I'm trying to come up with a fast approximation or solution to minimising E=(tanh(x)-a)^2 +bx^2 with respect to x, for a,b and x all real and positive.
Hence I'm trying to solve the derivative dE/dx=2(tanh(x)-a)sech^2 (x)+2bx=0. I guess that has no exact solution, but wolframalpha claims there is one. Does anyone know how it's doing it? (I don't have a paid subscription).
Alternatively, what is a good way to approximate my minimisation problem?
If a<1 and b is small (which is often the case), then x=artanh(b) is a good approximation, and then I could do one step of newton-raphson to improve it, and that would suffice. But if a>1, then I can't see what is a good approximation to start from.
Thanks for any suggestions!
https://imgur.com/a/8R10i24 Here is the link to the photo. I am having quite a hard time understanding the problem :<<< Thank youuu
Hey guys, high school senior here. As part of the maths course (level is about A-level Further Maths/ AP BC Calculus difficulty), I need to write a 4000 word "investigation" into any mathematical topic that's related to the course syllabus. I'm currently looking for topics, and was considering doing it on hyperbolic trig functions because they seem interesting. I've read up on the basics, and have been looking at their applications, although have not found too much. Any pointers on a topic, or on the uses of hyperbolic trig functions (in physics would be great, but I'm happy with any in general) would be much appreciated.
What is the latex command for tanh{x}? I tried \tanh{x} on PM solver but it doesn't work.
Just learned hyperbolic functions and I feel like I learned forbidden Calc. Like the dark side of the force or something.
I finished calc 2 this past semester and we never covered these topics. Are these topics necessary for a standard calc 3 course or should I be able to work through professor Leonardβs calc 3 without these?
https://www.geogebra.org/calculator/tqapvatv (edit. y is the angle from positive x-axis. Smaller circles radius is sinh(x) and larger one is cosh(x))
I was playing around with hyperbolic functions for complex numbers when I realize you can have sort of "unit circle" for trigonometric functions but for complex hyperbolic functions. By unit circle I mean having a simple geometric shape where you can see geometrically where the values for the functions comes from.
It is based on the equations:
cosh(x + iy) = cosh(x)cos(y) + i sinh(x)sin(y)
sinh(x + iy) = sinh(x)cos(y) + i cosh(x)sin(y)
If you keep x constant hyperbolic functions draw a ellipses where parameters are either cosh(x) or sinh(x).
I felt this was way too obvious but I did not find it on the internet and i thought it was neat so I shared it here.
What can I do with the hyperbolic function? Are there any explorable topics related to it? What would I need to know further to truly understand the hyperbolic function?
The only thing i know about them is that the inverse functions have good integrals
I'm trying to implement hyperbolic function, that could use any input signal as an `x` parameter.
taking simple hyperbolic function: y = A / xwhere y is output (`any` signal), A is just `A` signal in input, and x would be `Any` signal in input.
That would be really simple, if I could select `any` signal as an denominator in division in arithmetic combinator. But it is impossible.
Do you have any idea on workaround for this? Nothing comes to my mind honestly.
Hey! Calc 1 student here. As I understand it both are essentially the same but on a different shape. Trigs for Unit Circle and Hyperbolics for Unit Hyperbola. My question is that Hyperbolic functions have an equation like sinhx = (e^x - e^-x)/2 but what about Trigonometric functions? Do they have an equation like this one or is it all calculated geometrically?
I'm working with an analytical solution from a partial differential equation to measure a rate of fluid flow through soil in a cylinder. The numerator of the formula contains the inverse hyperbolic sine function. I do not understand for what this function is accounting.
Here is the exercise: https://www.pubs.ext.vt.edu/content/dam/pubs_ext_vt_edu/CSES/CSES-141/CSES-141-PDF.pdf
The formula is at the top of page 9 and is shown below:
K = Q[sinh^-1 (H/r) - (r^2 / H^2 + 1)^0.5 + r / H]/(2ΟH^2 )
Q = volume flow rate L^3 / T, H = height L, r = radius L
Thanks!
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What I've noticed recently is that many students don't really know what hyperbolic functions represent. In classes it's sort of just handed to you and your expected to know what it is and how to use it. When taught about sin, cos and tan we are introduced to them as ways of solving simple problems to do with triangles. However this isn't really done well for there hyperbolic siblings.
The way I realised what cosh and sinh actually represent was when looking at the following two expressions:
cos(x) = (e^ix + e^-ix )/2
cosh(x) = (e^x + e^-x )/2
These two functions do look very similar but when graphed the cosh(x) function is closer in shape to a parabola (although it definitely is not one!).
But lets suppose we instead plot w=cosh(z) where 0<z<iΟ (i.e. all z values are imaginary), all the output values for w will be complex. In fact this plot will look like a cos function.
In other words the cosh function is simply the complex analogue of the cos function. This can be extended to all hyperbolic functions.
I'm posting this as I find many people do not realise this and I hope this will help people understand hyperbolic functions as it has helped me.
Edit: formatting fixes
Edit 2: added a little clarity, thanks u/delcrossb
Problems: https://imgur.com/a/tQQRiNk
I also feel like my notes are missing out on something... Like I don't see how these problems have anything to do with inverse hyperbolic functions, even though answers I can find online (read: answers. the explanations are entirely nonexistent, which is frustrating). Here are my notes from this section: https://imgur.com/a/nKVS3Nr, am I missing anything? I took the notes straight from the textbook and I can't imagine the textbook left out anything I truly need.
I'm just having a really hard time trying to get this stuff to click. I've been able to solve all but three of the 21 problems on my assignment
