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Integrating and differentiating the number e.

  1. Jan 20, 2005 #1
    In calculus, my work has recently involved integrating and differentiating the numer e, of which I am very unsure of how to do. I set up some examples for myself to try to figure out, could anyone tell me if they are correct? Please correct me if I am wrong, or tell me where I have made a mistake.

    1) [itex]\int_{}^{} edx = e[/itex] Simple, or it should be. This is how I interpet it from what I have read. The text always makes basics sound confusing to me.

    2) [itex]\int_{}^{} e^xdx = e^x[/itex] The same as above because [itex] ln(e) = 1[/itex].

    3) [itex]\int_{}^{} e^{2x}dx = \frac{1}{2x+1}e^{2x+1}[/itex] No clue here, to me it looks completely wrong.

    4)[itex]\frac{d}{dx} e^{2x} = \frac{1}{2}e^{2x}[/itex] Chain rule applies here, correct? Or is it just [itex]\frac{d}{dx} e^{2x} = e^{2x}[/itex]?


    Also, I did a search for another thread that would explain this but did not find one. If one of you reading this could post the link if there is another thread discussing it, or if you know of a really good web page where all of this is explained in a simple, straight-foward way, I would appreciate it very much.

    I apologise if this belongs in the homework section, but this is not exactly homework.
     
    Last edited: Jan 20, 2005
  2. jcsd
  3. Jan 20, 2005 #2
    [tex]\int e dx = e \int 1 dx = e x[/tex]

    e is just a constant. Don't forget to treat it like one.

    2 is correct, but I don't understand your reasoning. The derivative of e^x is e^x, so the antiderivative of e^x should also be e^x. I don't see where logarithms come in, or why log(e) = 1 is useful.

    3 is wrong. Use the substitution u = 2x and du/2 = dx to make an integral you know how to deal with. Also, you can be sure your solution is not correct by differentiating your answer. You should get back the integrand. Also keep in mind that the equation you mistakenly used in this case is for x raised to a constant power, not a constant raised to the x power. That result is different.

    4 is fine.

    --J
     
    Last edited: Jan 20, 2005
  4. Jan 20, 2005 #3

    Hurkyl

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    (4)'s wrong -- he should've multiplied by two.


    Technically, they're all wrong. e.g.

    [tex]
    \int e^x \, dx = e^x + C
    [/tex]

    Where C denotes an arbitrary constant. Don't forget that indefinite integrals have many solutions, one for each choice of constant.

    And for (4), I think he meant to write:

    [tex]
    \frac{d}{dx} e^x
    [/tex]
     
  5. Jan 20, 2005 #4
    Yup.

    --J

    (extra characters added here.)
     
  6. Jan 20, 2005 #5

    HallsofIvy

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    Actually, he's doing everything wrong:

    [itex]\frac{dy}{dx}e^{2x}[/itex] doesn't even make sense. It should be either

    [itex]\frac{de^{2x}}{dx}= 2e^{2x}[/itex] or [itex]\frac{dy}{dx}= 2e^{2x}[itex] as long as [itex]y= e^{2x}[/itex].

    Looks to me like "theCandyman" has a serious problem with the basic concepts of functions and function notation.
     
  7. Jan 20, 2005 #6
    What does this mean? Sorry about the notation, and thank you for pointing it out. I am just too used to writing that whenever I do differentiation.

    Thank you, Hurkyl. I was so concentrated on trying to find what everything was that I forgot the constant. I wrote four correctly, is what HallsofIvy wrote the answer then?

    Justin Lazear, [itex]\int_{}^{}k^x = \frac{k^x}{ln(k)}[/itex], this is where I got [itex]ln(e) = 1[/itex].
     
    Last edited: Jan 20, 2005
  8. Jan 20, 2005 #7

    learningphysics

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    For part 4, is this what you wanted:

    [tex]\frac{d}{dx} e^{2x}[/tex]

    Or in other words the derivative of [tex]e^{2x}[/tex]?

    If so then the answer is [tex]2e^{2x}[/tex]

    But
    [tex]\frac{dy}{dx} e^{2x}[/tex] does not make sense.
     
  9. Jan 21, 2005 #8

    dextercioby

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    BTW,noone mentioned anything about the title of the thread.Quite interesting,i must say...
    Please compute the derivative and the antiderivative for the function:
    [tex] y(x)=e [/tex]

    Daniel.
     
  10. Jan 21, 2005 #9
    As Justin Lazear said, "e is just a constant. Don't forget to treat it like one." He already integrated e, the derivative is 0.
     
  11. Jan 21, 2005 #10
    Isn't e one of the only thing in calculus who's derivate is itself? I dont know if your asking a question or answering one, so ill answer :smile:
     
  12. Jan 21, 2005 #11

    learningphysics

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    e is just a constant. so its derivative is 0.

    But the derivative of [tex]e^x[/tex] is itself.
     
  13. Jan 21, 2005 #12
    Sorry I got confused, I thought I saw e^x, thanks for the clarification :smile:
     
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