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--- math104-s21:s:oscarxu [2021/05/11 12:32]
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+++ math104-s21:s:oscarxu [2026/02/21 14:41] (current)
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 Theorem: Suppose $f_n: X\to \mathbb{R}$ satisfies that $\forall \epsilon >0,\exists N>0$ such that $\forall x\in X, \lvert f_n(x) - f_m(x) \rvert < \epsilon$, then $f_n$ converges uniformly.
+Theorem: Suppose $f_n \to f$ uniformly on set $E$ in a metric space.  of $E$, and suppose that $\lim_{t\to x} f_n(t) = A_n$. Then $\{ A_n\}$ converges and $\lim_{t\to x} f(t) = \lim_{n\to\infty} A_n$. In conclusion, $\lim_{t\to x} \lim_{n\to\infty} f_n(t) = \lim_{n\to\infty} \lim_{t\to x} f_n(t)$.
+** After Midterm 2 **
+Definition of derivatives: Let $f: [a, b] \to \mathbb{R}$ be a real valued function. Define $\forall x \in [a, b]$
+$f'(x) = \lim_{t \to x} \frac{f(t) - f(x)}{t - x}$
+This limit may not exist for all points. If $f'(x)$ exists, we say $f$ is differentiable at $x$.
+Proposition: if $f:[a, b] \to \mathbb{R}$ is differentiable at $x_0 \in [a, b]$ then $f$ is continuous at $x_0$. $\lim_{x \to x_0} f(x) = f(x_0)$
+Theorem: Let $f, g: [a, b] \to \mathbb{R}$. Assume that $f, g$ are differentiable at point $x_0 \in [a, b]$, then
+$\forall c \in \mathbb{R}$, $(c \dot f)')x_0) = c \dot f'(x_0)$
+$(f + g)'（x_0） = f'(x_0) + g'(x_0)$
+$(fg)'(x_0) = f'(x_0) \cdot g(x_0) + f(x_0) \cdot g'(x_0)$
+if $g(x_0) \neq 0$, then $(f/g)'(x_0) = \frac{f'g - fg'){g^2(x_0)}$
+Mean value theorem: Say $f: [a, b] \to \mathbb{R}$. We say $f$ has a local minimum at point $p \in [a, b]$. If there exists $\delta > 0$ and $\forall x \in [a, b] \cap B_S(p), we have $f(x) \leq f(p).
+Proposition: Let $f: [a, b] \to \mathbb{R}$ If $f$ has local max at $p \in (a, b)$, and if $f'(p)$ exists, then $f'(p)=0$.
+Rolle theorem: Suppose $f: [a, b] \to \mathbb{R}$ is a continuous function and $f$ is differentiable in $(a, b)$. If $f(a) = f(b)$, then there is some $c \in (a, b)$ such that $f'(c) = 0$.
+Generalized mean value theorem: Let $f, g: [a, b] \to \mathbb{R}$ be a continuous function, differentiable on $(a, b)$. Then there exists $c \in (a, b)$, such that
+$(f(a) - f(b)) \cdot g'(c) = [g(a) - g(b)] \cdot f'(c)$
+Theorem: Let $f: [a, b] \to \mathbb{R}$ be continous, and differentiable over $(a, b)$. Then there exists $c \in (a, b)$, such that
+$[f(b) - f(a)] = (b - a) \cdot f'(c)$
+Corollary: Suppose $f:[a, b] \to \mathbb{R}$ continous $f'(x)$ exists for $x \in (a, b)$, and $|f'(x)| \leq M$ for some constant $M$, then $f$ is uniformly continous,
+Corollary: Let $f:[a, b] \to \mathbb{R}$ continuous, and differentiable over $(a, b)$. If $f'(x) \geq 0 $ for all $x \in (a, b)$, then $f$ is monotone increasing.
+If $f'(x) > 0$ for all $x \in (a, b)$, then $f$ is strictly increasing.
+Theorem: Assume $f, g: (a, b) \to \mathbb{R}$ differentiable, $g(x) \neq 0$ over $(a, b)$. If either of the following condition is true
+(1) $\lim_{x \to a}f(x) = 0, \lim_{x \to a}g(x) = 0
+(2) $\lim_{x \to a}g(x) = \infty$
+And if $\lim_{x \to a} \frac{f'(x)}{g'(x)} = A \in \mathbb{R} \cup {\infty, -\infty}$
+then $\lim_{x \to a} \frac{f(x)}{g(x)} = A$
+Higher derivatives: If $f'(x)$ is differentiable at $x_0$, then we define $f''(x_0) = (f')'(x_0)$. Similarly, if the $(n - 1)$-th derivative exists, n-th derivative.
+Definition of smooth: $f(x)$ is a smooth function on $(a, b)$ if $\forall x \in (a, b)$, if $\forall x \in (a, b)$, any order derivative exists.
+Taylor theorem: Suppose $f$ is a real function on $[a,b]$, $n$ is a positive integer, $f^{(n-1)}$ is continuous on $[a,b]$, $f^{(n)}(t)$ exists for every $t\isin (a,b)$. Let $\alpha, \beta$ be distinct points of $[a,b]$, and define $P(t) = \sum_{k=0}^{n-1} \frac{f^{(k)}(\alpha)}{k!} (t-\alpha)^k$. Then there exists a point $x$ between $\alpha$ and $\beta$ such that $f(\beta) = P(\beta) + \frac{f^{(n)}(x)}{n!}(\beta - \alpha)^n$.
+Taylor series for a smooth function If $f$ is a smooth function on $(a,b)$, and $\alpha \isin (a,b)$, we can form the Taylor Series: \\ $P_{\alpha}(x)=\sum_{k=0}^{\infty} \frac{f^{(k)}(\alpha)}{k!} (x-\alpha)^k$.
+Definition of Nth order taylor expansion:
+$P_{x_o,N}(x) = \sum_{n=0}^{N} f^{n)}(x_o) * \frac{1}{n!} (x-x_o)^n$.
+Definition of Partition: Let $[a,b]\subset \Reals$ be a closed interval. A partition $P$ of $[a,b]$ is finite set of number in $[a,b]$: $a=x_0 \leq x_1 \leq ... \leq x_n=b$. Define $\Delta x_i=x_i-x_{i-1}$.

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