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Math Help - Help with delta epsilon proofs...

  1. #1
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    Question Help with delta epsilon proofs...

    I need to prove that lim (mx+b) as x approaches c = mc +b. Any tips or help would be greatly appreciated.
    Also, What exactly is the relationship of the definite integral to the derivative?
    Again, help and or tips would be greatly appreciated.
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by bemidjibasser View Post
    I need to prove that lim (mx+b) as x approaches c = mc +b. Any tips or help would be greatly appreciated.
    Also, What exactly is the relationship of the definite integral to the derivative?
    Again, help and or tips would be greatly appreciated.
    For the second one, if f is a function and f ' is its derivative, we have :

    \int_a^b f'(x) ~dx=f(b)-f(a)

    Usually, we define an antiderivative of f, F and say \int_a^b f(x) ~dx=F(b)-F(a)

    Since f is an antiderivative of f ', we have the formula above.
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  3. #3
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    This actually is not that bad. I know that the epsilon-delta thing can be confusing.

    f(x)=mx+b, \;\ m\neq 0

    \lim_{x\to c}(mx+b)=mc+b

    |(mx+b)-(mc+b)|<{\epsilon}

    |m(x-c)|<{\epsilon}

    |x-c|<\frac{\epsilon}{|m|}={\delta}

    Therefore, for {\epsilon}>0, let {\delta}=\frac{\epsilon}{|m|}
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  4. #4
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    Thank you both. Is there a way to expand on the relationship of the derivative and the definite intergal? I think I can follow the formulas above, but how would you go about explaining more in words? Thanks again for help.
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  5. #5
    MHF Contributor Mathstud28's Avatar
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    An alternative to Moo's post would be that in a sense integration and differentiation are inverses. If F(x)=\int_{a}^x f~dx and f is continuous at some point x_0 then F(x) is differentiable at x_0 and F'(x_0)=f(x_0)
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  6. #6
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    Does the fundamental theorem have anything to do with this? Or the mean value theorem? Thank you again...
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  7. #7
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by bemidjibasser View Post
    Does the fundamental theorem have anything to do with this? Or the mean value theorem? Thank you again...
    These are the fundamental theorems of calculus (what me and Moo gave)
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