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--- math104-s21:s:vpak [2021/05/10 16:20]
68.186.63.173 [Summary of Material]
+++ math104-s21:s:vpak [2022/01/11 10:57] (current)
pzhou ↷ Page moved from math104-2021sp:s:vpak to math104-s21:s:vpak
@@ 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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