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

  • Thread starter ruby_duby
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  • #1
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Homework Statement



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

Homework Equations





The Attempt at a Solution



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

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

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

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

I would really appreciate any help/ guidance
 

Answers and Replies

  • #2
392
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Don't try series. Try Mean Value Theorem.

And don't forget the definition of sinh.
 
  • #3
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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
392
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Usually you don't, but if you do it's easy enough:

e^c=sinh(1), so c=ln(sinh(1))
 
  • #5
CompuChip
Science Advisor
Homework Helper
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thanks so much, ive managed to work out the answer, however, does anyone know if i need to find a specific value for c?

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).
 

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