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Math Help - Finding a point in common

  1. #1
    Senior Member polymerase's Avatar
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    Finding a point in common

    If the two curves y=e^{2x} and y=k\sqrt{x} (where k is a constant) have exactly one point in common, what much be the value of k?
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  2. #2
    Super Member angel.white's Avatar
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    It looks like
    e^{2x}=k\sqrt{x}

    which means
    \frac{e^{2x}}{\sqrt{x}}=k

    I think you would need to know the point where the two equations intersect to determine what the actual value of k is.
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  3. #3
    Senior Member polymerase's Avatar
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    Quote Originally Posted by angel.white View Post
    It looks like
    e^{2x}=k\sqrt{x}

    which means
    \frac{e^{2x}}{\sqrt{x}}=k

    I think you would need to know the point where the two equations intersect to determine what the actual value of k is.
    that is obvious info.....trust me this is not an easy question, my professor is always puts on one really hard question n this is one of them
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by polymerase View Post
    If the two curves y=e^{2x} and y=k\sqrt{x} (where k is a constant) have exactly one point in common, what much be the value of k?
    k = 2 \sqrt{e} is the answer

    Hint: Let one of the functions be y_1 and the other be y_2

    we must fulfill two conditions.

    (1) we must have y_1 = y_2

    (2) we must have y_1' = y_2'

    the first condition is obvious, but can you tell me why we need the second? after that, can you come up with the solution?
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  5. #5
    Super Member angel.white's Avatar
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    Quote Originally Posted by Jhevon View Post
    we must fulfill two conditions.

    (1) we must have y_1 = y_2

    (2) we must have y_1' = y_2'

    the first condition is obvious, but can you tell me why we need the second?
    I can't, and I really want to know, it may be a misconception on my part, but when I was thinking about it, I figured two different shaped curves could touch the same point, giving each a different tangent line. I figured if they were circles, they would have the same tangents, but since they weren't, they wouldn't.

    If I had to guess at an answer, I'd say it's because they don't cross each other, if they could cross each other they could clearly touch the same point with different tangent lines. But, thats just a guess.


    Also, how would you solve for x at this point? I can see that the point would be (\frac{1}{4}, \sqrt{e}) But I don't know how to solve for it without just being able to "see" the answer.
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  6. #6
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by angel.white View Post
    I can't, and I really want to know, it may be a misconception on my part, but when I was thinking about it, I figured two different shaped curves could touch the same point, giving each a different tangent line. I figured if they were circles, they would have the same tangents, but since they weren't, they wouldn't.

    If I had to guess at an answer, I'd say it's because they don't cross each other, if they could cross each other they could clearly touch the same point with different tangent lines. But, thats just a guess.


    Also, how would you solve for x at this point? I can see that the point would be (\frac{1}{4}, \sqrt{e}) But I don't know how to solve for it without just being able to "see" the answer.
    The answer is that you typically don't. This one happened to have a nice solution, but if you can't find one, the best you can do is a numeric approximation.

    -Dan
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