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--- math104-s21:s:antonthan [2021/05/07 12:20]
45.48.153.5 [Problem 4]
+++ math104-s21:s:antonthan [2026/02/21 14:41] (current)
@@ Line 165: / Line 165: @@
 ==== Problem 17 ====
+Consider $f(x)e^{g(x)}$, which is continuous and differentiable on the same intervals.  We take its derivative and apply MVT, and we are done.
 ==== Problem 18 ====
 === (a) ===
+Let partition $P_n = \{0, 1/n, 2/n, \ldots, n/n\}$.  Note that $mesh(P_n) = 1/n$ and $L(f, P_n) \leq R_n \leq U(f, P_n)$.
+The idea now is to use Ross 32.7.  Let $\epsilon > 0$, and since $f$ is integrable, there exists a $\delta > 0$ that satisfies $mesh(P) < \delta \implies U(f, P) - L(f, P) < \epsilon$ for all partitions $P$.  We choose $n$ large enough such that $mesh(P_n) < \delta$, so that we get the inequalities: $U(f, P_n) - \epsilon < R_n \leq U(f, P_n$ and $L(f, P_n) \leq R_n < L(f, P_n) + \epsilon$.  This gives us:
+$$U(f) \leq \sup_n(U(f, P_n)) = lim R_n = \inf_n(L(f, P_n)) \leq L(f)$$
+Since we also have $L(f) = U(f)$ by integrability, $lim R_n = \int_0^1 f(x)dx$, as desired.
 === (b) ===
@@ Line 387: / Line 396: @@
 ===== ඞ ඞ ඞ ඞ ඞ =====
-{{ :math104:s:amongus.mp4 |}}
+{{ math104-s21:s:amongus.mp4 |}}

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