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Homework Help: Proof of an inequality

  1. Mar 16, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove that

    [tex]\frac{1-h}{2}<\sum_{k=1}^{n}x_{2k}(x_{2k+1}-x_{2k-1})<\frac{1+h}{2}[/tex]

    where [tex]0=x_1<x_2<\cdots<x_{2n+1}=1[/tex] such that [tex]x_{k+1}-x_{k}<h[/tex] for [tex]1\le k\le 2n[/tex]


    2. Relevant equations

    How to prove? :-)


    3. The attempt at a solution

    I need to prove
    [tex]\left|-1+2\sum_{k=1}^{n}x_{2k}(x_{2k+1}-x_{2k-1})\right|<h[/tex]

    [tex]\left|-1+2\sum_{k=1}^{n}x_{2k}(x_{2k+1}-x_{2k-1})\right|=\left|-1+2\sum_{k=1}^{n}x_{2k}(x_{2k+1}-x_{2k}+x_{2k}-x_{2k-1})\right|\le\left|-1+4h\sum_{k=1}^{n}x_{2k}\right|[/tex]
     
  2. jcsd
  3. Mar 16, 2010 #2
    Draw a diagram. Each term in the sum corresponds to an approximation of the area under the graph in the interval [tex](x_{2k-1},x_{2k+1})[/tex]. For each interval, find the difference between
    [the approximation] and [the exact value for the area under the graph in the interval [tex](x_{2k-1},x_{2k+1})[/tex]]


    Add up the differences and you will get your result.
     
    Last edited: Mar 16, 2010
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