Calculus 2 Lecture 6.6: A Discussion of Hyperbolic Functions

Calculus 2 Lecture 6.6: A Discussion of Hyperbolic Functions

[failedfishermanBC]

Professor Leonard, how does it feel to know that you're the best educator on this planet right now?

[rumpelforeskin]

I find myself raising my hand in front of my computer on occasion when he asks "Show of hands if you're with me"

[thetroofteller]

Thanks Prof! This brings back so many great memories of my days in 1st year Maths. It's also really nice to have a Prof who calls you 'my friends' and doesn't treat you like an idiot like my lecturer did. If all your videos are like this I look forward to many wonderful and informative hours - cheers! 

[rmdir]

The reactions of the class are so wholesome. My professor just post long lectures of only him in front of a white board, not bad by any means. But something about the reactions of the class really helps me, I think people need to learn socially.

[adnanmohamed]

I am glad that the class says yes to Leonard when he asks: do you want to see a proof.

What is Math without proofs??!!

[madisongutierrez]

Thank you professor Leonard for actually explaining math in ways that makes it way easier to understand. Only wish I could have you as my actually professor.

[eddied]

Professor Leonard, I keep raising my hand! You never pick me lol

[apereed]

A serious conversation in Calculus II class today: Why are we paying thousands of dollars in tuition if we're all learning the material from Professor Leonard on YouTube? Something is broken in the education system. Luckily a solution can be found on YouTube with Professor Leonard.

[m.steven]

He's like the chadest and smartest of math professors out there

[hassanlaghbi]

Certainly, you are the best math teacher of all time.

[JiatingLin-cysc]

53:25
x−√x^2−1 won't be negative actually
The reason why we can dispose of the minus sign is that, firstly,
(x+√x^2−1)^(-1) = (x−√x^2−1)
Therefore
ln(x±√x^2−1) = ln(x+√x^2−1)^(±1) = ± ln(x+√x^2−1)
But arcosh(x) is defined as non-negative, so the minus sign is not required

[Hobbit]

You’re a hero :) u made my engineering life so much easier right now

[lseansiamz]

Best explanation ever!!!! if you agree 👍

[olgyish]

Thank you for your lectures, I finished watching your Calculus 1 Lectures, now I'm on my way to finish Calculus 2 with you.  Do you teach Linear Algebra and Multivariable calculus? I'm waiting for them.

[azrmuradl]

shout out to the greatest calculus teacher of all time after Newton

[athenenoctua]

For whatever reason we're taking this in our calculus 1 course, I haven't even been in college for a month and we're taking this.. and I'm still trying to figure calculus out and try not to be depressed from feeling really dumb.
I'm really grateful for people like you because I almost don't understand anything in my lectures.
Still lost
I know a lot of basics are lost for me but for now they don't even give us the opportunity to try and learn that, always quizzes and stuff (from the second week) so I kinda have to try and understand the topics we're taking and it's so difficult :(
It's all very overwhelming and I'd have probably changed what I'm studying if I didn't know it would help me reach something I really want

[georgesadler]

Professor Leonard thank you for a solid Discussion of Hyperbolic Functions in Calculus Two. Hyperbolic Functions and Inverse Hyperbolic Functions are not difficult to understand however they can be problematic. An application of Hyperbolic Function is a hanging cable between two towers. The hanging cable makes a shape called a Catenary. An example of a Catenary is the Gateway Arch in St. Louis, Missouri. Professor Leonard, you are missing equal signs in your final example in this video. Please correct this small error in the video.

[mxlexrd]

As a point of correction/clarification, when deriving the logarithm formula for inverse cosh, Professor Leonard states that the reason we need to take the positive root is that otherwise we'd get negatives. This isn't quite right, in fact the formula will still always be positive.

The reason we need to take the positive root is that cosh isn't a one-to-one function, so we need to restrict the domain to find an inverse - the restriction we choose is to take cosh(x) where x≥0. The result of this is that we require the range of our inverse to be y≥0, so e^y≥1.

Going back to the quadratic formula, choosing the negative root gives e^y between 0 and 1, so we must choose the positive root, which gives e^y≥1.

[georgegithinji]

Superheroes do exist in this planet. Thank you professor

[ahmedmuawiyah]

Happy new year professor, during Covid, your lectures saved a lot of university students.