25 degrees celsius here . and i want to ... | so in our lungs , we have 37 degrees celsius . and instead of water -- actually , i should n't be writing water -- instead of water , it 's blood , which is slightly different than water . the consistency is different . | so why do we care about the solubility of o2 in the blood ? |
let me do a little experiment . let 's say i have oxygen here , and we know that oxygen is about 21 % of the atmosphere . and i decide to take a cup , let 's say a cup like this -- simple cup of water . and i leave it out on the counter . and it 's about room temperature , about 25 degrees celsius here . and i want to ... | the consistency is different . and so these k sub h values are actually temperature dependent . and they 're going to change as you increase the temperature . | what does the subscript h represent in kh ? |
let me do a little experiment . let 's say i have oxygen here , and we know that oxygen is about 21 % of the atmosphere . and i decide to take a cup , let 's say a cup like this -- simple cup of water . and i leave it out on the counter . and it 's about room temperature , about 25 degrees celsius here . and i want to ... | because look , these partial pressures are basically the same . i mean , not even basically , they 're exactly the same . there 's no difference in the partial pressure . | when rishi says `` soluble '' throughout the video , does he mean more likely to enter the liquid ? |
let me do a little experiment . let 's say i have oxygen here , and we know that oxygen is about 21 % of the atmosphere . and i decide to take a cup , let 's say a cup like this -- simple cup of water . and i leave it out on the counter . and it 's about room temperature , about 25 degrees celsius here . and i want to ... | we only had a little bit of oxygen but now i 've got tons of carbon dioxide . and i do n't want to make it uneven . i mentioned before , we have nitrogen -- so let me still draw a bunch of nitrogen -- that will outnumber the carbon dioxide dramatically . | i am wondering why we want to have a higher co2 level in our blood than o2 ? |
let me do a little experiment . let 's say i have oxygen here , and we know that oxygen is about 21 % of the atmosphere . and i decide to take a cup , let 's say a cup like this -- simple cup of water . and i leave it out on the counter . and it 's about room temperature , about 25 degrees celsius here . and i want to ... | i could say , well , 769 divided by 29 equals about 26 . so that 's another way of saying that carbon dioxide is 26 times more soluble than oxygen . i 'll put that in parentheses -- than oxygen . | does the body deal with it in another way , or is it ok for a patient to handle high levels of co2 for a while in return for not dying of respiratory failure ? |
let me do a little experiment . let 's say i have oxygen here , and we know that oxygen is about 21 % of the atmosphere . and i decide to take a cup , let 's say a cup like this -- simple cup of water . and i leave it out on the counter . and it 's about room temperature , about 25 degrees celsius here . and i want to ... | let me do a little experiment . let 's say i have oxygen here , and we know that oxygen is about 21 % of the atmosphere . | what factors does the time to saturate/equilibriate depends , under such a experiment ? |
let me do a little experiment . let 's say i have oxygen here , and we know that oxygen is about 21 % of the atmosphere . and i decide to take a cup , let 's say a cup like this -- simple cup of water . and i leave it out on the counter . and it 's about room temperature , about 25 degrees celsius here . and i want to ... | so i 'm going to write the units of the kh down here . i can say , well , 769 liters times atmospheres over moles . and that 's something that i 've just calculated . | in from where does 769 come from ? |
let me do a little experiment . let 's say i have oxygen here , and we know that oxygen is about 21 % of the atmosphere . and i decide to take a cup , let 's say a cup like this -- simple cup of water . and i leave it out on the counter . and it 's about room temperature , about 25 degrees celsius here . and i want to ... | let me do a little experiment . let 's say i have oxygen here , and we know that oxygen is about 21 % of the atmosphere . and i decide to take a cup , let 's say a cup like this -- simple cup of water . | how can low partial pressure of oxygen shift the oxygen dissociation curve to the right ? |
so we 're told let g be a differentiable function defined over the closed interval from negative six to six . the graph of its derivative , so they 're giving the graphing the derivative of g , g prime is given below . so this is n't the graph of g. this is the graph of g prime . what is the x value of the left-most i... | and this is our actual , i guess you could call it , the original function . so an inflection point are points where our second derivative is switching sides . it 's going from positive negative or negative to positive . | can we still tell if there is an inflection point if there is a node in the first derivative ? |
so we 're told let g be a differentiable function defined over the closed interval from negative six to six . the graph of its derivative , so they 're giving the graphing the derivative of g , g prime is given below . so this is n't the graph of g. this is the graph of g prime . what is the x value of the left-most i... | so , if you look at the choices , if we want to answer the original question , well the left-most one is it x is equal to negative three ? it 's x equals negative three . x equals negative one is indeed a x value , where we have an inflection point . and let 's see , x equals two is one , and so is x equals four . | what do we call the plateaus in a f ' ( x ) or first derivative curve ( places where the curve briefly goes flat relative to the x axis similarly to a horizontal tangent of the f ( x ) curve but without creating a max or min point ) ? |
so we 're told let g be a differentiable function defined over the closed interval from negative six to six . the graph of its derivative , so they 're giving the graphing the derivative of g , g prime is given below . so this is n't the graph of g. this is the graph of g prime . what is the x value of the left-most i... | so , if you look at the choices , if we want to answer the original question , well the left-most one is it x is equal to negative three ? it 's x equals negative three . x equals negative one is indeed a x value , where we have an inflection point . and let 's see , x equals two is one , and so is x equals four . | before the point x= -3 , if the function was increasing suddenly how it can become 0 at x = -3 , is it not decreasing there ? |
so we 're told let g be a differentiable function defined over the closed interval from negative six to six . the graph of its derivative , so they 're giving the graphing the derivative of g , g prime is given below . so this is n't the graph of g. this is the graph of g prime . what is the x value of the left-most i... | and let 's see , x equals two is one , and so is x equals four . so they actually listed , all of these are inflection points . and they just wanted the left-most one . | so the change in direction of the curves define where the inflection points would be on the original problem ? |
beth harris : it 's often the case that when you 're traveling with the baby , it demands a lot of attention . and that 's what 's happening in this small painting by degas called `` at the races in the countryside . '' steven zucker : and that infant is clearly the center of this family 's attention . beth harris : an... | i mean , look at the way he 's cropped the wagon wheels . he 's cropped the horses . they do n't have the bottom of their legs . | why was some of the painting- such as the legs of the horses- cut off ? |
beth harris : it 's often the case that when you 're traveling with the baby , it demands a lot of attention . and that 's what 's happening in this small painting by degas called `` at the races in the countryside . '' steven zucker : and that infant is clearly the center of this family 's attention . beth harris : an... | steven zucker : and actually you can see her breast is exposed , and all of the figures are looking down at the child . so issues of class are very much built into this painting . beth harris : an upper class women would certainly not reveal herself in this way . | why is the painting so blurry ? |
beth harris : it 's often the case that when you 're traveling with the baby , it demands a lot of attention . and that 's what 's happening in this small painting by degas called `` at the races in the countryside . '' steven zucker : and that infant is clearly the center of this family 's attention . beth harris : an... | and there is all of that attention on the infant , and whether it 's going to eat , and whether it 's going to stop being fussy . historically , upper class women often did n't nurse their own children and would hire women who were known as wet nurses who would nurse infants . steven zucker : so a very intimate kind of... | why did upper-class women hire wet nurses ? |
beth harris : it 's often the case that when you 're traveling with the baby , it demands a lot of attention . and that 's what 's happening in this small painting by degas called `` at the races in the countryside . '' steven zucker : and that infant is clearly the center of this family 's attention . beth harris : an... | and there is all of that attention on the infant , and whether it 's going to eat , and whether it 's going to stop being fussy . historically , upper class women often did n't nurse their own children and would hire women who were known as wet nurses who would nurse infants . steven zucker : so a very intimate kind of... | do we have historical confirmation of the identity of the women in the carriage ? |
beth harris : it 's often the case that when you 're traveling with the baby , it demands a lot of attention . and that 's what 's happening in this small painting by degas called `` at the races in the countryside . '' steven zucker : and that infant is clearly the center of this family 's attention . beth harris : an... | beth harris : it 's really very sweet . and there is all of that attention on the infant , and whether it 's going to eat , and whether it 's going to stop being fussy . historically , upper class women often did n't nurse their own children and would hire women who were known as wet nurses who would nurse infants . | why did people stop wearing hats ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | so if i were to draw its derivative , its derivative would look something like this . its derivative looks something like this . but then something interesting happens at x equals negative 2 . | like can you write an equations showing how certain point will not be able to find a derivative ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | so we do not have the same limit of the secant slope as we approach from the left- and right-hand sides . so at both of these points we see the derivative jump , and it looks like f of x is not differentiable . | so , essentially , a function is not differentiable at points of discontinuity ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | the slope is 0 here . so right as x crosses 3 , the slope becomes 0 . so we immediately see there are points where it looks like the slope jumps . | the question is that if we divide the function in the graph into 3parts : y1 ( x < = -2 ) , y2 ( -2 < x < =3 ) , and y3 ( x > 3 ) , are the points ( -2 , -3 ) and ( 3 , 4.5 approximately ) still not differentiable ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | so we immediately see there are points where it looks like the slope jumps . and at these points we really do n't have a defined derivative . the slope jumps there as well . | then will the value of derivative be defined there ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | it looks like the slope is a constant negative 2 . so if i were to draw its derivative , its derivative would look something like this . its derivative looks something like this . | so is differentiability essentially continuity of the derivative of a function ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | the slope jumps there as well . and so at what arguments is f not differentiable ? well , it 's not differentiable when x is equal to negative 2 . when x is equal to negative 2 , we really do n't have a slope there . | so when you say when a function is not differentiable , is it the same thing as saying where the graph is not continuous ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . | how did sal get the lines of f ' ( x ) ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | and it looks like right out the get go , if i were to estimate the slope of its tangent line , it starts changing . it 's not a line anymore . it 's a curve . | what is a secant line ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . | at the points on that f ( x ) where the function is not differentiable ( when x = -2 and 2 ) , do limits still exist at those points ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | and it looks like it continues to get lower all the way until you get to x equals 3 . so it looks like the slope of the line is -- it looks like it 's getting lower at a constant rate , i guess i could say . so it looks like it 's doing something like this over this interval . | and could someone please explain the difference between continuity and differentiability ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . | how will you know the value of f ' ( x ) ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | so if we look at f of x right over here , it looks like its slope is pretty much consistently negative 2 over this interval between x equals , i guess , it 's like negative 8 and 1/2 all the way up to x equals negative 2 . it looks like the slope is a constant negative 2 . so if i were to draw its derivative , its deri... | example seconds , how did you know the derivative is a constant -2 ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its slope is pretty much consistently negative 2 over this interval between x equals , i guess , it 's like negative 8 and 1/2 all the way up to x equals negative 2 . it looks like the slope is a constant neg... | should n't the line representing the derivative move upwards ( positive ) towards x=2 ( where is would be 0 ) then downwards ( negative ) to x=3 ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | so i 'm just trying to , obviously , estimate it . so it looks like the slope goes up to 3 and 1/2 right when i cross that point . and then the slope becomes lower and lower and lower all the way until i get to this point right over here , all the way until i get to x equals 2 . | i thought derivatives are suppose to be tangents of a function at a point ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | when x is equal to negative 2 , we really do n't have a slope there . remember , when we 're trying to find the slope of the tangent line , we take the limit of the slope of the secant line between that point and some other point on the curve . if we did that as we approached from the left , it looks like the derivativ... | could someone explain how sal calculates exactly the start and end point of the line ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . | the function f ( x ) in this video is continuous at every point ( at least every point we can see in this graph ) , but it 's not differentiable , because for that , not the function f ( x ) itself must be continuous , but instead the function g ( a , b ) = ( f ( a ) - f ( b ) ) / ( a-b ) ) but g ( a , b ) is not the d... |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its slope is pretty much consistently negative 2 over this interval between x equals , i guess , it 's like negative 8 and 1/2 all the way up to x equals negative 2 . it looks like the slope is a constant neg... | derivative of any point between -8.5 to -2 is -2 but the slope is 0 ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | it 's a curve . the slope of the tangent line right at this point looks like it 's around -- i do n't know -- it looks like it 's around 3 and 1/2 . because if i were to draw a tangent line right over here , it looks like if i move 1 in the x direction , i move up about 3 and 1/2 in the y direction . so i 'm just tryin... | around should n't the line ( pink ) of the derivative be a constant 3.5 from x= -2 , to x=-1 and then start decreasing after -1 ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | it 's a curve . the slope of the tangent line right at this point looks like it 's around -- i do n't know -- it looks like it 's around 3 and 1/2 . because if i were to draw a tangent line right over here , it looks like if i move 1 in the x direction , i move up about 3 and 1/2 in the y direction . so i 'm just tryin... | around should n't the line ( pink ) of the derivative be a constant 3.5 from x= -2 , to x=-1 and then start decreasing after -1 ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | the slope is 0 here . so right as x crosses 3 , the slope becomes 0 . so we immediately see there are points where it looks like the slope jumps . | why not stop a x = 0 and y = 3 instead of x=-1 and y=0.5 ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . | how do i find all values of x for which the function is differentiable ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | but then right as x crosses 3 , this becomes a flat line . the slope is 0 here . so right as x crosses 3 , the slope becomes 0 . | should n't sal be drawing a line with a slope of -2 ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | it looks like the slope is a constant negative 2 . so if i were to draw its derivative , its derivative would look something like this . its derivative looks something like this . | considering derivative as the rate of change , how can we define that the rate in 'undefined ' at those point ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | because if i were to draw a tangent line right over here , it looks like if i move 1 in the x direction , i move up about 3 and 1/2 in the y direction . so i 'm just trying to , obviously , estimate it . so it looks like the slope goes up to 3 and 1/2 right when i cross that point . | obviously we see there is a lot of change , drastic , in fact , but yet rate of change is undefined ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . | tan ( x ) is differentiable at which intervals ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | the slope jumps there as well . and so at what arguments is f not differentiable ? well , it 's not differentiable when x is equal to negative 2 . when x is equal to negative 2 , we really do n't have a slope there . | if limit l exists at a where there is a removable discontinuity at a , is the function still differentiable ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | so we immediately see there are points where it looks like the slope jumps . and at these points we really do n't have a defined derivative . the slope jumps there as well . | sorry to clarify my question , a function with a removable discontinuity is not differentiable because there is no defined derivative at that point ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | so it looks like it 's doing something like this over this interval . but then right as x crosses 3 , this becomes a flat line . the slope is 0 here . so right as x crosses 3 , the slope becomes 0 . so we immediately see there are points where it looks like the slope jumps . | how u take the derivative a straight line parallel to x-axis ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | it looks like the slope is a constant negative 2 . so if i were to draw its derivative , its derivative would look something like this . its derivative looks something like this . | is sal simply graphing the values of the function 's derivative at each point ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | but then right as x crosses 3 , this becomes a flat line . the slope is 0 here . so right as x crosses 3 , the slope becomes 0 . | when we have a non differentiable function , due to a cusp , why do we consider that the slope is approaching infinity ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | so we do not have the same limit of the secant slope as we approach from the left- and right-hand sides . so at both of these points we see the derivative jump , and it looks like f of x is not differentiable . | so a function is not differentiable when the derivative has jump discontinuities ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | right as we cross x equals negative 2 , it looks like the slope goes from being negative to being positive . and it looks like right out the get go , if i were to estimate the slope of its tangent line , it starts changing . it 's not a line anymore . | how did sal construct the second slope of the tangent line without calculating ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . | is the function not differentiable only at -2 or on all x values ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . | however , if we were to have a function g ' ( x ) ( the derivative of g ( x ) ) , and there was a sharp turn at some point x=c , but g ' ( x ) is continuous , would g ( x ) be differentiable x=c ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | so i 'm just trying to , obviously , estimate it . so it looks like the slope goes up to 3 and 1/2 right when i cross that point . and then the slope becomes lower and lower and lower all the way until i get to this point right over here , all the way until i get to x equals 2 . | essentially what i 'm saying is that , if there is a sharp turn at some point in the derivatives graph , is the function differentiable at the point ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | well , it 's not differentiable when x is equal to negative 2 . when x is equal to negative 2 , we really do n't have a slope there . remember , when we 're trying to find the slope of the tangent line , we take the limit of the slope of the secant line between that point and some other point on the curve . if we did t... | if the derivative is the slope of the tangent line to any point , would n't it briefly become 0 at x = -2 ? |
consider f which is defined for all real numbers . at what arguments x is f of x not differentiable ? so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its sl... | so to think about that , i 'm actually going to try to visualize what f prime of x must look like . so i 'm going to do f prime of x in this purple color . so if we look at f of x right over here , it looks like its slope is pretty much consistently negative 2 over this interval between x equals , i guess , it 's like ... | i do not understand the differential when the x is -3. why is the purple line crossing ( 2,2 ) ? |
speaker 1 : we 're on the second floor of the metropolitan museum of art , and we 're looking at a painting by degas , `` the dance class . '' and this is a painting that was , according to the wall text , originally intended for the very first impressionist exhibition in 1874 , but not actually shown until two years ... | speaker 1 : -- of that first impressionist exhibition . speaker 2 : well , this a pretty outrageous painting , really . speaker 1 : it does't look so outrageous to us now , does it ? | why was degas so obsessed with painting ballerinas ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . | how can two line segments be parallel ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . | can two line segments or two rays be parallel lines , or can only lines be parallel ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | drag the ray so it has an endpoint at a , so we want to make its endpoint at a where the ray terminates and goes through one of the other black points . the ray should also be parallel to the pink line . so i have two options . | a ray can also be a line segment right ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | any pair of points can be connected by a line segment . that 's right . | how and why do lines intercept ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | well , yes , when i do that , it does indeed look like my ray is parallel to the pink line . and this is a ray because it has one endpoint . this is where the ray terminates . it 's an endpoint . | can a ray become parralel with another ray ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | any pair of points can be connected by a line segment . that 's right . | i know a line segment is finite , but could it be stretched so that it is almost infinite ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . | how can two segments be parallel ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point . and they look pretty parallel . | another question ; is a 'ray ' only classified as one if it starts at a point , then runs through yet another point ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | let me make these two points parallel . and these are line segments because they have two end points . they each have two end points . | when does a line truly end ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | in fact , i think this is the right answer . if we did it another way , if we had connected that point to that point and this point to this point , then it would n't look so parallel . these clearly , if they were to keep going , they would intersect at some point . | does a ray always have 1 end point ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | so let 's try to make it go through this point . well , yes , when i do that , it does indeed look like my ray is parallel to the pink line . and this is a ray because it has one endpoint . | so you said that parallel lines can never touch , well can there be a parallel and perpendicular line put together ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | let me make these two points parallel . and these are line segments because they have two end points . they each have two end points . | what are some real life line segments , lines or rays that we might see everyday ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | let 's do one more . drag the ray so it has an endpoint at a , so we want to make its endpoint at a where the ray terminates and goes through one of the other black points . the ray should also be parallel to the pink line . so i have two options . | the ray has 2 points but it still goes through so it is still a ray ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | any pair of points can be connected by a line segment . that 's right . | how do i name the intersection of line and plane ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | so let me set it back up the way i did it the first time . let me make these two points parallel . and these are line segments because they have two end points . | but is there proof given that the two lines are parallel ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . | why two line segments are not parallel ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | any pair of points can be connected by a line segment . that 's right . | how many endpoints does a line have ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | it continues forever to the right . so it continues forever in one -- continues forever in one direction . | hello every one , does any one of you know what is perpendicular ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | any pair of points can be connected by a line segment . that 's right . | what points are included on a line segment ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point . and they look pretty parallel . | why is the line only have one end point because there are 2 dots ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | in fact , i think this is the right answer . if we did it another way , if we had connected that point to that point and this point to this point , then it would n't look so parallel . these clearly , if they were to keep going , they would intersect at some point . | does a ray always start at a point ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | any pair of points can be connected by a line segment . that 's right . | so does every shape have a symbol ? |
any pair of points can be connected by a line segment . that 's right . connect two pairs of black points in a way that creates two parallel line segments . so let 's see if we can do that . so i could create one segment that connects this point to this point and then another one that connects this point to this point ... | well they do n't continue forever . they continue forever in no directions , in zero directions . if it was a ray , it would continue forever in one direction . | why do lines point in zero directions ? |
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