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I wrote this on an exam, is it correct?

  1. Sep 22, 2011 #1
    F(x) always exists and is differentiable as long as f(x) is continuous.

    Do you agree?
     
  2. jcsd
  3. Sep 22, 2011 #2
    No, you have it wrong

    Continuity and Differentiability
    D => C but the converse C => D is not true

    y = |x| is continuous at x=0 but not Differentiable there.


    A function is Continuous ...
    Informally..... if you can trace its graph without lifting your pencil
    Formally .......if Limit f(x) as x-> a = f(a)

    A function is Differentiable if
    Informally..... if it can be approximated linearly (by a tangent line) at the point in question
    Formally....... if the Limit of the definition exists
    f ' (x) = Limit as dx -> 0 [ f(x+dx) - f(x) ] / dx

    Existence and Differentiability are the same thing
     
  4. Sep 22, 2011 #3
    You misunderstood his question. He was talking about primitives and integrals.

    1MileCrash: yes, a continuous function [itex]f:[a,b]\rightarrow \mathbb{R}[/itex] always has a differentiable primitve function.
     
  5. Sep 22, 2011 #4
    If by F(x) you mean an anti-derivative of f(x), you are correct.
     
  6. Sep 22, 2011 #5
    Alrighty, another question on the same subject but not from the exam.

    Of course the nonelementary antiderivatives inspired that question. Is the lower limit in the integral sign of a nonelementary antiderivative kind of like the + C arbitrary constant for elementary antiderivatives?
     
  7. Sep 22, 2011 #6

    Mark44

    Staff: Mentor

    If I understand what you're asking, there's no connection between the lower limit of integration and the arbitrary constant.

    What do you mean by "nonelementary antiderivative?" I get the feeling you're really asking about definite (w. limits of integration) versus indefinite (wo limits of integration) integrals.
     
  8. Sep 22, 2011 #7
    I can tell I wasn't clear, we're running on two different terminologies, what you're referring to, we call improper integrals. By elementary antiderivative, I mean an antiderivative that is just a normal polynomial/logarithmic/what have you function.

    Now that I'm on an actual PC, I can show you.

    The question was:

    [itex]\int 8\sqrt{\frac{3}{4} sin^{2}\theta} d\theta[/itex]

    This function has no elementary antiderivative, but it does have an antiderivative. Show one, and explain why it has an antiderivative.

    An antiderivative is:

    [itex]\int^{\theta}_{0} \sqrt{\frac{3}{4} sin^{2}t} dt[/itex]

    Because that's a function of theta increasing at the rate of:
    [itex]\ 8\sqrt{\frac{3}{4} sin^{2}\theta}[/itex]

    An my answer "why" was the opening of this thread.

    So, my question is, since

    [itex]\int^{\theta}_{0} \sqrt{\frac{3}{4} sin^{2}t} dt[/itex]

    Is increasing at the same rate and therefore has the same derivative whether I replace that 0 with 6, 22, 1000, is it like an arbirary constant for a normal antiderivative, since they are all antiderivatives of the original function no matter what the lower limit of the integral is?
     
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