Calculus explained through a story

Calculus explained through a story

Hi, I’m Toby and welcome to another episode of the joys of mathematics today I’m not going to throw any formula or equations at you. I’m just going to tell you a story Now behind the story you might be able to find hidden away in there the essence or some of the intuition behind Calculus, but don’t worry because this is not going to be a mathematical story This is going to be a story about a guy called Bob who lives in a little Hut in some snowy mountains So I want you to just sit back and relax and enjoy listening to the story Don’t stress too much about trying to turn your brain on but hopefully you might learn something along the way So, like I said, this story is set in the mountains now, it’s a very windy and stormy day in the mountains We’ve got a lot of wind. A couple of things happen though, which are not ideal The first one is that one of the trees one of these little happy trees on the mountain over here is blown over It falls, not such a happy little tree now the second thing that happens in the windstorm Is that a little section of the bridge which is made of logs here and connects one hill to another Over a river it breaks away and falls What it’s left in its place is just a couple of wires that used to hold the logs together The logs that had broken off don’t fall straight down into the river But they are blowing sideways a little bit to this grassy meadow down here Okay, so all of this goings on sort of concerns Bob who lives in his little Hut and he comes out to check that Everything is okay Let’s draw Bob here Now this story is about the lengths that Bob goes through to do the right thing now the reason he’s so concerned that he comes out of his Hut is he knows that down here in the meadow where the Wood has fallen, in this Meadow are a couple of deer who are very dear to him they are a couple of friends of his and he’s concerned he wants to make sure that Everything is alright and that no one’s been hurt down in this meadow Now to be able to get down there and check he’s going to have to cross this bridge of logs now Obviously the problem with that is that a section of the bridge is missing Bob’s gonna have to come up with a plan He does have one idea in mind and that is that looking at one of these fallen trees beside him on the hill He thinks he might be able to use that tree lay it down along the wire that’s left in place on the bridge And create a little ramp for himself to walk down Now there’s only one problem that Bob really needs to consider about this plan And that is that you know We’ve got a curved bridge and if we lay a flat tree down on this wire that part of the bridge there is going to be straight and He doesn’t want it to end up being so steep that he just slides down it and isn’t able to climb back up to get Home, you know bob has also some very old shoes that are completely worn away on the bottom So he doesn’t have much friction coming from his shoes one thing that Bob knows from past Experience is what kind of slopes and ramps he is able to walk up with these shoes of his now Obviously walking along flat ground is going to be easy Walking straight up and down a wall is going to be impossible. But what Bob does know is that exactly in between vertical and horizontal if I was to draw a line exactly in between them there that Is the limit of how steep a ramp Bob can walk up with his shoes without slipping okay, so if we know that this little flat tree lying down here on the ramp is going to be as Steep as this or less it’ll be fine. There’ll be no problem So Bob needs to check and the way that he’s going to measure how steep the little tree ramp would be once It’s on the bridge is that he picks up a pebble. Now I’ve got one here by the camera And he holds the pebble to his eye and he looks at the hole in the bridge and he just uses the pebble to measure As sort of a measuring stick how many pebbles he think could fit in the area vertically between the missing logs so if we had a little triangle This is one log, this is the other log These are the missing logs that would have fit in along here. And this is the wire that’s just in their place holding them together He measures how many pebbles he thinks with his eye could fit down here and how many could fit across here Now what Bob measures just by using his little pebbles is that he thinks he could fit three pebbles in This height and three pebbles in this width. What it means for us Is that our slope is Exactly as tall as it is wide it also means that it’s exactly the same as this line here because this line here is As tall as it is wide So if everything is fine, and our pebbles were a good little measuring stick Then a little straight ramp here should be just steep enough for Bob to be able to handle with his shoes So let’s go ahead with it. Bob’s pretty strong and he carries the tree down to the slope and puts it in here So now a section of the bridge is our little tree but it’s pretty easy to cross over so if you’re here to learn a thing or two about mathematics then what we just did is differentiation we had a curved line Which is our entire bridge and we found the slope at a point on our curved line Just by zooming in on an area of interest and treating that area as a straight line. In fact in this case We had a tree in there So this section of our curve was treated as a straight line and we found the slope at that point So let’s keep going Bob walks along the bridge and comes down into the Meadow What are in the meadow are a couple of deer. Let’s try and draw maybe one of those on The head might be a bit hard to see, some little legs four I don’t know If deer have a tail? So Bob has gone down into the meadow And the first thing he notices is that all the deer are alive The piece of the bridge hasn’t fallen directly on to any of them. So that’s really good news However, what he does start to get concerned about is the fact that quite a lot of the grass down here is now being covered by the section of the bridge now it’s kind of concerning because he knows there’s not too much grass down here to begin with and he starts to think, you know, maybe with this grass gone and being Inedible to the deer because it’s covered in a part of the bridge Maybe the deer are going to run out of food Bob knows exactly how much grass a single deer will go through in a season so he knows how much grass is needed to be down here in this meadow and Because the meadow is just a rectangle. He knows how much grass was there originally what Bob really wants to know to make sure his little friends are going to be safe is he wants to know how much of The area of the grass is now gone He wants to subtract this amount from the total amount of grass And make sure it’s still enough to be able to feed the deer If it turns out there isn’t enough grass to keep them alive Bob will probably have to call for help to move the section of the bridge out of their meadow or look for some other solution So Bob gets down here onto the meadow and he thinks okay. I’ve got another problem This part of the bridge that fell into the meadow is a really irregular shape It’s not a circle and it’s not a square. Otherwise, I would be able to find its area quite easily So he thinks well, maybe I can approximate this as a circle Or I can use my trusty little pebble to try and measure how many circles would fit inside this Irregular shape. Now I’m going to draw it up here to make it a bit easier for us see So this is the shape of the part of the bridge that’s fallen into the meadow Like I said, it’s quite irregular because it’s not just one log It’s a few section of different logs that have fallen kind of looks like the shape of Australia now that I look at it But that’s just a coincidence. So Bob thinks okay all I know is how to find the area of something like a pebble if it’s Approximated to be perfectly round. So I’ve got a few of these pebbles I can lay them out I can get them from the riverbed and lay them along the shape and if I take my pebbles and lay them around Maybe I can lay them on top of the shape And I can just sort of count how many Fit into this shape Now that’s one way you could do it you could count up the how many pebbles Approximately equal the area of our unknown shape. Here we have one two, three, four, five six, seven eight nine pebbles This might be an OK approximation. If you find the area of these nine pebbles, you can just say that’s the area of the shape But you know, it’s not perfect. We have some overhang here so we’re saying that this area is part of the shape when it isn’t we’ve got Some gaps in here between the pebbles because they don’t Perfectly fit together and whilst this is an approximation. It’s very far from perfect so what Bob can do to get a better approximation of this area is To get some smaller pebbles now if he lays some smaller pebbles on here There’s going to be less of a gap between the pebbles so less area that’s not being measured and when a pebble does Overhang, it doesn’t go too far off our area. It’s still not perfect. We still have a bit of overhang We still have a bit of gap between our pebbles, but we’re getting a bit better So there we are. That’s what our new measuring technique would look like We’ve still got a bit of area in here. That’s not being measured correctly by these pebbles We’ve still got a tiny bit of overhang, but it’s definitely a lot better than what we had before Yeah This might be good enough People but Bob he really cares about these deer and he doesn’t want to take any chances with approximations. He wants to know exactly Will these creatures be OK, will they have enough grass, how much of their grass has been taken away from them by the bridge? We really want to know exactly okay, so this isn’t good enough for Bob He looks around him and he sees the riverbed and he sees it’s made up of sand now sand I guess you could say is a very tiny tiny pebble Bob knows that the smaller the size of the item he uses To approximate this area the more accurate it’s going to be so he thinks to himself, you know, it’s worth it I’m going to get the sand Bob lays individual grains of sand and little columns Down the shape and he has to count exactly how many grains of sand he’s placed he knows the area of each grain of sand because they’re perfectly round grains of sand and He takes his time to lay them out one by one grain by grain Along the shape it takes so long for him to do this that while he’s laying them down It’s no longer day It turns to night And then it turns to day again And then it turns to night but Bob keeps going because he is really dedicated to these creatures and He wants to do what he can to be as accurate as possible. A couple of days later we are all done it’s back to Sun and we’ve Completely filled our shape with grains of sand if you ask Bob how many grains of sand he laid down on that shape He might say it seemed like an infinite number But we were just getting very close to infinity There’s certainly hundreds of thousands of little grains of sand on that shape But the good news is that bob has an almost perfect knowledge of the area of this shape here So he pulls out his notepad. He works out the area of the original garden He subtracts his now known area of the shape here He works out how much grass would have been needed to feed a couple of these deer down in the meadow he works it all out. Luckily, he finds that even with the amount of grass that’s missing There is still going to be just enough grass to feed everyone in the meadow So this is a really good news and Bob feels really great With this peace of mind that people are going to have enough to eat with that Bob climbs back up the hill walks across the bridge across our little ramp made from a tree and he walks home for the night He had a pretty big excursion, but at least he’s happy that everyone is ok now what we did here when we calculated the area of the shape was actually integration or the essence of integration. What integration means is that we can find the area of any shape by Slicing it up into infinitely small pieces, and this is a pretty useful thing to be able to do To find the area of any shape and to find the slope of any line, even if it’s curved So I hope you learned a thing or two from Bob’s little adventures in the mountains today Thank you for watching This is your invitation to subscribe to my channel if you’d like to see more content like this And I hope you have an absolutely mathematical day

100 thoughts on “Calculus explained through a story

  1. Differentiation,an operation used in calculus in which the derivative of a function or variable is determined.Professional jargon that is of little use to students.No wonder people give up when trying to learn when confounded with terminology that itself needs to be deciphered.She is very nice all the same.

  2. i'm not going to watch this vid no more. I am feeling like im falling in to something but not Calculus.. G… this girl can make you feel a person that you are not Extremely Dangerous.

  3. Bob introduces wolves to the meadow. Because Bob knows the Lorenz equations, he knows the predator/prey relationship is dependent on initial conditions and so without absolute control he’s happy that he’s not God, can’t predict nor control the outcome and the ecosystem will do whatever it does down there on the Mandelbrot meadow.

  4. but but but upper part of bridge too steep for Bob to keep traction on worn out shoes; he slip off bridge while carrying tree land on deer's heads and KILLS THEM

  5. Hum? Bob Ross music ! In the background a nature scene. Up front and center a calming voice explaining mathematics. Boy, Donald Duck in Mathematics Land sure seems old fashioned now! Can't wait for Differential Equations and Linear Math.

  6. The only way to learn calculus is to do the homework. Hearing a story won't cut the mustard; it won't even butter the parsnips.

  7. Yes, the smile is nice.., but the cacophonous birds don't do much for my serenity.
    In truth the whole thing was quite aggravating. Just not good teaching, imho.
    The drawing was too small and complicated, as was the story.
    Her illustrations were way out of scale, and no one can walk up a 45 degree climb in the first place, much less navigate a fallen tree. I think a simple, clear math lesson would have been far preferable.
    Holding up a pebble in front of your eye to guesstimate something a hundred yards away?? What the blazes was THAT all about?
    But yeah…, how did I get here ??!!
    Just three clicks ago I was watching Hayley Westenra singing Pie Jesu.

  8. i say shoot the frigging deer and make jerky, then who cares how much grass is needed, Poor Bob day and night, night and day, day and night…. then,..hey there's enough grass after all, i wasted days and days, i could have been eating jerky.. DAMN

  9. I really enjoyed watching her pretty smile, and I learned about differentiation and integration and everyone in the meadow has enough to eat!

  10. while counting the sand Bob remember he left his happy little  bathtub running at a rate of x gal/min and the happy little stopper was partly plugged allowing an y amount gal/ min to exit… he wonders if he has time to keep counting sand or if his happy little cabin will be floating down a happy little river when he is done….

  11. Hey Bob, don't make it bad
    Take a sad song and make it better
    Remember to let the deer into your heart
    Then you can start to make it better

  12. Bob has major OCD .. i love it…
    Wait, what? @ 13:45 she says 'people'… Oh, so they were 'people' … That changes everything!

  13. Your presentation was as attractive as you! If we had more teachers like this in the schools, I would stop demanding the demise of public education…….

  14. Why is Bob living alone on a high mountain? , can't Bob call for help on his smart phone or ask the other happy trees to help him ?.

  15. When looking for the area of the meadow, couldn't Bob figure it our by taking a rope, walking around the perimeter of the fallen bridge? He could then take that length of rope and image it as a square box. The length divided by 4, then square it to find the area of the square. Is there anything similar to that in calculus? I remember an old story about a guy who bought a barrel of wine, but the barrel was not round, it was oval. He thought he was getting cheated by the wine merchant and so figure out a way to calculate the area of rectangles that proved the wine merchant was fair. Loved that story as well since anything involving wine made calculus easier to swallow ;-}

  16. Frankly , you are better than many professional teachers in explaining things. You may have a good future as an educator I believe. Good luck.

  17. While bob was laying down the grains of sand Deer never came close to him for weeks and they all died of hunger, problem solved.


    You are answering a great calling!

    How refreshing it is to see the way you have undertaken the sacred role of the educator… and how learned you have become in so few years! (I am tempted to borrow further from others by stating – "I never knew so young a body with so old a head"). Your delivery and the obvious pleasure the process provides you are worthy of remark. You are a natural teacher… a born enthusiast… and, clearly, you cause a viewer to be all the more attentive due to such attributes.

    Furthermore, by offering-up your instructive videos on the worldwide Internet, you are bringing much to average man.

    Rather late in his life, I knew an ex Harvard student who left early… and was often described as having brought science, math, complex and critical thought to average man… everywhere and every-when… due to his lifelong worldwide travel for speaking engagements, his extensive writings, and certain films that will live forever. "The Leonardo of our times" – R. Buckminster Fuller – in his youth must have shared much in his nature with you. Ultimately, he was nominated for the Nobel Peace Prize.

    I can not help but wonder where your callings will carry you… and how you will serve those who connect with you. This is approximately what I said last year to a man named Bezos at a small gathering in a private home. I framed my comments to him to (hopefully) cause a new type of contemplation about a calling to the role of the humanitarian (because it would be intellectually proper and because his further pursuit of commercial success could not materially advance his human growth nor the welfare of many the way a "Fullerian" redirection of his life would). I asked about what his dreams involved citing the title of a book that his company sold among millions of other titles, Einstein's Dreams. During the depth of night when one's imaginings might reach well beyond what was possible in a typical dream… where might soar the greatest of minds/the possessor of the greatest of tools (e.g. wealth of a scale never before seen)? In the most extreme, evolved form of one's envisionings, what might be accomplished in the future by such a dreamer?

    And, I can assure you that it is not overly melodramatic to ask those questions of you. You have many gifts and a truly great start. I do hope to learn about where you will go… and to what altitudes you will soar!

    With sincere compliments,

    [email protected]

  19. Happy Little Trees?? "Bob" in the cabin?? Its Bob Ross!! That's where he has been all this time! He probably was painting those happy trees!! He probably was very sad when one of his happy little trees was no longer happy!!

  20. A faster way might have been to weigh a single grain of sand. Then cover quickly the area in a single layer of sand. Then collect the sand in a bucket and weight it. Subtract the weight of the empty bucket and divide the sand total weight by the weight of the single grain of sand. Beats counting them one by one. LOL Oh then he could take the sand and poor it onto the ground and move the sand around till it formed a rectangle, square or circle in a single layer. Then do the appropriate simple measurements. Or he could have just burned the logs. Or rolled them into the river. 😉

  21. This is so sweet .. thank you ….. one thing to note which still requires integration is that the cumulative area of the gaps between sand grains may be bigger compared to the area of the gaps between the big stones, this is known as porosity. In short porosity is a ratio measure (in %) of the material to the volume of its container

  22. If any of my calculus professors had started the semester off with this story i would have walked out and dropped the class.

  23. The opening music sounded like a version of the Swedish Jul story running on TV every year called "Karl-Bertil Jonssons julafton" is it so?

  24. Great ideas explained very very very well passionately. Kids wanna wanna attend your lectures … thank you for doing this!

  25. Bob is strong as an ox, patient as a saint, yet he still hasn't learned how to jump over a gap in the bridge that would lend more sun to the grass below for the deer to feed on? Sounds like Bob is kind of stupid to me…kidding….good show.

  26. Bob is strong as an ox, patient as a saint, yet he still hasn't learned how to jump over a gap in the bridge that would lend more sunlight to the grass below for the deer to feed on? If he can carry a log on 45 degrees and not slip…Why waste the energy to carry a log in the first place to find out if he will? …Sounds like Bob is kind of stupid to me…kidding….good show.

  27. Hi, the explanation was a little slow for me, plenty of fluff, so I quit somewhere before the half way point. Sounds a bit like trig to me, but at 6¼ minutes, the problem is not clear. Your diagram looked like 6 pebbles wide, and 3 pebbles tall, so 3 in 6, I use feet, so a 6" rise in 12 inches, or 6 in 12 slope. But, to u called it 3 in 3, a 12 in 12 slope, 45°. Bye.

  28. This did help calculus seem more tangible. I aced geometry, algebra, and trig because I could picture them working in the real world. What I always said about calculus was something can’t actually get infinitely closer to zero. It all seemed too theoretical. I know they use it in business a lot for models and predictions but I always worked better with stuff that seemed more real

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