Intersections or Asymtotes of Sinh(x) and Cosh(x)

[Mt Gargas]

Hi,

Is there a boundary or asymptote type of line limiting the upward path of either Sinh(x) or Cosh(x) and if so what is its cartesian-x-y type formula?

Also since the respective upward-continuations of the Sinh(x) and Cosh(x) curves would seem to be bringing them closer together i was wondering does Sinh(x) ever intersect Cosh(x), and if so at what value of x?

Matthew

[Pubkeybreaker]

On Monday, June 9, 2014 10:17:47 AM UTC-4, Mt Gargas wrote:
> Hi, Is there a boundary or asymptote type of line limiting the upward path of either Sinh(x) or Cosh(x) and if so what is its cartesian-x-y type formula? Also since the respective upward-continuations of the Sinh(x) and Cosh(x) curves would seem to be bringing them closer together i was wondering does Sinh(x) ever intersect Cosh(x), and if so at what value of x? Matthew

I suggest that you start by learning and understanding the definitions
of cosh() and sinh() as real-valued functions. Do you know their
representations as elementary functions? The questions you ask would be answered
immediately from this knowledge.

[Dirk Van de moortel]

Mt Gargas <mtga...@gmail.com> wrote:
> Hi,
>
> Is there a boundary or asymptote type of line limiting the upward
> path of either Sinh(x) or Cosh(x) and if so what is its cartesian-x-y
> type formula?

Yes, as you can see from the definitions, the line
y = 1/2 exp(x)
is an asymptote for both functions on the positive x-side.

On the negative x-side the line
y = 1/2 exp(-x)
is an asymptote for cosh(x), and the line
y = - 1/2 exp(-x)
is an asymptote for sinh(x).

>
> Also since the respective upward-continuations of the Sinh(x) and
> Cosh(x) curves would seem to be bringing them closer together i was
> wondering does Sinh(x) ever intersect Cosh(x), and if so at what
> value of x?

sinh(x) = cosh(x)
<==> 1/2 (exp(x) + exp(-x)) = 1/2 (exp(x) - exp(-x))
<==> exp(-x) = - exp(-x)
<==> 2 exp(-x) = 0
<==> exp(-x) = 0
<==> never

So sinh(x) and cosh(x) never intersect

Dirk Vdm

[Pubkeybreaker]

On Monday, June 9, 2014 10:32:48 AM UTC-4, Dirk Van de moortel wrote:
> Mt Gargas <mtga...@gmail.com> wrote: > Hi, > > Is there a boundary or asymptote type of line limiting the upward > path of either Sinh(x) or Cosh(x) and if so what is its cartesian-x-y > type formula?
> Yes, as you can see from the definitions, the line y = 1/2 exp(x) is an asymptote for both functions on the positive x-side.

BZZT. WRONG. Thank you for playing.

y = exp(x)/2 is not a line. It is a curve.

[Pubkeybreaker]

On Monday, June 9, 2014 10:32:48 AM UTC-4, Dirk Van de moortel wrote:

<snip>

> bringing them closer together i was > wondering does Sinh(x) ever intersect Cosh(x), and if so at what > value of x?



>sinh(x) = cosh(x) <==> 1/2 (exp(x) + exp(-x)) = 1/2 (exp(x) - exp(-x)) <==> exp(-x) = - exp(-x) <==> 2 exp(-x) = 0 <==> exp(-x) = 0 <==> never So sinh(x) and cosh(x) never intersect Dirk Vdm

A GOOD TEACHER would not simply give the answer. Instead, a good teacher would
guide the student to find the answer for himself.

[Dirk Van de moortel]

A good student would not come here.

Dirk Vdm

[Pubkeybreaker]

On Monday, June 9, 2014 11:49:49 AM UTC-4,

>> A GOOD TEACHER would not simply give the answer.
>> Instead, a good > teacher would guide the student to find the answer for himself.

>A good student would not come here. Dirk Vdm

Agreed.

[Dirk Van de moortel]

Pubkeybreaker <pubkey...@aol.com> wrote:
> On Monday, June 9, 2014 11:49:49 AM UTC-4,
>
>>> A GOOD TEACHER would not simply give the answer.
>>> Instead, a good teacher would guide the student to find the
>>> answer for himself.

But agreed :-)

Dirk Vdm

[Pubkeybreaker]

Obligatory question for the OP.

Explain why they are called hyperbolic functions.

[Mt Gargas]

> So sinh(x) and cosh(x) never intersect

Thank you for that clear explanation.

However despite others hinting at something supposedly obvious I unfortunatedly can not 'see' what others can readily 'see' in this other part of my query answered here:-

[Dirk Van de moortel]

Take the definitions
cosh(x) = 1/2 ( exp(x) + exp(-x) )
sinh(x) = 1/2 ( exp(x) - exp(-x) )

For ever increasing x (positive), the term 1/2 exp(x) grows
larger, while the term 1/2 exp(-x) gets closer to 0, so both
their sum and difference get closer to 1/2 exp(x).

For ever decreasing x (negative), the term 1/2 exp(-x) grows
larger, while the term 1/2 exp(x) gets closer to 0, so their
sum (cosh) gets closer to 1/2 exp(-x), whereas their difference
(sinh) gets closer to -1/2 exp(-x).

If you have a software where you draw curves, you can really
"see" it. Try for instance
http://fooplot.com
and have it show all the functions
y = sinh(x)
y = cosh(x)
y = exp(x)/2
y = exp(-x)/2
y = - exp(-x)/2

Have fun :-)

Dirk Vdm

[Mt Gargas]

> Take the definitions
>
> cosh(x) = 1/2 ( exp(x) + exp(-x) )
>
> sinh(x) = 1/2 ( exp(x) - exp(-x) )
>
>
>
> For ever increasing x (positive), the term 1/2 exp(x) grows
>
> larger, while the term 1/2 exp(-x) gets closer to 0, so both
>
> their sum and difference get closer to 1/2 exp(x).
>
>
>
> For ever decreasing x (negative), the term 1/2 exp(-x) grows
>
> larger, while the term 1/2 exp(x) gets closer to 0, so their
>
> sum (cosh) gets closer to 1/2 exp(-x), whereas their difference
>
> (sinh) gets closer to -1/2 exp(-x).
>
>
>
> If you have a software where you draw curves, you can really
>
> "see" it. Try for instance
>
> http://fooplot.com
>
> and have it show all the functions
>
> y = sinh(x)
>
> y = cosh(x)
>
> y = exp(x)/2
>
> y = exp(-x)/2
>
> y = - exp(-x)/2
>
>
>
> Have fun :-)
>
>
>
> Dirk Vdm

Thank You Dirk for that good idiot-proof explanation that lets me too 'see'

M.