Consider $f(x)=1 / x$ over the interval $[1,4]$. Let $P$ be the partition consisting of the points $\{1,3 / 2,2,4\}$. \begin{enumerate}[label=(\alph*)] \item Compute $L(f, P), U(f, P)$, and $U(f, P)-L(f, P)$. \item What happens to the value of $U(f, P)-L(f, P)$ when we add the point 3 to the partition? \item Find a partition $P^{\prime}$ of $[1,4]$ for which $U\left(f, P^{\prime}\right)-L\left(f, P^{\prime}\right)<2 / 5$. \end{enumerate}

Exercise 7.2.2

Consider $f (x) = 1 ∕ x$ over the interval $[1, 4]$ . Let $P$ be the partition consisting of the points ${1, 3 ∕ 2, 2, 4}$ .

(a): Compute $L (f, P), U (f, P)$ , and $U (f, P) - L (f, P)$ .
(b): What happens to the value of $U (f, P) - L (f, P)$ when we add the point 3 to the partition?
(c): Find a partition $P^{'}$ of $[1, 4]$ for which $U (f, P^{'}) - L (f, P^{'}) < 2 ∕ 5$ .

TODO

ulissemini

2022-01-27 00:00