The MVT, Higher Order Partial Derivatives, and Taylor's Formula Review

# The Mean Value Theorem, Higher Order Partial Derivatives, and Taylor's Formula Review

We will now review some of the recent material regarding the Mean Value Theorem, higher order partial derivatives of functions, and Taylor's formula for functions from $\mathbb{R}^n$ to $\mathbb{R}$.

• Recall from The Mean Value Theorem for Differentiable Functions from Rn to Rm page that if $S \subseteq \mathbb{R}^n$ is open, $S \subseteq \mathbb{R}^n$, and $f : S \to \mathbb{R}^m$ is differentiable on all of $S$ then if $\mathbf{x}, \mathbf{y} \in S$ are such that the line segment joining these points is contained in $S$, i.e., $L(\mathbf{x}, \mathbf{y}) \subset S$, then for every $\mathbf{a} \in \mathbb{R}^m$ there exists a point $\mathbf{z} \in L(\mathbf{x}, \mathbf{y})$ such that:
(1)
\begin{align} \quad \mathbf{a} \cdot [ \mathbf{f}(\mathbf{y}) - \mathbf{f}(\mathbf{x}) ] = \mathbf{a} \cdot [ \mathbf{f}'(\mathbf{z}) (\mathbf{y} - \mathbf{x})] \end{align}
• Note that the notation "$L(\mathbf{x}, \mathbf{y})$" denoting the line segment joining the points $\mathbf{x}$ and $\mathbf{y}$ can be written as:
(2)
\begin{align} \quad L(\mathbf{x}, \mathbf{y}) = \{ (1 - t)\mathbf{x} + t\mathbf{y} : t \in [0, 1] \} \end{align}
• We then looked at some corollaries to the Mean Value theorem on the Corollaries to the Mean Value Theorem for Differentiable Functions from Rn to Rm page. We proved the less general multivariable real-valued function Mean Value Theorem which states that if $S \subseteq \mathbb{R}^n$ is open, $f : S \to \mathbb{R}$, and $\mathbf{x}, \mathbf{y} \in S$ are such that $L(\mathbf{x}, \mathbf{y}) \subset S$ then there exists a point $\mathbf{z} \in L(\mathbf{x}, \mathbf{y})$ such that:
(3)
\begin{align} \quad f(\mathbf{y}) - f(\mathbf{x}) = \nabla f(\mathbf{z}) \cdot (\mathbf{y} - \mathbf{x}) \end{align}
• We also proved that if $\mathbf{f} : S \to \mathbb{R}^m$ is differentiable on $S$ then for any two points $\mathbf{x}, \mathbf{y} \in S$ with $L(\mathbf{x}, \mathbf{y}) \subset S$ there exists a point $\mathbf{z} \in L(\mathbf{x}, \mathbf{y})$ such that:
(4)
\begin{align} \quad \| \mathbf{f}(\mathbf{y}) - \mathbf{f}(\mathbf{x}) \| \leq M \| \mathbf{y} - \mathbf{x} \| \end{align}
• Where $\displaystyle{M = \sum_{k=1}^{m} \| \nabla f_k(\mathbf{z}) \|}$.
• On the A Sufficient Condition for the Differentiability of Functions from Rn to Rm page we looked at a very important result which gives us a sufficient condition for the differentiability of a function from $\mathbb{R}^n$ to $\mathbb{R}^m$. We saw that if $\mathbf{f} : S \to \mathbb{R}^m$ and $\mathbf{c} \in S$ then if one of the partial derivatives of $\mathbf{f}$ at $\mathbf{c}$ exist, i.e, one of $D_1 \mathbf{f}(\mathbf{c})$, $D_2 \mathbf{f}(\mathbf{c})$, …, $D_n \mathbf{f} (\mathbf{c})$ exist, and the remaining $n - 1$ of these partial derivatives are continuous on an open ball centered at $\mathbf{c}$ then $\mathbf{f}$ is differentiable at $\mathbf{c}$.
• On the Higher Order Partial Derivatives of Functions from Rn to Rm page we defined higher order partial derivatives of functions from $\mathbb{R}^n$ to $\mathbb{R}^m$. Essentially, they are the partial derivatives of partial derivatives, etc… We looked at a couple of examples in computing these higher order partial derivatives.
(5)
\begin{align} \quad \quad f(x, y) = \left\{\begin{matrix} xy \frac{x^2 - y^2}{x^2 + y^2}& \mathrm{if} \: (x, y) \neq (0, 0)\\ 0 & \mathrm{if} \: (x, y) = (0, 0) \end{matrix}\right. \end{align}
• We proved that $D_{1, 2} f(x, y) = 1 \neq -1 = D_{2, 1} f(x, y)$ which shows that mixed partial derivatives need not equal. Nevertheless, we stated an important result which gives us a sufficient condition for when second order mixed partial derivatives equal. We saw that if $\mathbf{f} : S \to \mathbb{R}^m$ and $\mathbf{c} \in S$ then if the partial derivatives $D_j \mathbf{f}$ and $D_k \mathbf{f}$ exist on an open ball centered at $\mathbf{c}$, $B(\mathbf{c}, r)$ and if $D_j \mathbf{f}$ and $D_k \mathbf{f}$ are differentiable at $\mathbf{c}$ then:
(6)
\begin{align} \quad D_{j, k} \mathbf{f}(\mathbf{c}) = D_{k, j} \mathbf{f} (\mathbf{c}) \end{align}
• On the Taylor's Formula for Functions from Rn to R page we said that if $f : S \to \mathbb{R}$, $\mathbf{x} \in S$, and if all of the second order partial derivatives of $f$ at $\mathbf{x}$ exist, i.e., $D_{i, j} f(\mathbf{x})$ exist for $i, j \in \{ 1, 2, ..., n \}$ then the Second Order Directional Derivative of $\mathbf{f}$ at $\mathbf{x}$ in the Direction of $\mathbf{t} \in \mathbb{R}^n$ is defined as:
(7)
\begin{align} \quad f''(\mathbf{x}, \mathbf{t}) = \sum_{i=1}^{n} \sum_{j=1}^{n} D_{i, j} f(\mathbf{x}) t_j t_i \end{align}
• We similarly defined the Third Order Directional Derivative of $\mathbf{f}$ at $\mathbf{x}$ in the Direction of $\mathbf{t}$ (which exists provided that all of the third order partial derivatives of $f$ exist) and is defined as:
(8)
\begin{align} \quad f'''(\mathbf{x}, \mathbf{t}) = \sum_{i=1}^{n} \sum_{j=1}^{n} \sum_{k=1}^{n} D_{i, j,k} f(\mathbf{x}) t_k t_j t_i \end{align}
• An analogous definition was also made for even higher order directional derivatives of $\mathbf{f}$ at $\mathbf{x}$ in the direction of $\mathbf{t}$.
• We then stated a very nice formula known as Taylor's formula which states that if $S \subseteq \mathbb{R}^n$, and if $f : S \to \mathbb{R}$ and all of the partial derivatives of $f$ of order less than $m$ are differentiable on $S$ then for all $\mathbf{a}, \mathbf{b} \in S$ with $L(\mathbf{a}, \mathbf{b}) \subset S$ there exists a point $\mathbf{z} \in L(\mathbf{a}, \mathbf{b})$ such that:
(9)
\begin{align} \quad f(\mathbf{b}) - f(\mathbf{a}) = \sum_{k=1}^{m-1} \frac{1}{k!} f^{(k)} (\mathbf{a}, \mathbf{b} - \mathbf{a}) + \frac{1}{m!} f^{(m)} (\mathbf{z}, \mathbf{b}-\mathbf{a}) \end{align}