Please help with these concepts, I would like to understand so I can do them on my own.
I honestly have no idea how to begin with these problems..
Write cosh^2(x) + sin^2x in terms of e^x, then write in terms of a hyperbolic trig function if possible.
Find the difference quotient at -1 for cosh(x)
Develop/determine the relationship between tanh(x) and sinh(x)
Develop/determine the relationship between tanh(x) and sech(x)
Is sech^2(x) the difference quotient for tanh(x)? What is the difference quotient for tanh(x)? I've been trying to find out what it is in terms of a hyperbolic trigonometric function but I can't figure it out
See the attached pdf. I did give you the definitions of sinh and cosh. On that first one, cosh^2(x) + ... I assume you meant to write sinh^2(2). It's all by analogy to the trig functions. Do you know Euler's formula? Maybe not, I seem to remember I learned the hyperbolic functions first. As for difference quotient, I'm not familiar with that nomenclature, what have you been told about it?
A difference quotient is essentially for a function f, the formula is f(x+h)-f(x)/h
That's how my teacher defines it
We haven't talked about Euler's formula yet, unfortunately
Assuming that's "all over h," that's what we used to call "the Newton ratio," its limit as h goes to 0 is the derivative, right?
Usually when the hyperbolic functions are introduced, the student will be told right away that they are linear combinations of e^x and e^-x. That's what my pdf gives. I just wanted to remind you of that ... or were you supposed to figure that out yourself in some way?
Euler's formula tells us that the "standard" trig functions sine and cosine are linear combinations of e^ix and e^-ix. That seems odd to a lot of people, but when you get it, you'll find it's a beautiful fact. Euler's formula unifies the exponential and trig functions. You'll probably learn this in diffyq.
So he didn't tell you what sinh(x) is in terms of e^x? W'hat clues did he give you?
The difference quotient thing, what he's driving at is that if you differentiate a hyperbolic function you'll get another. These functions are connected by differentiation, just as the trig functions are. Pretty much it's the same, except for an occasional difference of sign.
Maybe he gave you a few identities?