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Math Help - Upper Lower Sum

  1. #1
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    Upper Lower Sum

    Consider f(x)=2x+1 over [1,3]. Let P be the partition consisting of points {1,3/2,2,3}. Find L(f,P), U(f,P).
    I'm having trouble calculating. I have L(f,P)=17/2 and U(f,P)=23/2 but don't know how to get there.
    L(f,P)=sum(inf f(x)*2)
    U(f,P)=sum(sup f(x)*2)
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mathematic View Post
    Consider f(x)=2x+1 over [1,3]. Let P be the partition consisting of points {1,3/2,2,3}. Find L(f,P), U(f,P).
    I'm having trouble calculating. I have L(f,P)=17/2 and U(f,P)=23/2 but don't know how to get there.
    L(f,P)=sum(inf f(x)*2)
    U(f,P)=sum(sup f(x)*2)
    Recall that in general for a partition \displaystyle P:a=x_0\leqslant\cdots\leqslant x_n=b one has that \displaystyle U(P,f)=\sum_{j=1}^{n}\sup_{x\in[x_{j-1},x_j]}f(x)\Delta_x_j. So, the only possible ambiguity here is how to compute \displaystyle \sup_{x\in[x_{j-1},x_j]}f(x) but since your f is strictly increasing this is just f(x_j). Make sense?
    Last edited by Drexel28; March 10th 2011 at 08:04 PM.
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  3. #3
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    so the xj is 3 right?
    so f(3)=2(3)+1=17(2)
    L(f,p)=34?
    U(f,p)=f(1)(2)=[2(1)+1]2=3*2=6
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mathematic View Post
    so the xj is 3 right?
    so f(3)=2(3)+1=17(2)
    L(f,p)=34?
    U(f,p)=f(1)(2)=[2(1)+1]2=3*2=6
    No. Look again at the definition. We have \underbrace{1}_{x_0}\leqslant \underbrace{\frac{3}{2}}_{x_1}\leqslant \underbrace{2}_{x_2}\leqslant \underbrace{3}_{x_3}.
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    So f(1)*2+f(3/2)*2+f(2)*2+f(3)*3
    =6+8+5+17
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mathematic View Post
    So f(1)*2+f(3/2)*2+f(2)*2+f(3)*3
    =6+8+5+17
    Why are you multiplying by the things you did? Look again at the definition of the upper sum. You should, for example, by multiplying f(1) by \frac{3}{2}-1=\frac{1}{2}
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  7. #7
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    so f(1)(1/2)+f(3/2)(2-3/2)+f(2)(3-2)
    3/2+4(1/2)+5
    3/2+4/2+5
    7/2+10/2=17/2

    f(3/2)(1/2)+f(2)(2-3/2)+f(3)(3-2)
    4/2+5(1/2)+7
    2+5/2+7
    9/2+14/2
    23/2
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