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Calculus theorem/proof my teacher posted -- Not sure what it is...

  1. Nov 10, 2015 #1
    My teacher mentioned this was a very important thing to know in calculus, he didn't explain too much about it but tried to emphasize how important is it.

    If $$P=\{a,x,...,x_{k-1},x_{k},...,x_{n}=b\} , P^*=\{a,x,...,x_{k-1},x^*_{k},x_{k},...,x_{n}=b\}$$
    Then, $$L_{f}(P)≤L_{f}(P^*)≤U_{f}(P^*)≤U_{f}(P) $$

    "So adding pts to partition to get P* raises lower sum & decreases upper sum"

    Any ideas guys?

  2. jcsd
  3. Nov 10, 2015 #2


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    I'm not sure it's particularly important. It's also fairly obvious.
  4. Nov 10, 2015 #3


    Staff: Mentor

    Moved this post to the technical math sections, as it doesn't appear to be homework. (Aside: @Niaboc67, you do realize that when you post in the HW sections, you have to use the template?)

    Let's look at this with a specific example.
    ##f(x) = x^2## on the interval [0, 2]
    P = {0, 1, 2}, and P* = {0, 1, 1.5, 2}
    The difference between P and P* is that P* has one more element.
    What are ##L_f(P), L_f(P^*), U_f(P^*)##, and ##U_f(P)##?
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