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Math Help - Partition of Interval

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    Partition of Interval

    Let P = {x0,x1,x2, ... ,xn} be a regular partition of the interval [a,b]. (Show that if f is continuous and decreasing on [a,b], then...

    Uf(P) - Lf(P) = [f(a) - f(b)]▲x
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by WartonMorton View Post
    Let P = {x0,x1,x2, ... ,xn} be a regular partition of the interval [a,b]. (Show that if f is continuous and decreasing on [a,b], then...

    Uf(P) - Lf(P) = [f(a) - f(b)]▲x
    Uhh? What are these?I'm not sure I agree with the question. But, I don
    t know what these are? Is U\left(P,f\right)=\sum_{j=1}^{n}\sup_{x\in[x_j,x_{j+1}]}f(x)\text{ }\Delta x_j?
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    Quote Originally Posted by WartonMorton View Post
    Let P = {x0,x1,x2, ... ,xn} be a regular partition of the interval [a,b]. (Show that if f is continuous and decreasing on [a,b], then... Uf(P) - Lf(P) = [f(a) - f(b)]▲x
    Notice that this is a regular partition. So \Delta_x=\frac{b-a}{n}.
    Also this is a decreasing function which means Uf(P) = \sum\limits_{k = 0}^{n - 1} {f(x_k )\Delta_x }~\&~ Lf(P) = \sum\limits_{k = 1}^n {f(x_k )\Delta_x } .
    Can you see how to finish?
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