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Find f'(0), if it exists and is f continuous at x=0

  1. Oct 14, 2011 #1
    *couldn't edit the title so Find*

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

    f(x) = x+1 , x<0
    f(x) = 1 , x=0
    f(x)= x2-2x+1 , x>0




    3. The attempt at a solution

    for a (find f'(0), if it exists) i did as followed

    Lim h->0- (x+h)+1-(x) /h giving me in the end 1

    as for the third equation, I did as follow:
    Lim h->0+ (x+h)2-2(x+h)+1 - (x2-2x+1) /h

    in the end giving me 0
    So I answered that f'(0) doesn't exsit

    for b) is f continuous at x=0
    i just solved the limit of x +1 which gave me 1 and the limit of x2-2x+1 which gave me 1. Hence I answered that f is continuous at x=0

    If I need to add more information, let me know =)
     
  2. jcsd
  3. Oct 14, 2011 #2

    SammyS

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    Re: Fin f'(0), if it exists and is f continuous at x=0

    That all looks good. --- except I inserted some parentheses, where needed.
     
  4. Oct 14, 2011 #3

    HallsofIvy

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    Re: Fin f'(0), if it exists and is f continuous at x=0

    By the way, for general functions the fact that [itex]\lim_{x\to a^-}f(x)\ne \lim_{x\to a^+}f(x)[/itex] does not prove that f(a) does not exist- only that f is not continuous at x= a. However, the derivative of a function, while not necessarily continuous, does have the "intermediate value property". That is, if f'(a)= A and f'(b)= B then, for any C between A and B, there exist c between a and b such that f(c)= C.

    That implies that, whether f' is continuous or not, if f'(a) exists, we must have [itex]\lim_{x\to a^-}f(x)= \lim_{x\to a^+}f(x)[/itex] so your argument is valid.
     
  5. Oct 14, 2011 #4
    Re: Fin f'(0), if it exists and is f continuous at x=0

    I don't understand why you brought the "intermediate value" theory in this problem. Isn't used to find if f(x) as a root between a and b considering n isn't equal to a or b?
     
  6. Oct 14, 2011 #5

    vela

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    You evaluated the second limit wrong somehow. It should equal -2, not 0.
    You need to show
    [tex]\lim_{x \to 0} f(x) = f(0)[/tex]What you've written shows that
    [tex]\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x)[/tex]which means
    [tex]\lim_{x \to 0} f(x)[/tex]exists. You should probably state explicitly that the limit equals f(0) so the grader knows you understand what continuity means.
     
  7. Oct 14, 2011 #6

    SammyS

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    How did I miss that ? DUH !
     
  8. Oct 14, 2011 #7
    Lim h->0+ ((x+h)2-2(x+h)+1 - (x2-2x+1)) /
    is then x2+2xh+h2 -2x+2h+ 1 - x2+2x-1
    which can then be simplified to 2xh+h^2+2h
    so Lim h->0+ (2xh+h2+2h)/h
    And here its just basic simplifying however................. I think its where I usually mess up (small mistakes...) and I did Lim h-->0+ 2xh + h + 2h giving in the end 2x(0) + (0) + 2 (0)
     
  9. Oct 14, 2011 #8

    vela

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    You flipped a sign.
     
  10. Oct 15, 2011 #9

    HallsofIvy

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    Re: Fin f'(0), if it exists and is f continuous at x=0

    Mathematical theorems can be used for more than one purpose! Ignoring the reference to the "intermediate value theorem" do you understand my point that finding [itex]\lim_{x\to a^+}f(x)[/itex] and [itex]\lim_{x\to a^-} f(x)[/itex] and finding that they are the same does not necessarily tell you what f(a) is?
     
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