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--- math104-s21:s:oscarxu [2021/05/11 15:05]
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 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 $\Delta x_i=x_i-x_{i-1}$.

Lecture Notes