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Homework Help: How to use the graph of the distance between any two points

  1. Feb 9, 2010 #1
    1. The problem statement, all variables and given/known data

    Write an expression for the distance between the point P(1,2) and an arbitrary point [tex]( x, \sqrt{x} )[/tex] on the curve [tex]y = \sqrt{x}[/tex]. Graph this distance versus x, and use the graph to find the x-coordinate of the point on the curve that is closest to the point P.

    2. Relevant equations


    3. The attempt at a solution

    Well, here's the expression I wrote for the distance:

    [tex]d = \sqrt{ (x - 1)^2 + ( 2 - \sqrt{x} )^2 }[/tex]

    I've graphed this on a graphic software. Now, how can I use this graph to answer the question? What do I have to look for on the graph?
  2. jcsd
  3. Feb 9, 2010 #2


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    Homework Helper

    Since this is the in the pre-calculus forum I'll assume you won't be using derivatives, so you'll be finding an approximate solution to the shortest distance.

    When you graphed the distance versus x, at what (approx) x value is the distance the shortest? In other words, where is d the smallest?
  4. Feb 9, 2010 #3
    d is smallest when x is approx equal to 1.35296

    The answer is correct as given at the back of the book.

    Yes, I didn't want to use derivatives for this question; but if I were to use derivatives, I would have differentiated the function and equated it to zero, and then would've solved for x, right?

    EDIT: Also note that in the graph there's only one minimum and no maximum extrema, so differentiating twice won't be necessary.
    Last edited: Feb 9, 2010
  5. Feb 9, 2010 #4


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    Homework Helper

    Yes that's correct :smile:

    May I ask how you found that answer with such precision?
  6. Feb 9, 2010 #5
    I used a graphing software to create the graph, and then I zoomed in many times on the part of the graph where d was the smallest. I used http://www.walterzorn.com/grapher/grapher_app.htm".
    Last edited by a moderator: Apr 24, 2017
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