the last question on my calc 2 final
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- Опубліковано 8 чер 2024
- This is the last question on my calculus 2 final exam! I asked my calculus students to find the slope of the tangent line, the area under the curve, and the arc length of the hyperbolic cosine function cosh(x). Notice the graph of y=cosh(x) is very similar to the parabola y=x^2+1 because the power series expansion for cosh(x) is 1+x^2/2+x^4/4+... The graph of a hyperbolic cosine function is called a catenary (the shape of a hanging chain). I hope my calculus 2 students enjoy this equation and appreciate the cool properties that cosh(x) has! Dear calculus teachers, feel free to put this question on your calculus tests and let me know your students' reactions! : )
#calculus #catenary
the last lecture in my calc 1 class 👉 • finding the volume of ...
why arc length = area for cosh(x) 👉 • Area under the curve e...
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0:00 the story behind this question
0:33 find the slope of the tangent line to y=cosh(x) at x=4
2:17 find the area under y=cosh(x) from x=0 to x=4
3:31 find the arc length of y=cosh(x) from x=0 to x=4
5:31 why cosh(x) is super cool
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the last lecture in my calc 1 class 👉 ua-cam.com/video/ybPJzU4ZlEY/v-deo.html
From the point of view of a student: I'm pretty sure that the majority of the class would think that they've made a mistake.
Hehehe 😆
As someone who's had several professors give this kind of question: Yup. Every time I end up quadruple checking my answer because "that can't be right...."
I would've freaked out and put ≈ 27.285 for the last one... assuming I did the math right, of course.
I'd put some other integral into my calculator, just to make sure it hadn't gotten stuck somehow.
I would one of those students at least double checking my work.
As a college student who took his Calc 2 final last week. If I saw this on my exam I would be simultaneously relieved and stressed out. The question was not very hard, but someone once told me the only function whose derivative and area are equal is e^x. This question shows that is not exactly the case. I also wish our math department let us use calculators on the exam, but that’s separate issue.
Keep in mind, cosh is composed of e^x terms, so this actually makes sense!
Calculators!? Bah humbug.
Captain, actually that is not correct. e^x has the same derivative and antiderivative, but not the same derivative and area because e^0=1, not =0. So, for example, the derivative of e^x for x=1 is e^1=e. But the area under e^x between 0 and 1 is integral \0, 1\ e^x dx = e^x ](0, 1) = e^1-e^0 = e-1, not e.
Either the person that told you that was confused or you misunderstood what they said ; what's true is that exp is the only differentiable function equal to its derivative that is equal to 1 at point 0.
Using or not using calculators is a matter of how a class is taught: calculators may be great at illustrating some concepts and doing story problems fast.
It’s not about complexity but more about fun with math Which is pretty cool
Yup!!!
Not only that, if you get these results you also kind of know that you did it right which makes you feel better as well. Just great if teachers do these kind of things!
@2D ANIMATOR I think it's the microphone
@@deltav9784 ya right
Hi @@blackpenredpen I have a math challenge for you, factor x²+x+1 without using complex numbers
cosh vs. josh, who wins?
me
sinch
Crunch
putin
27.29
This is actually a really important property of catenary curves, which is the shape hyperbolic cosine makes. It looks like a parabola to the naked eye until you see them plotted together. The fact that the area, slope, and length are all the same means that a catenary is the most stable natural curve (except maybe e^x).
When a cable hangs under its own weight without any other forces, it follows a catenary curve. If you want to build an arch or suspension structure where the forces travel exactly along the curve without deforming, this is the curve you want, and this property is why.
Cool
@Aditya Chavarkar Yeah, e is weird like that. It just keeps popping up in places you wouldn't expect. I always figured it was "natural" because it models exponential growth in nature. Populations of organisms tend to grow following e^x.
I think an actual suspension bridge isn't exactly this form though. As you point out, this is a cable under its own weight. But a suspension bridge weight is two components, the cable weight which is linear with it's length, and the bridge deck. But the deck length is linear with horizontal length and cable length is not. Maybe that's why bridge decks are built with an arch of their own?? Don't know for sure, not a civil engineer but I studied a little bit of it in school. 😀
@@mikefochtman7164 actual bridges are more complex for sure, because they have to support not only their own weight but the dynamic load of whatever's crossing them as it moves. That plus wind means the stresses are always changing. From what I just looked up, it ends up being somewhere between a parabola and a catenary.
That is true of a rope or chain with constant mass per length along the chain and not carrying a separate load
In contrast, a weightless rope or chain supporting a bridge with a deck having constant weight along the horizontal will fall into a parabola.
A real bridge has some mass in both the chain and the deck, so the chain will fall along a curve somewhere between the two. They are not so different so it is not obvious.
For engineering a bridge with large tolerances and where the deck is a lot heavier per unit length than the chain, the parabola is a better approximation.
The above assume the vertical load is evenly spread along the chain. Real bridges and overhead wiring for electric trains or trams actuall have discrete verticals at intervals which is different again.
You could model the chain as a series of straight lines as you would with a pinjointed truss chain. That is a better approximation than a smooth curve but still not exact.
In the complete model for static loading each segment of chain is a separate catenary curve, but there is a gentle kink ie an angle where the verticals meet it (ie a discontinuity in dy/dx and ď^2y/dx^2 is undefined) Note that neither a parabola nor a catenary has angles like that...
The do the complete model for all the different loading possibilities (traffic has passive weight and if accelerating or decelerating imparts a horizontal load to the deck, ice on chain and or deck, wind loading at various speeds and directions, movement of the anchor points in an earthquake, XR protesters climbing the chains to hang banners, terrorists flying planes into the verticals, etc etc)
In practice wherever you stop it is still an approximation: it's more about knowing how far it makes sense to go. Most engineers would stop before modelling meteor impacts for example ;)
I never learned about the hyperbolic trig functions at school or college, but this is too cool! Now I need to learn more! Thank you bprp!
Thank you. I am glad that you like it!
yeah, i just finished my math for my degree last year, and it only came up at all once, in differential equations. he was like "as you know, cosh=blah"... i said "literally this is the first time seeing this, and i've done the entire math series."
I am studying Computer science(Cyber security) and I can confirm that this happened with my batch too. When the teacher found out that we are unaware of hyperbolic functions, she just shared a youtube video.
@@pharynx007only time ive seen hyperbolic functions was in statics. Once you give cables mass they form catenary curves.
@@jacksonmagas9698 we never covered them in our statics course either. 😂
I love when teachers do things like this. It's not only trolly, its a lesson in self confidence. Smart people who doubted probably went back and wasted time on trying to get a different answer even though they were more than capable of getting it correct.
Well it's the last question, so assuming they went in order, they'd have the rest of the exam time to check their work.
Absolutely. As a retired physics lecturer to undergrads I have huge respect for BPRP's confidence building skills. I know the maths but learn a lot in terms of teaching style from his channel. My students are so unlucky that I retired before y-t became a thing.
As someone who has horrible self confidence in their math this question would have given me such bad anxiety and I definitely would of flipped if I had it on an exam lol I
Self doubt in math is a side effect of not fully understanding the material. I've never been accused of being a particularly confident person, but I'm good at math, and this exam question wouldn't even make me blink.
People like you are what's wrong with this generation of kids lmao. If you truly understood the material you would be able to understand immediately that your answer is correct. If you truly know the material, you wouldn't be using the calculator, this is not a hard problem at all. Hyberbolic derivatives and the hyperbolic version of the pythagorean's theorem was taught throughout multiple chapters. And if you fail to realize that cosh is just the average of e^x, then just drop the class lmao, shit was taught in Calc 1 and I'm pretty sure it was lightly covered in high school trig. Since is a question on the FINAL, students should have already covered Series and Sequences, chapter 11, and should easily recognize coshx & sinhx
i like that you cover hyperbolic trig stuff in your class. usually gets skipped
We did the hyperbolic, which was cool, but we skipped the Jacobians 😭
@@reidflemingworldstoughestm1394 Jacabians are better left for calc 3.
@@RichardJohnson_dydx That would be where you find them.
@@RichardJohnson_dydx Just finished Calc 3 and never heard of a Jacobian
It did for my calc BC class
Thanks for the shoutout professor! I had a fun time in class!
You’re welcome! As I said it in the video, great job in the class!
wholesome moment right here
a) 27.29
b) refer to "a)" for the answer
c) refer to "b)" for the answer
Hahaha
This reminds me of my 5th grade math teacher. In one of the exams, the answers would follow an arithmetic progression. I couldn’t help smiling when handing in my exam, as I knew I had got a perfect score. She smiled back, "you realized what I did". 😂
😆
That’s pretty cool
4 points, not bad, huh (but out of 200)
I just died laughing😂
😂
He was laughing and having a good time
Shit got real when he said out of 200...
I wouldn't even need to calculate part c because I have seen your video in which you proved that for cosh(x), area and arc length are same.
_Writes down video url as proof of work_
@@sashimanu actually in India in most of the competitive exams we have OMR so we have to only tick the option. No need for method.
I am a calc 1 student and yet this legend makes things seem so fun and easy, honestly you are a work life saver.
Why can't our teachers provide us with such easy questions during our exams 😭? It was fun to learn through this video. Loved it 🥰
One of my favorite things about calculus is that you can use *simple* operations such as the derivative or integral (both are defined in terms of limits) to relate various analytic functions to each other (like relating sinh to cosh, relating log to 1/x, relating arcsin to square roots, etc). Hard to make those relationships with plain old arithmetic, you need the idea of the limit
I have always learnt to calculate somerthing exact, so using a calculator where you just yeet the integral into was not allowed on my school/uni. So when you were writing the questions I saw they were all equal to sinh(x) for all x. Cool little property of the hyperbolic functions I guess
If I saw this when I was doing an exam Id be INSANELY happy. Not often do you see answers lining up so perfectly. I'd still probably double check to make sure but I'd be happy
I have a vague recollection from ca 56 years ago that the tension at any link in a chain forming a catenary is directly related to the height of that link above the ground. If the chain passes over frictionless small pulleys at each end, and hangs vertically downwards, so that the overhang at each end just balances the weight of the chain between the pulleys, and the ends of the overhang just touch the ground, then the height from the ground to any link equals the tension in that link. I could have remembered this incorrectly of course being a while ago.
If I was at the student's place, I definitely would have written :-
y = cos(hx)
y' = -sin(hx) × h
y' (4) = -hsin(4h) 😂
I should definitely plug this into the final of my Calculus 2 students!
Hahaha be my guest!!
@@blackpenredpen No, enforce your copyright ;D
This video was a great holiday gift. Thanks. It reinforces something I’ve spent my career stressing and my college life long ago rebelling against - just blindly performing the perfunctory manipulations (crank turning) to achieve an answer, symbolic or numeric, is of little use and is actually dangerous. By having a basic understanding of the trig functions referenced in the question and what they represent, the question is answerable almost by inspection and provides confidence that the actual answers are correct (QED). I have no use for manipulators and calculator jockeys because they lack any insight into what they’re doing and therefore cannot justify their results. Now, that said, 40 years ago taking that exam I can only imagine the angst it created for those that cranked the correct 3 answers and how many times it would have been recalculated. For those, only a computational error during recalculation would provide them the validation they mistakenly sought. Truly elegant, bravo sir.
Thank you! And I wish you a great holiday season!
We had a homework problem like that in 3rd semester, something involving the normal vector of a trig function. Each step reversed the previous one, from sin to cos, to sin, to cos, and on it went to the final answer. I couldn't believe it.
I love how you found a way to troll the students on the final
I've been a college math teacher for my whole adult life. I find your cosh(x) question super cool as well.
Wow this is actually super cool! Never thought about this until now. Nice design for final questions!
I undergraduated in 2018 and graduated (Msc) in Electrical Engineering last month, and i remember i used to like to solve many exercises about Calculus. But it's so wonderful to see Calculus from another point of view. Greetings from Brazil.
So glad to see someone teaching hyperbolic trig in intro calc. It has so many useful applications, and yet so many teachers never even mention it.
Mine didn't teach hyperbolic trig, among many other things, but I was really interested so I went back and picked it up myself. I live in Texas though and we lost a month of the semester due to the winter storm. No electricity or water for a lot of people in Austin. So always short on time. Learned calculus 3 myself because I was afraid I would have a lot of gaps missing. Catenary curves are really cool! Even found in soap bubbles.
27.29?
I love how passionate you are. A teacher colleague at the high school I work told me recently that she thinks, that Maths is only taught to train certain areas of the brain. She teaches phys Ed. No, it is also taught for the beauty of it. Doing maths enriches your life, opens your eyes and opens your mind.
I love your videos. I can tell how much fun you have, your class must be fun!!
you should have given 9 points each= 9*3= 27 points to totally mindfk them
Actually part a equals part c is the key insight of the famous catenary problem in physics.
That's how I knew the answer. I saw cosh and thought, "hyperbolic cosine is a catenary curve. All the answers are going to be the same for any given x."
Video in a nutshell: bprp procrastinates grading finals and makes a video about it
😂!!!
I learned more from this video than my calc 2 class right now... Kep up the good work!
Love hyperbolics. Just saw your merch and I love them! Definitely buyin.
Wow! Very impressive that you came up with this question
Thanks. I actually have solved area = arc length previously so I know how cool cosh(x) is. And then I just realized that the slope of the tangent line gives the same numerical value so I had to include it there 😆
This is devious. I surely would have spent a fat minute just redoing and redoing the question, seeing if I did it wrong since I get the same answer.
I just recently started learning about hyperbolic trig functions, so this was a nice practice for me!
Forget about the math!! His penmanship on a dry erase board and his ability to keep his lines straight are both absolutely impressive!!!!!!
i’ve never done calc 2, but this was very interesting to watch
That’s a really mind blowing integral
It's so wholesome seeing him being proud and happy
BEAUTIFUL! THIS IS THE BEAUTY OF MATHEMATICS
Haha I already knew that cosh has the same arc length and area under the curve over any finite interval! I remember seeing it on the wikipedia page on hyperbolic functions. That's probably the coolest property that cosh has.
I think Michael Penn did a video on it somewhat recently so when I saw part 3 I instantly knew what was up
My favorite property of cosh is how it's the shape that all ropes hang in when supported at their ends. Idk if this has some hidden relation to the same derivative/area/arclength property, but I think it's pretty cool
To be really fussy about wording (which is usually a good idea in mathematics!): "over any interval of a given finite length", right?
@@RolandHutchinson thanks, you're right, that does make a bit more sense than what I said. And I agree with you, clarity and precise wording in writing about mathematics is very important. Too often do resources about math overwhelm the reader with unnecessarily complicated descriptions of ideas which could be described more elegantly and simply. Even if it means sacrificing a bit of the exactness of an idea I think it is most of the time better to explain things as simply as possible.
@@violintegral One is pleased to have been of service. And IMHO, you are absolutely right about the importance of writing clearly and as simply as possible. One part of keeping it simple is to bear in mind the audience you have. Precalc students will not want the level of detail or sophistication that would be appropriate for professional mathematicians at a research conference. But both need writing or speaking to be accurate and clear (and if possible, elegant and simple).
You teach calculus far better than my previous professors since 11th grade. I'm currently in 2nd year college. I'm still hoping to have a teacher like you in calculus someday.
Em toda sua simplicidade, a matemática é linda!
Your joy right before the 5-minute mark is infectious!
This is so cool. Honestly wish my math professors would have done this for my classes. Though I don't know if I find it cool because it is, or if its because of my math degree.
Wow 🤯 increíble vídeo!!! Incredible video!!!
Sir how beautifully you have adjusted the whole board till the end without rubbing anything
Love it. Great work with the question.
The three part question you presented is really fun, and I really wish I had a professor like you for my final exam because our class average was a 41 (which happens to be failing) and our professor takes off massive points for accidentally missing some writing. So it shocked me to see that you took off no points for the student having the write answer, but they forgot to write the dx on part 2 of the three part question 😭😭 he would’ve been marked off 4 points for that and the “same lol” would’ve been marked off as points too 😭😭😭😭😭😭😭😭😭😭
I’m just jealous that you were a better professor then the one I had
That's just plain unfair. The "dx" thing might be understandable, but the "same lol" is completely separate from the actual answers; it shouldn't have any effect on grade.
Yes I also am surprised no points were lost for not writing dx. I would’ve personally removed 1 point.
I liked this video very much. Amazing. Years ago I've heard teachers talking about how beautiful math is. I'm not sure I will ever fully understand them. Perhaps a little though. Today I'm a math teacher myself.
Before studying math I have always thought (up to my degree) that the figure of a chain is a parabol indeed (y=x²). But later when i studied it and also after studying complex analysis I loved it and I loved its connection with trigonometry.
That’s why cosh is the best: it’s the only function that has an equal derivative integral and arc length. They all turn into sinh. This is because cosh is a solution to the positive wave equation and therefore only requires 2 differentiations to cycle rather than 4 for regular trig.
Man I can't wait to learn this in two years, currently in algebra 2.
amazing sinQ/cosQ for the great content and the love you spread for maths!
Smart joke
I was scared and amazed at the same time. Good job with the question
This was super, thank you!
Very easy question! Free points for your students! Yay!
Matthew must be feeling so great atm
i love watching these like i know what i’m doing
My best professors always wanted the student to learn something from the exams. Unfortunately, few of my math profs had that attitude. Nice work.
Hello! I’m an IB student and an aspiring engineer, I just wanted you to know that your videos inspired me and made me like math which made it possible to pursue an engineering career!!!!
👍 I am glad to hear. Thank you.
I'm guessing this is because (e^x) has the same value, tangent & area for all points on the curve.
I’d probably say that since cosh^2 - sinh^2 = 1 then
the integral of is equal to the integral of , which is the area!
I’m in calc II right now and this is the teacher I need
That's pretty cool. I can't think of another function that has that interesting property. I may steal your idea and put it on my calculus final!
I really appreciate this exercise. Technically speaking, the three results are different because all of them have different units though.
Yep, but the _amount_ of the units is the same...
Measures, lenghts, distances, etc are simply functions. If you really were technical you'd never bring up something like units.
Lets say the area of a set of R^2 is defined as a certain integral, like m(A) = integral on A of a non negative function f(x,y).
This would be a standard definition in measure theory, where do you think "units" get involved? Yes, you're right, nowhere.
@UCLQTi7fpQV1TyrXMEvPJFtA Its a matter of definition, lenghts, etc are real valued functions, there's no way around it. What you are talking about isnt math, math doesnt work on units. I hope this time you get it: real valued functions, its not that hard.
@@VraxxTheEmperorYou should learn to express your opinions in a more respectful way. You sound too arrogant and opinionated.
@@VraxxTheEmperor you're right, seeing people speaking of units in mathematics is just painful lol
This would make me question everything I answered, even if I was 100% confident 😅
😆
This is amazing! I loved it so much
Happy hollyays, christmass and new year for you too!!
The answer is incredible! TQ for making these questions lmao 😂
😆
I'll be honest. If I'm doing my exam and see that all three questions give me the same answer, I'll freak out.
Remarkable! It's a fascinating property. I now wonder how many others there are like this.
Idk if it's because it involves e, but I love how it all comes crashing in.
Really highly graduated teacher
This seems really easy for an assessment question.
Fun question. Your students are lucky to have you!
This is a man that loves math-and that’s awesome
boi i didn't take calc 2 with you. i do love this problem though. i also really love this function because of its interesting graph. that shape is called a catenary, which is the category of shapes that a chain would naturally assume when hung by its ends orthogonal to the direction of gravity. additionally, if you rotate the catenary about the y-axis to form a surface, the resultant dome is perfectly balanced to support its own weight even when built out of heavy material. the domes on many buildings assume this shape and hold themselves up with no additional help. if you wanted to try and mess with this to graph the exact shape of domes and chains you see out there in the wild, the general form for any flattened catenary is f(x) = Acosh(Bx). mess with the constants to customize your experience
that evil laugh lol
😆
That’s hilarious. Don’t recall much work with hyperbolic functions back in school but I retained just enough memory of the identities to guess what was going on.
A hanging cable is a hyperbolic cosine (cosh). Water coming from a drinking fountain is a parabola. And the concave mirror in a flashlight or headlight is a paraboloid of revolution.
That was really cool but I would think I did something wrong if I got the same value for all three sub questions XD
Good job Matthew
Hahaa
You are a really fun instructor I really wish that you're my mentor in Calculus
I would've absolutely loved this question on my exam. Very cool :)
My brother had in an examen the integral of (arctg(0.2x))^2 and he couldn solve it, could you try it? Love your vids
I don't think it has an antiderivative that is expressible in terms of elementary functions. Was it a definite or indefinite integral?
@@violintegral Indefinite
Back in my day-damn I’m feeling old-calculators weren’t allowed in exams. But people didn’t carry 1000 (circa 1980s) supercomputers in their pockets to take selfies everywhere they went.
Also, I don’t recall calculators doing integration either. 🤔
I had a five function calculator with no memory function, a slide rule for back up and a CRC manual. When I graduated I bought a programable HP 25 for $275.
Very fun! Excellent presentation 😊
The fact that he is so INTO the subject should majorly rub off on his students and make them even better. Great job, sir! And as AndryCraft69 said, I for one would be thinking I messed up something along the way.
I think it's also important to realize that the area is actually not really comparable to the other two - it is measured in different units! But I gotta say, that's an evil question to put on a test :)
The derivative would also be in different units if this were science.
Hallooo can you help me answer this question?
integral du / u²+a²
integral du / a²-u²
integral x⁴-x²-x-1 / x³-x²
For the first two integrals, he already made a video :
ua-cam.com/video/e_3zVCaQc9Q/v-deo.html
And for the last, I think you need partial fractions.
@@darkkevindu6982 ow thanks bro
@ Hm -- All three of those are written wrong because you do not have required grouping symbols.
That's pretty cool. I'm remembering that calculus is awesome because of you :)
This is a great example of a case where you can pretty trivially analytically show an intuitively surprising property.
It's got me trying to find some geometric intuition on why the area under cosh ought to be equal to the arclength along the same segment. And then why either of those ought to be equal to the slope. Since cosh can be defined geometrically, I'd expect there to be some obvious "link" that I haven't thought of yet LOL
Is this an actual test for calculus 2? I could answer just by watching some of your videos and I'm 16. Are the other questions easy like these?
cool guy
I dont know about other schools, but no, cal 2 is usually not this easy. At my institution, Calc2 is actually the most failed class, probably because its required for most of the majors and for some of the non-mathy majors, the highest required (meaning its the hardest they'll come across). I think the point of this question was just to test some basic principles, and to have a bit of fun while doing so.
@@novan8r thank you
If I were you're student, I would have 100% lost my mind doing that question
I really liked that question! Im having calculus 2 next semester and will come back here to remind me haha
I’m a precalculus student so none of this makes sense to me. But this makes me excited because it seems like the more advanced you get in math the more interesting it gets. I’m sure it’s quite difficult but I’m always up for a good challenge.