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Math Help - Need Some Calc. Help

  1. #1
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    Need Some Calc. Help

    Find the value of k so that f(x)=xe^x/k has a critical value at x=10.

    I am pretty sure I have to find the derivative here, using the product rule? But I'm not exactly how to find the derivative of e^x/k and after that I'm not sure what to do. Set the derivative equal to 10 perhaps? Someone help me clear this up, please.
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  2. #2
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    k is a constant, so we can pull it out, getting \frac{dy}{dx}=\frac{1}{k}(\frac{d}{dx}(xe^x)). Use the product rule to differentiate xe^x, set \frac{dy}{dx}=0, and then plug 10 in for x. Now, solve for k.
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  3. #3
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    Quote Originally Posted by redsoxfan0825 View Post
    Find the value of k so that f(x)=xe^x/k has a critical value at x=10.

    I am pretty sure I have to find the derivative here, using the product rule? But I'm not exactly how to find the derivative of e^x/k and after that I'm not sure what to do. Set the derivative equal to 10 perhaps? Someone help me clear this up, please.
    assuming you mean ...

    f(x) = xe^{\frac{x}{k}}

    f'(x) = x \cdot \frac{1}{k}e^{\frac{x}{k}} + e^{\frac{x}{k}}

    f'(x) = e^{\frac{x}{k}}\left(\frac{x}{k} + 1\right)

    critical values occur where x = 0 ...

    \frac{x}{k} = -1

    at x = 10 , k = -10
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  4. #4
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    Thank you very much. There are a couple of other problems I'm having trouble with as well.

    "The quantity of a drug in the bloodstream t hours after a tablet is swallowed is given by: q(t) = 351(e^-t - e^-2t) in milligrams
    When is the quantity of the drug in the bloodstream maximized? That is, find the value of t that maximizes q(t). Round your answer to two decimal places."


    This is a surge function problem, but it's not in the normal format of ate^-bt that I'm used to working with. I can do (1/b) to find t, but I don't know how to find b. My first thought would be to combine e^-t - e^-2t to find b but I'm not sure if I can combine those just because they have the same base.


    Also, for part of a problem I need the derivative of e^-kt, would that just be -ke^-kt? Or is k positive?
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  5. #5
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    Quote Originally Posted by redsoxfan0825 View Post
    Thank you very much. There are a couple of other problems I'm having trouble with as well.

    "The quantity of a drug in the bloodstream t hours after a tablet is swallowed is given by: q(t) = 351(e^-t - e^-2t) in milligrams
    When is the quantity of the drug in the bloodstream maximized? That is, find the value of t that maximizes q(t). Round your answer to two decimal places."


    This is a surge function problem, but it's not in the normal format of ate^-bt that I'm used to working with. I can do (1/b) to find t, but I don't know how to find b. My first thought would be to combine e^-t - e^-2t to find b but I'm not sure if I can combine those just because they have the same base.


    Also, for part of a problem I need the derivative of e^-kt, would that just be -ke^-kt? Or is k positive?
    you're making this much harder on yourself than it really is ...

    q(t) = 351(e^{-t} - e^{-2t})

    q'(t) = 351(-e^{-t} + 2e^{-2t})

    q'(t) = 0 when 2e^{-2t} - e^{-t} = 0

    e^{-t}(2e^{-t} - 1) = 0

    solve the single correct factor than can equal 0 for t


    next time start a new problem with a new post.
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  6. #6
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    I solved for 2e^-t - 1 = 0 and got t = .69, but when I put that back into the equation I get error so I think I made a mistake somewhere.
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  7. #7
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    Quote Originally Posted by redsoxfan0825 View Post
    I solved for 2e^-t - 1 = 0 and got t = .69, but when I put that back into the equation I get error so I think I made a mistake somewhere.
    that is the correct value of t.

    the maximum value of q(t) occurs at t = \ln(2) \approx 0.693...

    q(\ln{2}) \approx 87.75 mg
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