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Use tangent to find area of triangle

  1. Jul 19, 2012 #1
    1. The problem statement, all variables and given/known data
    the tangent of the graph 1/x^2 at P(2,1/4) forms a triangle with the x and y axis. Find area of triangle.


    2. Relevant equations



    3. The attempt at a solution
    so f'(x)=-2/x^3

    so the slope of the tangent at point 2, is f'(2)=-2/8 = -1/4=mt

    now i use the equation of the line to determine x + y intercepts

    y=mtx+b plug in point P(2,1/4)
    1/4=-1/4(2) + b
    b=3/4 (y intercept_)

    y=-1/4x+3/4
    let y=0
    0=-1/4x+3/4
    x=3

    therefore AreaΔ=bxh/2

    where Base = |b| = 3/4
    where height = |x| = 3

    A=(3/4)(3)/2
    A=9/8 units^2


    I beleive this is the correct solution, to me everything makes sense, but i'm just kind of nervous about the area.. 9/8 units i mean i know its possible but i dunno... I'm worried i'm messing my numbers up or something.

    ***i found a mistake where my slope was 1/4 and it was supposed to be -1/4
     
    Last edited: Jul 19, 2012
  2. jcsd
  3. Jul 19, 2012 #2

    Ibix

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    Science Advisor

    Have you tried plotting your tangent line and the curve on graph paper? Does the tangent line look plausible, and does the old counting-the-squares-inside-the-triangle method give you something like 9/8? I think you are right, but checks you can do yourself always reassure.

    You switch between x and t for the horizontal coordinate, and m and mt for the gradient. You seem to come out of it with the right answer (now - I spotted your sign error but when I clicked "quote" it had gone, which caused a moment of doubt of my sanity), so I guess these are typos. Proofreading needed when you're going to hand in for real...
     
    Last edited: Jul 19, 2012
  4. Jul 19, 2012 #3

    eumyang

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    Homework Helper

    Looks fine to me. Just a couple of things to nitpick:
    Change the "-2" to "2" and "mt" to "m".

    Change "mt" to "mx".
     
  5. Jul 19, 2012 #4
    okay thanks, i already did the recalculation when x=2, not -2, i just forgot to change the function but i edited it back. and i made it y=Mtx+b is that okay?
     
  6. Jul 19, 2012 #5
    yea sorry about that, and yea i did graph it, it just was looking a little tight on if the tangent could reach point 3 but it's definitly possible. Thanks for the help and the ideas for checking
     
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