Notes - Continuous Mathematics HT23, Taylor's theorem


Flashcards

Can you state the one-dimensional version of Taylor’s theorem in full?


Let $k \ge 0$ be any integer. As long as $f : D \to \mathbb{R}$ satisfies smoothness conditions,

\[f(x) = \sum_{i=0}^k \frac{(x-x_0)^i}{i!}\frac{\text{d}^if}{\text{d}x^i}(x_0) + \frac{(x_0-x)^{k+1}\space}{(k+1)!}\frac{\text{d}f^{k+1}\space}{\text{d}x^{k+1}\space}(\xi)\]

for some $\xi \in (x _ 0, x)$.

How is the

\[\frac{(x_0-x)^{k+1}\space}{(k+1)!}\frac{\text{d}f^{k+1}\space}{\text{d}x^{k+1}\space}(\xi)\]

term in Taylor’s theorem notated?


\[e_{k+1}(x, x_0)\]

Let $D \subseteq \mathbb{R}^n$ and $f : D \to \mathbb{R}$ where $f$ satisfies necessary smoothness conditions. Given vectors $\pmb x, \pmb x _ 0$ and some $k \ge 0$, what does Taylor’s theorem say $f(x)$ is equal to not in terms of the Jacobian or Hessian?


\[\begin{aligned} f(\pmb x) &= f(\pmb x_0) \\\\ &+ \left[ \left(h_1 \frac{\partial}{\partial x_1} + \ldots + h_n \frac{\partial}{\partial x_n}\right)f \right](\pmb x_0) \\\\ &+ \frac{1}{2!}\left[ \left(h_1 \frac{\partial}{\partial x_1} + \ldots + h_n \frac{\partial}{\partial x_n}\right)^2f \right](\pmb x_0) \\\\ &+ \ldots \\\\ &+ \frac{1}{(k+1)!}\left[ \left(h_1 \frac{\partial}{\partial x_1} + \ldots + h_n \frac{\partial}{\partial x_n}\right)^{k+1}f \right](\pmb x_0 + \xi \pmb h) \end{aligned}\]

where $x - x _ 0 = h = (h _ 1, h _ 2, \ldots, h _ n)^\intercal$.

Taylor’s theorem for a multivariate $f$ contains an error term that looks like

\[\frac{1}{(k+1)!}\left[ \left(h_1 \frac{\partial}{\partial x_1} + \ldots + h_n \frac{\partial}{\partial x_n}\right)^{k+1}f \right](\pmb x_0 + \xi \pmb h)\]

What’s the geometric interpretation for $x _ 0 + \xi \pmb h$, where $h = x - x _ 0$?


Some value on the line between $x$ and $x _ 0$.

Let $D \subseteq \mathbb{R}^n$ and $f : D \to \mathbb{R}$ where $f$ satisfies necessary smoothness conditions. Given vectors $\pmb x, \pmb x _ 0$ and some $k \ge 0$, what is the $0$-th order Taylor expansion of $f(\pmb x)$?


\[f(\pmb x) = f(\pmb x_0) + (\pmb x - \pmb x_0)^\intercal \frac{\text df}{\text d\pmb x} (\pmb x^\star)\]

for some $\pmb x^\star$ on the line between $\pmb x _ 0$ and $\pmb x$.

Let $D \subseteq \mathbb{R}^n$ and $f : D \to \mathbb{R}$ where $f$ satisfies necessary smoothness conditions. Given vectors $\pmb x, \pmb x _ 0$ and some $k \ge 0$, what is the first order Taylor expansion of $f(\pmb x)$?


\[\begin{aligned} f(\pmb x) &= f(\pmb x_0) \\\\ &+ (\pmb x - \pmb x_0)^\intercal \frac{\text df}{\text d\pmb x} (\pmb x_0) \\\\ &+ \frac{1}{2}(\pmb x-\pmb x_0)^\intercal \pmb{\text H}(f)(\pmb x^\star)(\pmb x-\pmb x_0) \end{aligned}\]

for some $\pmb x^\star$ on the line between $\pmb x _ 0$ and $\pmb x$.

Let $D \subseteq \mathbb{R}^n$ and $f : D \to \mathbb{R}$ where $f$ satisfies necessary smoothness conditions. Given vectors $\pmb x, \pmb x _ 0$ and some $k \ge 0$, what is the second order Taylor expansion of $f(\pmb x)$?


\[\begin{aligned} f(\pmb x) &= f(\pmb x_0) \\\\ &+ (\pmb x - \pmb x_0)^\intercal \frac{\text df}{\text d\pmb x} (\pmb x_0) \\\\ &+ \frac{1}{2}(\pmb x-\pmb x_0)^\intercal \pmb{\text H}(f)(\pmb x_0)(\pmb x-\pmb x_0) \\\\ &+ \pmb e_3 \end{aligned}\]

for some $\pmb x^\star$ on the line between $\pmb x _ 0$ and $\pmb x$.

Proofs




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