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--- math104-s21:s:vpak [2021/05/10 23:20]
68.186.63.173 [Summary of Material]
+++ math104-s21:s:vpak [2026/02/21 14:41] (current)
@@ Line 92: / Line 92: @@
 If $\bold{f}$ is differentiable at x, then $\bold{f}$ is also continuous at x. \\
 If $\bold{f}$ is differentiable on interval $\bold{I}$, and $\bold{g}$ is differentiable on range($\bold{f}$),  then $\bold{h}$ $=$ $\bold{g(\bold{f})}$ is differentiable on $\bold{I}$
+A real function $\bold{f}$ has a //local maximum// at point p if there exists $\delta$ $>$ 0 such that $\bold{f(y)}$ $\leq$ $\bold{f(x)}$ for any y where d(x,y) $<$ $\delta$. \\
+If $\bold{f}$ has a local maximum at x, and if $\bold{f'(x)}$ exists, then $\bold{f'(x)}$ $=$ 0.
+**Mean Value Theorem.** If $\bold{f}$ is a real continuous function on [a,b], and is differentiable on (a,b), then there exists an x $\in$ (a,b) such that \\
+$\bold{f(b)}$ $-$ $\bold{f(a)}$ $=$ (b $-$ a) $\bold{f'(x)}$ \\ The generalized theorem for $\bold{f}$ and $\bold{g}$ continuous real functions on [a,b] is \\
+($\bold{f(b)}$ $-$ $\bold{f(a)}$) $\bold{g'(x)}$ $=$ ($\bold{g(b)}$ $-$ $\bold{g(a)}$) $\bold{f'(x)}$
+**Theorem 5.12.** Suppose $\bold{f}$ is real differentiable function on [a,b], and $\bold{f'(a)}$ $<$ $\lambda$ $<$ $\bold{f'(b)}$. Then there exists x $\in$ (a,b) such that $\bold{f'(x)}$ $=$ $\lambda$.
+A function $\bold{f}$ is said to be //smooth// on interval I if $\forall$ x $\in$ I, $\forall$ k $\in$ $\N$, $\bold{f^k}$ exists.
+**L'Hopital Rule.** $\lim\limits_{x \to a}$ $\frac{\bold{f(x)}}{\bold{g(x)}}$ $=$ $\lim\limits_{x \to a}$ $\frac{\bold{f'(x)}}{\bold{g'(x)}}$ if either
+  * $\bold{f(x)}$ $\to$ 0 and $\bold{g(x)}$ $\to$ 0 as x $\to$ a
+  * $\bold{g(x)}$ $\to$ $\infin$ as x $\to$ a
+**Taylor's Theorem.** Let $\bold{f}$ be a real function on [a,b], assume $\bold{f^{n-1}}$ is continuous and $\bold{f^n}$ exists, and for any distinct $\alpha$, $\beta$ $\in$ [a,b] define \\
+P(t) $=$ $\displaystyle\sum_{k=0}^{n-1}$ $\frac{\bold{f^k(\alpha)}}{k!}$ (t $-$ $\alpha$)<sup>k</sup> \\
+Then there exists a point x between $\alpha$ and $\beta$ such that \\
+$\bold{f(\beta)}$ $=$ P($\beta$) $+$ $\frac{\bold{f^n(x)}}{n!}$ ($\beta$ $-$ $\alpha$)<sup>n</sup> \\
+Note Taylor Series on smooth functions may not converge, and may not be equal to original function f(x).
+A //partition// P of [a,b] is the finite set of points where a$=$x<sub>0</sub>$\leq$x<sub>1</sub>$\leq$...x<sub>n</sub>$=$b \\
+Let $\alpha$ be a weight function that is monotonically increasing. Define \\
+U(P, f, $\alpha$) $=$ $\displaystyle\sum_{i=0}^{n}$ M<sub>i</sub> $\Delta{\alpha}$<sub>i</sub> \\
+L(P, f, $\alpha$) $=$ $\displaystyle\sum_{i=0}^{n}$ m<sub>i</sub> $\Delta{\alpha}$<sub>i</sub> \\
+where M<sub>i</sub> is the $\sup$ and m<sub>i</sub> is the $\inf$ over that subinterval. \\
+If $\inf$ U(P, f, $\alpha$) $=$ $\sup$ L(P, f, $\alpha$) over all partitions, then the //Riemann integral// of f with respect to $\alpha$ on [a,b] exists \\
+$\int_a^b f(x)d{\alpha}(x)$
+A //refinement// Q of P contains all the partition points in P, with additional points. \\
+If U(P, f, $\alpha$) $-$ L(P, f, $\alpha$) $<$ $\epsilon$, then U(Q, f, $\alpha$) $-$ L(Q, f, $\alpha$) $<$ $\epsilon$. In other words, refinements maintain the condition for integrability.
+**//Key Theorems: //** \\
+  * If f is continuous on [a,b], then f is integrable on [a,b].
+  * If f is monotonic on [a,b] and if $\alpha$ is continuous on [a,b], then f is integrable on [a,b].
+  * Suppose f is bounded and has finitely many discontinuities on [a,b]. If $\alpha$ is continuous at every point of discontinuity, then f is integrable.
+  * If f is integrable on [a,b] and g is continuous on the range of f, then h $=$ g(f) is integrable on [a,b].
+  * If a$<$s$<$b, f is bounded, f is continuous at s, and $\alpha$(x) $=$ I(x-s) where I is the //unit step function//, then $\int_a^b fd{\alpha}$ $=$ f(s)
+  * Suppose $\alpha$ increases monotonically, $\alpha$' is integrable on [a,b], and f is bounded real function on [a,b]. Then f is integrable with respect to $\alpha$ if and only if f$\alpha$' is integrable: \\ $\int_a^b fd{\alpha}$ $=$ $\int_a^b f(x){\alpha}'(x)d(x)$
+  * Let f be integrable on [a,b] and for a$\leq$x$\leq$b, let F(x) $=$ $\int_a^x f(t)dt$, then \\ (1) F(x) is continuous on [a,b] \\ (2) if f(x) is continuous at p $\in$ [a,b], then F(x) is differentiable at p, with F'(p) $=$ f(p)
+**Fundamental Theorem of Calculus.** Let $f$ be integrable on $[a,b]$ and $F$ be a differentiable function on [a,b] such that $F'(x)$ $=$ $f(x)$, then $\int_a^b f(x)dx$ $=$ $F(b)$ $-$ $F(a)$
+Let $\alpha$ be increasing weight function on $[a,b]$. Suppose $f_n$ is integrable, and $f_n$ $\to$ $f$ uniformly on $[a,b]$. Then $f$ is integrable, and \\
+$\int_a^b fd{\alpha}$ $=$ $\lim\limits_{n \to \infin}$ $\int_a^b f_{n}d{\alpha}$
+Suppose {$f_n$} is a sequence of differentiable functions on $[a,b]$ such that $f_n$ $\to$ $g$ uniformly and there exists p $\in$ $[a,b]$ where {$f_n(p)$} converges. Then $f_n$ converges to some $f$ uniformly, and \\
+$f'(x)$ $=$ $g(x)$ $=$ $\lim\limits_{n \to \infin}$ $f'_{n}(x)$ \\
+Note $f'_{n}(x)$ may not be continuous.
+==== Questions ====
+**1. What **

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