Dear all calculus students, This is why you're learning about optimization
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- čas přidán 30. 05. 2024
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Hope you guys enjoy! Two things to mention here.
1) The video that explains the last method for the 'lost fisherman problem' is already up on patreon and will be out in just a few days from this one's release.
2) Most of these examples/stories actually came from 2 books I recently read and those are linked in the description if you want to learn more math/optimization.
5 days ago?
wait how is it 5 days ago
where can i find that video about the most optimal solution to the boat problem?
@@naswinger It's on patreon now and will be out on the channel in 3 days. And for everyone asking about how this comment was posted 5 days ago, videos are posted early on patreon (plus I need to make them unlisted in order to get approval from the sponsor).
@@user-ox3ov2qt5o definitely does, thought that was a given
Put the boat in reverse dummy.lmao
Let's frame the question better: you travelled 100 km away from the shore. Suddenly, it becomes foggy, and a violent gust tilted your boat for an unknown angle. Your fuel tank is leaking, and the engineer cannot stop it completely. All communication stopped. Survive the day.
Y. Z. Survive the day =/= Go back to shore
@@joelmiller2601 go back to shore == survive the day
ninja lame Sorry; but no.
@@y.z.6517 Then there is a high chance that we go far and far away from the shore
Ahhh eliminating your enemies while using the least amount of cannon resources
Then use the rest of the powder as fireworks to celebrate victory.
czcams.com/video/XPCgGT9BlrQ/video.html 💐
@@ililililili9726 😂😂
And they say war is bad..
@@noop9k It is, because now a new generation hates you.
zach star: we can do better but I wont explain it
me: aww
zach: just kidding
me: yay
zack: but not in this video
me: aww
I think the solution is a spiral. Not sure tho.
@@abhijanwasti7991 hmm i think you could start doing the 1km circle and then move to the 1.04km towards the end
i did some drawing on paint and came to the conclusion that you do 1km circle normally until you completed 3/4 of it and then stop turning and move in a straight line
-> 1 + 3pi/2 + 1
Gotta tease a little bit
turned off notifications, disliked, unsubscribed, unfunded, demonetized, and reported for child abuse
Oof
Optimization was one of my favorite parts of Calculus. I really appreciated how applicable it was to the real world
tedskins2000 same!
same!
same
!emaS
@FBI ! I'm not under arrest am I? 🤔
6:35 Delta fixed a triangle problem, how ironic Δ
I wouldn't say that's ironic. It's just coincidental and funny.
czcams.com/video/XPCgGT9BlrQ/video.html 💐
Just recently got interested in math. Wish I had before. Like in school, everyone asked "why do we need to learn this?" and no one could ever answer. "Because you should". It's as if they tried their best to make it as uninteresting and boring as possible. Had someone talked about humanity going to Mars or like the golden ratio... what a difference it would have made.
Yep. One of the failures about education is that teachers often can't communicate why is the thing important, or how it may be fun or interesting. I was never interested in biology for example, and yet I caught myself watch full series about evolution of life by Aron Ra right from the single cell organisms up to humans, and thats 40 something episodes 5-15 minutes each. Which is a far too much for a guy who is not interested in biology =)
I was the same way in high school but learned to actually think math is pretty cool. You can almost tell the future with it. For example, everybody places an x where they guess a ball is going to land where you would calculate it with math and predict it perfectly every time. The look on their faces when you accurately predict where EXACTLY that ball will land 10 times 50 times in a row.
To be fair, you never asked yourself how a plain can fly, how a computer works, how wireless communications works, lack of curiosity is the main issue
czcams.com/video/XPCgGT9BlrQ/video.html 💐
@@gabrielbarrantes6946 I have done a ton of research into the current US education system and thought a lot about how to fix it and I can 100% definitively tell you that lack of curiosity is not at all an issue when first learning about these topics. School has a way of taking things you like and forcing you into the most uninteresting and uninspiring part of those subjects especially in the beginning. Talk to any child in grades 1st to 6th and you will generally find they like things like math. There are even alternate education systems that you can put children in right now that are completely driven by the child's curiosity as in they have no set curriculum and those systems have proven to even give better results than regular school on tests.
Dear calculus students, enjoy it while you can! Calculus was such a cool course and in my opinion the math only gets messier from here! Thanks Zach!!
Math always get messier as you progress.
@@pouzivateljutube2995 It's called entropy 😁
Differential equations is a relief 🥵
Damn things get bad after Fourier sequences...
As for the Taylor's theorem and what not, it still doesn't penetrate my head.
Everything after 1+1 is messier.
Very very fun boat problem, thank you! The best I managed to get was ~6.4
The strategy I came up with is:
- go in a straight line for some distance r
- draw the two tangents to the unit circle that pass through your current location, label them A and B
- follow A until you're on the unit circle
- follow the arc of the circle until your direction of motion is perpendicular to B
- go in a straight line towards B
The total distance in the worst case scenario is given by
r + sqrt(r²-1) - 2arccos(1/r)+ 3π/2 + 1
The optimal solution is for r ≈ 1.16
This was really clever! Only thing I'm not seeing is where you got the 3arccos(1/r), the way I drew this out I'm getting that it would be 2arccos(1/r), assuming the worst case is where you barely miss the shore and drive in an arc until you reach basically where 3pi/2 is on the unit circle and then drive perpendicular to the shore until you hit it (which would be 1km). In that case you'd drive the entire 3pi/2 arc - 2arccos(1/r). Or maybe I just didn't interpret your method correctly.
@@zachstar yep! You're right, shouldn't be doing maths at 12am I guess! The actual value becomes 6.4 then, at r≈1.16 oops
This is super cool!!
How do you actually come up with these amazingly cool solutions?...wow ...is it easier to come up with such knowing optimizations ?
P.S.
I have not yet learnt calculus ... probably will be learning them this year...
~ A sophomore (10th grade)
@@sudheerthunga2155 for me it didn't involve much calculus, the key was to reimagine the problem as stretching a rope around a circle with a couple of extra conditions. It was a really fun process and not tedious at all. The calculus only came at the very end for finding the optimal angle, but there you might as well just plot the graph.
This was definitely and exactly what I was suggesting. This absolutely makes for an awesome and informative series man! Keep it up
czcams.com/video/XPCgGT9BlrQ/video.html 💐
I love these videos where you show the application of different kinds of math, I will be studying this this year and I'm so exited for it
czcams.com/video/XPCgGT9BlrQ/video.html 💐
i´ve benn watching this channel for the last 2 months and love it. Great job man. I really like your content. Keep the good work. Cheers.
For anybody curious about the answer to the minimum spanning tree question, the algorithm they use is called Kruskal's Algorithm. Essentially, they sort the distances and, going over the sorted list, connect any two currently unconnected nodes.
this is the first time I can say that graph theory helped me solve a problem in any circumstance other than school
Hey man. I just wanted to thank you for the resource this channel is. At my high school we’re required to right a 4000 word essay on anything. I chose math and just had a bunch of trouble finding a topic within that. Thanks to videos like this I settled on optimizing baysean search theory. I honestly could not of done it without this channel and would of failed a class. Just wanted to say thank you
ib xdd ?
1:08 *solves by logic*
Stop. Since it's day, the wind breeze should go to the land, I feel the air hitting my face, so it should be the other way right?
move in a spiral pattern, so that the distance between the lines of that pattern (measuring at an angle that goes through the centre) never exceeds the view distance in the fog. you'll also find survivors that fell out of the boat, if they're lucky and you don't reach shore first
also what i thought first, till he mentioned the "what if we go the 1km at an angle approaching 0" then i realise, ok we gotta think in maths, not practical reality
3:28 Angry pacman
Yes
Le joie de vivre is watching your fascinating videos!! Thanks!
gotta love the idea to "drive a perfect circle" around your starting point before you went for about 1km trough thick fog .. xD
One of my friends figured out how to find the Fermat point for math club in high school. One of the rest of us in the room (might have been me, but I think it was another kid) realized that you could take weights on strings and hang them through holes, and the stable point was the Fermat point. Bell should have hired a 10th grader to help them.
Love this one. Sharing with my class (I'm a prof teaching calculus and physics). Best wishes Zach
I started this channel to help my class. Thanks for the inspiration Zach.
As always, a job well done!
I love this channel! Thank you.
Imagine if you used this ancient technology called a compass.
Or better yet, just look for waves, they likely lead back to the shore
Polar inversion.
@@otheraccount5252 Just remember which way is north before going out into the water. If, somehow, this happens, just go "south".
@@thetimelords911 You clearly don't understand the concept of being lost.
*compbutt
This is one video, I waitet for!!! More of optimization please! x3
6:41 Fermat point is shortest distance task, if you have extra variable let say bandwidth, minimum length is dominant, but bandwidth can change location point.
9:18 Basketball was invented in 1891, so this problem from 1686 probably did not involve a basketball.
EDIT: 11:10 nvm you got me there
Watch the whole video before commenting.. Why don’t people do this more often?
@@joelmiller2601 That's put me in my place...
Now I'm off to watch the rest of the video...
Joel Miller Thats like the teacher telling you to ask your questions after the lecture
Like I’d even remember
@@ThePenisMan You probably have a note. Write your question and read it out if it's not answered by the end of the lecture.
Bastion Barrick that is an incredible waste of paper
I never understood the use of calculus until i started machine learning, Everything makes sense now
I love seeing my city Girona in a video in english!!! ty!!
This is awesome! Thank you!
Just half way into the video..
And I say awesome..
Keep it up , man
love your videos man!
Very cool concept. Please keep making such videos?
@5:00 Also, if you had turned right like in the figure, don't start your circle turning left, turn right again. Much more efficient. You will reach in (half the distance-1.04) or whatever distance.
Okay video! Thanks for uploading!
I love videos like this. Now to learn calculus... As soon as I can afford it (both time and money.)
Am I high or does the first solution not account for travelling away from the shore?
If you traveled away from the shore, then you would only have to travel half a circle to get back. The problem was concerned with optimising the maximum distance.
@@Max-ww7iz But that case would never happen because the radius of your circle is the distance you travel before starting to move in an arc
No both of you still don't get it. You can drive in *any* direction and hit shore with this method. You take your point of lostness as the center of a circle. Then you drive your recorded distance to shore or more in any direction, for example straight ahead. Now circle around, done. Try it with a piece of paper!
@@lucastotheju start 1 km away from the shore, lets say you move 1.04km further away from shore (at first looks like u made it worse), from there you make a circle with center point being ur starting point (that was 1km off shore, this is important - dont envision a circle with the wrong center point), you will hit the shore because 1.04km is longer than 1km and the circle diameter will be 2.08km, so even if u move exactly away from shore, it was still only 1km+1.04km. In this case u only travel half a circle anyway, so it's still shorter than other situations where u ended up close to the shore after your initial move.
Pause the video at 2:49 and observe the black circle. It's a circle with the center point at the 1 km spot around which you're supposed to be sailing to find the coast. In this case, he drew a 1.5 km line towards the shore. However, notice that sailing 1.5 kn in ANY possible direction still lands you on that same circle.
I remember James Grimes on his old channel did a similar problem to the Fermat point with four locations instead. His solution was to use a soap film to simulate how it should look like.
Voronoi diagrams could also help to solve minimum distance problems like the triangle one. Given 3 points as the example, the point were the three Voronoi regions meets is the point that has the minimum distance between all three. There is a similar problem were a company has N werehouses that serve a certain region of a city and you have to find the best way to split these regions. The solution is given by the Voronoi diagram. Anyway, thanks for the great video Zach, have a good one!
To find the Fermat Point isn’t that the same as drawing a line from each vertex so it is perpendicular to the opposite line for two vertexs then where the lines intersect that would be the Fermat point?
Please answer
What you described is the orthocenter of the triangle and, sadly, it's not the point with the desired property.
i literally coded Dijkstra's algorithm just two days ago and it is fun to look at the result
Inspiring, thank you.
Calculus was my favorit math course after Linear Algebra . i really enjoyed Optimization problems .❤
It was my favorite yet most frustrating part of Calc 1 in high school. I loved it's direct applicability. The ones I hated the most was like filling up a cone, determining the rate at a certain height
12:56 Certain triangles (BCD, BCF, EFJ, GHI and HIJ) just fill my heart with so much pain...
For the airport communication probem is what you did with the equilateral triangles just finding the middle point of two edges?
Or is it more then that? (Sorry for mistakes I didn't learn maths in english)
I love your videos! :)
Thank you for such an informative video on why maths is actually useful! Now I can convince my friends too... 😁
I have 2 videos asking this exact question in my recommended.
Hi Zach. On optimization problems. How about variable situations like a factory making something out of a variety of parts, materials, etc. Does the optimization solution for something change under a different viewpoint.
Very ‘Presh Talwalker’-like, enjoyed it!
Brings me so many memories. Why can't professors use these type of examples?
I think the lost boat problem can be optimized further by moving with a square path inside a (one kilometer circle plus some small distance.)
Yup, but use a triangle instead of a square. The square inside the circle has a perimeter of 6.93 km and the triangle has a perimeter of 5.2 km, both are which are lower than the circle minimum distance of 6.995.
Get out of my head! My Dad was an Engineer. I was thinking about this concept the other day. Thanks!!!!
Thank you
Can’t determine what direction it came from, can draw a perfect circle. Nice!
My shortest path I calculated is 6.45km for the whole journey, but I'm not sure about the solution being perfect (nor correct) so looking forward to your next video :)
thanks for the 24 sound effect
This is what I hope to learn in an msc in supply chain management
How is it going?
This is so awesome
in the boat problem how do you know that you circle is big enough to reach the shore at all? wouldn't the worst case scenario be that you end up going directly away from shore and your circle ends up being too small to reach it?
You can slightly decrease the velocity for the basketball problem. This is only because a basket ball can hit the front of the rim and still go in. I don’t know how much but you could decrease it by a bur
For the first problem:
What i dint rlly get. What if ur going 1 km out on the sea? Like if i lost orientation it msy happen so how does it help me to make a circle
Before watching a solution. I ended up with this equation:
S(x) = x(1+ 2pi - 2acos(1/x)). Solving for the minimum gives roughly 6.995... at x = 1.044... I think that's a definite improvement compared to 7.28... at x=1.
*EDIT:* ok, 30 seconds passed, and he repeated my solution word by word. And it's not even the best one D: gotta try more.
*EDIT2:* Before I watch further. I got another equation:
S(x) = x + 2sqrt(x^2-1) + 2pi - 4acos(1/x), this one gives the minimum of 6.459.. km at x = 1.242... that's my second guess.
Thank you for this video. Could you include links to the derivations in the basketball problem?
Had a test on this recently good shit
I learned optimization and other than calculate the size of the structure built with certain prices of certain materials, I used it for calculating optimum stadium ticket price in a football manager game when I knew the current audience numbers, the current price and the size of the stadium. For maximum profit that is.
But like for what wouldn't use optimization!
But so we just assume that we can't see the shore until we're on the beach with the boat?
I will say a great video to explain the basic concept of optimization
About the lost fisherman, in nature similar problems often cause spiral shaped trajectories.
for optimization problems, LP can be put to use since that's what it's for.
A good current real life explanation of functions and calculus: a pandemic. All the infection rate models are based on functions, and add, subtract and multiply hundreds of functions (factors) that influence infection rate. These growth curves then are analysed using calculus to measure rate of changes : too high, we have a problem so increase safety measures. Low : open up some activities to find the balance point. And yes, they probably factor in extra measures to counter the 20% of people (or states) who won't follow measures. The other interesting problem was asymptomatic disease: we are used to seeing runny noses when people are sick. This disease you can show nothing for 2 weeks and spread it. That means cases you measure today are actually indicating spread 2 weeks ago, and you have to factor that in (you're actually measuring data that is 2 weeks behind what is actually happening today), another reason for calculus to determine what IS happening today.
Hi Zach Star, awesome video. I had a question about text books because I'm going to highschool soon and wanted to get ahead in math and science, so I was wondering if you could tell me how to find the right textbooks to learn a subject
When I'm looking for a good textbook for math/science class I usually search quora and reddit for that same question and I always find some kind of response. Often there are good amazon reviews but they can be misleading as well depending on how technical or rigorous you are looking for a book to be.
This is pure genius. I read a book with similar content, and it was pure genius. "The Art of Saying No: Mastering Boundaries for a Fulfilling Life" by Samuel Dawn
If you’re using the best value d for the boat would a polygon be better or worse?
Regarding the boat problem, what if you drove into the wrong direction, so not towards the shore, but away from it. Thereby ending up, say 2 kilometers from shore, thus not reaching the shore using the 1km circle. Is it just an assumption that the rough direction of the shore is known or am I missing something?
Enjoyed watching something that might have some use.
How does Ant Colony Optimization fit into the idea of calculus optimization? How would one attempt to compare different optimization algorithms?
was kinda hoping for something on parameter adjustment like the LM (levenberg maquardt)
Do you know why we take a whole unit about vectors and how can we use them in the real world. Also great video as always please please keep on doing these types of videos
Surely for the first puzzle your given solution only works if you go in the rough direction of the shore - if the random angle you choose is further out to sea, your circle no longer hits the shore?
you said you can't remember where the shore is. What if you go relative further out into the ocean, instead of towards the shore. Your area you will ancher around, and do a full circle will all be within the water
You travel *at least* the same distance away from your anchor, as your anchor is from the shore. This will guarantee that you always hit the shore before completing a full circle. (e.g. your anchor is 5km away from shore, then travel *at least* 5km in any direction).
Does the solution of the lost fisherman problem have to do anything with a regular polygon like shaped path, which stops before the next edge collides with the first one?
Not that I know of. I should note that I found this example in a book and the author provided 3 methods (2 mentioned in this video), but there was no indication that even the third method (that will be discussed in the next video) was the most efficient so there could be something else for all I know.
In the boat example, if you are entirely unsure of where shore is and the fog hinders vision, then why wouldnt the most efficient way back be to turn the boat exactly 180 degrees and travel that path 1.05 km then do the circle thing if you dont hit shore?
What software do you use to make these kind of videos?
I think better d is a d such that the integral of the path value (sum) is minimum rather than just the worst case path to be minimum
What if we decide to go further into the sea? Then we wouldn`t hit the shore, right? Am I missing something?
Yes you are. We left an anchor with rope at our "Lost Point" and drive in any direction as far or further than our distance to shore. Now drive in circle. You will always hit the shore because a circle with radius equal or greater than the distance from shore will intersect with the shore, no matter what.
You were thinking of taking our guess drive as center, but we take it as our start around the perimeter.
The only thing to think about then is to optimize *how far* we drive in any direction.
He always showed the worst case, which is bad for understanding the problem, granted.
I’m confused doesn’t the ship have radar and staff locating where the ship is at what time or whatever?
I know math is great and all but I’m pretty sure sailor have like water maps and stuff.
A better solution is to go in a distance of "d", then go through an OUTWARD SPIRAL. Suppose the distance from the spiral from the original to the point that the spiral goes one complete circle is "f", i.e. f>d, r = (theta/2pi) * (f-d) + d. The final solution is in terms of both f and d, while under the constraint f>d. Let's say L is the worst-case scenario distance of travel, the resultant f and d are found by performing partial derivatives on L with respect to f and d, set the partial derivatives to zero.
Oh well, I give up on the calculation..... and I assume the rate of radius growth of that spiral is linear to the angular distance of travel. If that is not the case, the ultimate solution perhaps requires variation calculus...... and I don't know what I am talking about.
I was just thinking about spiral
Hi Zach, I'm researching on teaching and real-life examples. For the boat: mention the boat is lacking a functioning compass to catch a more pragmatic audience. Is this a classical greek example? From which books did you get it?
In the description, it's called 'when least is best'
How do you make your animations?
Using test taking strategy, I would say nautilus spiral? Depending on the domain the answer is always choice C or the golden ratio (or e if it's larger than 2) or "they're the same" or Fibonacci or nautilus or there exists no such foo.
Where is the video link for the solution to the first puzzle ?
please do reply when you get the link
I too, was looking for the link..
It's not available yet! Will be out in 3 days and is available on patreon now.
@@zachstar Thanks
booksc.xyz/dl/78506901/ac16a2
7:00 wouldnt it be easier to just take the angle bisectors and let them intercept? This method also gives you the abstract solution for any given triangle...
For those coming to this video late the link for the follow-up video for the Lost Fisherman Problem, and jumping to past the recap, is: czcams.com/video/h60zE9QDkXo/video.html
I just wrote my optimization exam this morning. Why didn’t I see this before then 😫
I don't get 13:26 though. why can't you connect EB instead of EF? EB seems like a shorter path
4:04 whoa this graph looks like the angle of deviation vs angle of incidence graph for a thin prism. Even the equation is the same
So you're saying you'd putt he anchor down where you were at 1km out, and then go 1km out and in a circle around that anchor and you're guaranteed. But that if you made any small error ( the 1 plus 2 pi thing) then you'd miss the shore by a smidge?
On the boat problem: what if you go one direction 90° and back 180° isn't this more efficient?
@Zach what if boat 🚣 moves unfortunately in the straight directions. 🤔