# Prove that e^c = sinh(1) for some c in [-1,1]

1. Apr 17, 2009

### ruby_duby

1. The problem statement, all variables and given/known data

Assuming that (ex)’ = (ex) at all point x, prove that ec
= sinh(1) for some point C $$\in$$ (-1,1). Give the full proof.

2. Relevant equations

3. The attempt at a solution

I honestly dont know how to attempt this. i can show that (ex)’ = (ex) by:

(ex) = 1 + x + x2$$/2!$$ + x3$$/3!$$ + x4! + x5$$/5!$$ +...

(ex)’ = 1 + x + x2$$/2!$$ + x3$$/3!$$ + x4! + x5$$/5!$$ +...

I just dont know how to tackle the sinh part of the equation.

I would really appreciate any help/ guidance

2. Apr 17, 2009

### Billy Bob

Re: Analysis

Don't try series. Try Mean Value Theorem.

And don't forget the definition of sinh.

3. Apr 19, 2009

### ruby_duby

Re: Analysis

thanks so much, ive managed to work out the answer, however, does anyone know if i need to find a specific value for c?

4. Apr 19, 2009

### Billy Bob

Re: Analysis

Usually you don't, but if you do it's easy enough:

e^c=sinh(1), so c=ln(sinh(1))

5. Apr 19, 2009

### CompuChip

Re: Analysis

You were only asked to show that it exists, not to actually give it (often in mathematics, it's easier to prove that you can construct or find it, than to actually do it).