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Homework Help: Derivatives and exponential function

  1. Dec 26, 2009 #1
    1. The problem statement, all variables and given/known data
    attachment.php?attachmentid=22710&stc=1&d=1261854223.jpg

    I don't understand why they took M(e)=1 , and how the proceed on with the proof.
    Thanks in advance(=

    2. Relevant equations



    3. The attempt at a solution
     

    Attached Files:

  2. jcsd
  3. Dec 26, 2009 #2
    Re: Derivatives

    Try to write the derivative of the exponential with the limit definition:

    f'(x)=lim h->0 (f(x+h)-f(x))/h

    and try to factor out e^x
     
  4. Dec 27, 2009 #3
    Re: Derivatives

    yes. i understand . However, why did they take M(e)=1?
    I can't really accept their explanation : M(2)<1 and M(4)>1 ,thus they allow M(e)=1..
    My question is how about other real number between 2 to 4?
     
    Last edited: Dec 27, 2009
  5. Dec 27, 2009 #4

    HallsofIvy

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    Science Advisor

    Re: Derivatives

    It's NOT so much "how" as "why". I presume that just before the section you quoted they showed that the derivative of [itex]e^x[/itex], for e any number, "[itex]M(e)e^x[/itex]" where M(e) is a number depending on e only, not on x.

    It is not too difficult to show that M(2) is less than one and that M(3) is larger than 1 (by numerically approximating the limits). "Choosing" M(e) to be 1 is really choosing a specific value of e such that M(e)= 1. My point about M(2) and M(3) is that there is such a value of e, between 2 and 3. You could, then, by a succesion of numerical approximations, show that M(2.7) is less than 1 but that M(2.8) is greater than 1 so "e" is between 2.7 and 2.8. Or that M(2.71) is less than 1 but that M(2.72) is greater than one so that "e" is between 2.71 and 2.72, etc.

    Choosing "e" to be the number such that M(e)= 1 means that the derivative of the function [itex]y= e^x[/itex] is just [itex]e^x[/itex] again.
     
  6. Dec 28, 2009 #5
    Re: Derivatives

    Oh! Thanks hallsofivy.(= I was rather taken aback by the fact that M(e)=1 and not any other real numbers between probably 2.71 to 2.72 . once again thanks!
     
  7. Dec 28, 2009 #6
    Re: Derivatives

    Hi icystrike ;
    M(e)=1 since e is defined to be the real number such that the area under the function
    f(x)=1/x and x=1 ,x=e and the x-axis is equal to 1.
    Best Wishes
    Riad Zaidan
     
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