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Math Help - Evaluating a limit with an integral in it?

  1. #1
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    Question Evaluating a limit with an integral in it?

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  2. #2
    Super Member Anonymous1's Avatar
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    Wolfram Mathematica Online Integrator

    yikes.

    It may be a change of variables type of limit. You know, something like \lim_{h\rightarrow 0} \frac{1}{h} \int f dx = \lim_{y\rightarrow \infty} y \int f dx.

    Do you have any notes on this?
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  3. #3
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    *sigh* No. And I already tried using wolfram. But I think I might have an idea using the second fundamental theorem of calculus... maybe.
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    Super Member Anonymous1's Avatar
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    Are you taking Real Analysis? Or calc?
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  5. #5
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    I'd evaluate the integral, substitute 2 and 2+h for the t values, and multiply that by 1/h. Then I'd evaluate the limit.
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  6. #6
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    Quote Originally Posted by Anonymous1 View Post
    Are you taking Real Analysis? Or calc?
    Calculus 1
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    Super Member Anonymous1's Avatar
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    Quote Originally Posted by SethP View Post
    I'd evaluate the integral, substitute 2 and 2+h for the t values, and multiply that by 1/h. Then I'd evaluate the limit.
    *edit* You would, if you COULD.
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  8. #8
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    Actually I think I got it figured out... I'm not sure how I did it, but I got the right answer.

    Thanks anyway, guys.
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  9. #9
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    Oh, yeah. It's been a while since I've done limits; my brain sort of gets rid of stuff I haven't thought of in a while.
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    Quote Originally Posted by lauren72 View Post
    Actually I think I got it figured out... I'm not sure how I did it, but I got the right answer.

    Thanks anyway, guys.


    You used the 2^{nd} FTC?
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  11. #11
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    Yup. And got sqrt(15)
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  12. #12
    Super Member Random Variable's Avatar
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    Let  F(x) be an antiderivative of  \sqrt{7+t^{3}}

    then  \lim_{h \to 0} \ \frac{1}{h} \int^{2+h}_{2} \sqrt{7+t^{3}} = \lim_{h \to 0} \frac{F(2+h)-F(2)}{h} = F'(2)= \sqrt{7+2^{3}}  = \sqrt{15}

    so your answer is correct
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