The idea that this video explains is so profound to me. Calc2 is providing the color to the picture that Calc1 drew in black and white. I am absolutely loving this stuff. This will all end once I get to sequence and series though.....I know this in advance lol.
why did he choose to call the small interval length ds. Where does the derivative or differential fit in to all of this, what was the purpose for putting in ds, because it represents more then just the arc length but a differential ( i heard him say it in the video). Why didn't he just put an arbitrary letter to represent the small intervals of arc length instead. I feel like there's something important I'm missing here. Finally, what does a differential even mean, what does it represent?
so to clarify we are breaking the distance between those 2 points into many small streght lines and summing up their distances and to express everything in terms of dx we break ds into dx/dy ( which is its slope and if we squre we get ds ( the diagnol of a triangle)
Good but hesitates to call ds a tangent since he like others thing tangent touches at a point thus has zero length. Tries to escape by using term "loosey goosey" (ha ha). But in fact ds is a tangent and a tangent DOES touch two ADJACENT points on a curve (otherwise it could not have a definite direction). What are ADJACENT points on a curve? They are 2 points only ds apart. Otherwise its a secant.
as always brilliant, however there is an assumption that is made. The assumption is that the distance of a small curve length ds approaches the distance of the straight line connecting the endpoints of ds. Obviously this assumption must be true or the formula would not hold but I don’t know why it is the case. As Sal says, it might get a little hairy as we are speaking in loosie goosy terms here.
Isaac Rozental any function that is differentialable will satisfy this requirement. A curve that does not approach a straight line as you zoom in would necessarily have an undefined derivative.
It's amazing how simple it seems when you see the proof... Like we could've thought of that!
"Loose goosie" "The math gets hairy"
Oh how I love Khan Academy
This video makes so much more sense than my Calc 2 textbook. Thanks again Sal!
This guy who ever invented integrals is smart af.
Once in a millennium genius
The idea that this video explains is so profound to me. Calc2 is providing the color to the picture that Calc1 drew in black and white. I am absolutely loving this stuff. This will all end once I get to sequence and series though.....I know this in advance lol.
If you say it like that, Calc 3 is like the shadows or the depth of the drawing to create just a bit more complexity.
Unbelievable, may Allah Almighty bless you ever
Thank you so much. I asked for a video about this not too long ago and here it is. You made it so much clearer thanks a ton.
Thank you Sal! It seems so yeezy when u do it
Thank you so much for this video, even the small proof helped me understand what this is about. I appreciate it :)
Thanks!!
Awesome!!
Thank you
This is a dirty proof. I like it.
Very good. Thanks
bro knows his stuff
thank you very much
brilliant
He forgot to close a parent there ....4:22
NICE VID!!!!!
Thank you very much
Its pretty confusing because dx/dx is always said to be only a notation but now we use it for calculation...
tnx
why did he choose to call the small interval length ds. Where does the derivative or differential fit in to all of this, what was the purpose for putting in ds, because it represents more then just the arc length but a differential ( i heard him say it in the video). Why didn't he just put an arbitrary letter to represent the small intervals of arc length instead. I feel like there's something important I'm missing here. Finally, what does a differential even mean, what does it represent?
+purplefire5 s is arc length, ds is difference in arc length
i neeeeedddd the answer to this question :(
so to clarify we are breaking the distance between those 2 points into many small streght lines and summing up their distances and to express everything in terms of dx we break ds into dx/dy ( which is its slope and if we squre we get ds ( the diagnol of a triangle)
Good but hesitates to call ds a tangent since he like others thing tangent touches at a point thus has zero length. Tries to escape by using term "loosey goosey" (ha ha). But in fact ds is a tangent and a tangent DOES touch two ADJACENT points on a curve (otherwise it could not have a definite direction). What are ADJACENT points on a curve? They are 2 points only ds apart. Otherwise its a secant.
may Allah bless u..keep it up
Uncalled for
as always brilliant, however there is an assumption that is made.
The assumption is that the distance of a small curve length ds approaches the distance of the straight line connecting the endpoints of ds. Obviously this assumption must be true or the formula would not hold but I don’t know why it is the case.
As Sal says, it might get a little hairy as we are speaking in loosie goosy terms here.
Isaac Rozental any function that is differentialable will satisfy this requirement. A curve that does not approach a straight line as you zoom in would necessarily have an undefined derivative.
thank you very much