Evaluate where F(x) is differentiable

  1. Hi there, I cannot seem to figure this question out.

    1. The problem statement, all variables and given/known data
    Let f: [0,3] -> R be defined as follows

    x if 0≤x<1,​
    f(X)= 1≤x<2
    x if 2≤x≤3​

    obtain formulas for F(x) = for 0≤x≤3 and sketch the graphs of f(x) and F(x). Where is F(x) differentiable? Evaluate F(x) where differentiable.


    I missed a day of class and am now totally lost. I've read through the sections in Intro to real analysis by Bartle that cover this section and I get no where. Any help on where to go would be greatly appreciated!
     
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,785
    Staff Emeritus
    Science Advisor

    You are missing the definition of f(x) for x between 1 and 2.

    Hopefully, you know that f(x)= x is differentiable for all x. I suspect that the missing formula for x between 1 and 2 also defines a differentiable function. If that is the case, the only problem is whether the function is differentiable at the "joints", x= 1 and x= 2. Apply the definition of the derivative,
    [tex]\lim_{h\to 0}\frac{f(a+h)- f(a)}{h}[/tex]
    with a= 1 and then with a= 2. Look at the one sided limits.
     
  4. shoot. I missed that when I checked it over. it is f(x) = 1 if 1≤x<2

    With that information, would it really make much of a difference to what you said previously?
     
  5. SammyS

    SammyS 8,876
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    You have f(x) and F(x). What relationship is being assumed between those two functions?
     
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