# Prove that sinhkt is of exponential order |k|?

## Homework Statement

Prove that sinh(kt) is of exponential order |k|.
Find M>0, c>=0 and T>0 to show that
|f(t)|= Me^(ct), t>T

## Homework Equations

|f(t)|= Me^(ct), t>T

## The Attempt at a Solution

I'm looking at the graph of sinhkt (i graphed a few values of k) and indeed it is of an exponential order.

Now,
|sinhkt|<=Me^(ct), t>T

sinhkt isnt bound. So c=/=0=/=t

oh boy... help?

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Dick
Homework Helper

Sigh...ok so,

|e^kt-e^-kt|<=2Me^ct
|e^kt-1/e^kt|<=2Me^ct
|(e^3kt-e^kt)/(e^2kt)|<=2Me^ct
|e^3kt-e^kt|<=2Me^[t(c+2k)]
......

ln|e^3kt-e^kt|<=ln|2Me^[t(c+2k)]|
ln|e^2kt - 1|+ tk <= (c+2k)*t+ln|2M|
ln|[e^2kt - 1]/(2M)]| <= (c+2k)t+kt
ln|[e^2kt - 1]/(2M)]| <= (c+3k)t

OK....????? uhh... now what?

Last edited:
Sigh...ok so,

OK....????? uhh... now what?
heh...its interesting how you were able to express emotion in your posts.

heh...its interesting how you were able to express emotion in your posts.