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

[Niaboc67]

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?

Thanks

 

[PeroK]

I'm not sure it's particularly important. It's also fairly obvious.

 

[Mark44]

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?

Thanks

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 $U_f(P)$?