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Math Help - Mean Value theorem frustration

  1. #1
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    Mean Value theorem frustration

    My question is: Prove that 1/9 <= sqrt(66)-8 <= 1/8, without solving for sqrt(66). The only hint I have is to use the mean value theorem. Please don't cheat, as I have to show my work, and this question is due in 24 hours.
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  2. #2
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    Re: Mean Value theorem frustration

    I'm not sure how to do this using the 'mean value theorem', I'm pretty sure the mean value theorem goes something like: (roughly) when considering an arc between two endpoints of a function, at least one point exists where tangent to the arc is parallel to the line connecting the endpoints.

    However, this is how I would approach your problem:







    With the expression in this form it's easier to argue the following:





    As well as:





    So:



    It must be true that:



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  3. #3
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    Re: Mean Value theorem frustration

    Thanks a lot! I imagine that you're still on the site, so here's another:

    The area between two varying concentric circles is at all times 9pi inches^2. The rate of change of the area of the larger circle is 10pi inches^2/sec. How fast is the circumference of the smaller circle changing when it has area 16pi inches^2?
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  4. #4
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    Re: Mean Value theorem frustration

    Im using large letters for larger circle and small letters for smaller circle
    A-a=9pi dA/dt-da/dt=0 dA/dt=10 So da/dt=10 Whena=16pi pi(r^2)=16pi So r=4
    We want dc/dt
    dc/dt=da/dt*dc/da c=2pir and a=pir^2 so get dc/da from dc/dr*dr/da and finish off
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