Taylor's theorem

-

The essence of differentiation is to approximate an arbitrary function by a linear function. We can extend this idea by using higher-order derivatives to obtain better approximations.



Theorem (Taylor's theorem)

Let \(f(x)\) be a function of class \(C^{n}\) on an open interval \(I\). Let \(a \in I\). Then for all \(x \in I\), there exists \(c_x\) between \(x\) and \(a\) such that

\[ \begin{eqnarray} f(x) &=& f(a) + f'(a)(x-a) + \frac{1}{2}f''(a)(x-a)^2 + \cdots\nonumber\\ && + \frac{1}{(n-1)!}f^{(n)}(a)(x-a)^{n-1} + \frac{1}{n!}f^{(n)}(c_x)(x-a)^{n}.\nonumber\\ \label{eq:taylor} \end{eqnarray} \]

Proof. If \(x=a\), then we can set \(c_x = a\) and Eq. (eq:taylor) clearly holds.

Suppose \(b \in I\), \(b\neq a\). We need to show

\[ \begin{eqnarray} f(b) &=& f(a) + f'(a)(b-a) + \frac{1}{2}f''(a)(b-a)^2 + \cdots\nonumber\\ && + \frac{1}{(n-1)!}f^{(n-1)}(a)(b-a)^{n-1} + \frac{1}{n!}f^{(n)}(c_b)(b-a)^{n}.\nonumber\\ && \label{eq:taylorb} \end{eqnarray} \]

for some \(c_b\). Let us define the constant \(A\) by

\[ A(b-a)^{n} = f(b) - \left[f(a) + f'(a)(b-a) + \cdots +\frac{1}{(n-1)!}f^{(n-1)}(a)(b-a)^{n-1}\right] \]

and the function \(g(x)\) on \(I\) by

\[ g(x) = f(b) - \left[f(x) + f'(x)(b-x) + \cdots + \frac{1}{(n-1)!}f^{(n-1)}(x)(b-x)^{n-1} + A(b-x)^{n}\right]. \]

\(g(x)\) is a differentiable function on \(I\) and \(g(a) = g(b) = 0\). Thus, by Rolle's theorem, there exists \(c_b\) between \(a\) and \(b\) such that \(g'(c_b) = 0\).

[See also: Mean Value Theorem for Rolle's theorem.]

Let us calculate \(g'(x)\). Noting that, for \(k \geq 1\),

\[\left[\frac{1}{k!}f^{(k)}(x)(b-x)^k\right]' = \frac{1}{k!}f^{(k+1)}(x)(b-x)^k - \frac{1}{(k-1)!}f^{(k)}(x)(b-x)^{k-1},\]

the first several terms of \(g'(x)\) are

\[-f'(x) +f'(x) - f''(x)(b-x) + f''(x)(b-x) - \frac{1}{2}f'''(x)(b-x)^2 + \cdots\]

so that most terms are canceled. After all, we have

\[ g'(x) = -\frac{1}{(n-1)!}f^{(n)}(x)(b-x)^{n-1} + nA(b-x)^{n-1}. \]

Since \(g'(c_b) = 0\), we have

\[ 0 = -\frac{1}{(n-1)!}f^{(n)}(c_b)(b-c_b)^{n-1} + nA(b-c_b)^{n-1} \]

or

\[ A = \frac{1}{n!}f^{(n)}(c_b). \]

Substituting this \(A\) into \(g(a) = 0\), we have Eq. (eq:taylorb). ■

Let's see what Taylor's theorem means. Let us define the polynomial of degree \(n\) by

\[ P_n(x) = f(a) + f'(a)(x-a) + \frac{1}{2}f''(a)(x-a)^2 +\cdots + \frac{1}{n!}f^{(n)}(a)(x-a)^{n}. \]

Taylor's theorem states that the difference between \(f(x)\) and \(P_{n}(x)\) is given by

\[ f(x) - P_{n}(x) = \frac{1}{(n+1)!}f^{(n+1)}(c_x)(x-a)^{n+1} \]

for some \(c_n\). If \(n\) is sufficiently large and \(x\) is close to \(a\) (i.e., \(|x - a|\) is small), then \(|(x-a)^{n+1}|\) is very small so that the above difference is very small. Thus, Taylor's theorem indicates that the function \(f(x)\) can be approximated by polynomial functions.

Example. Consider \(f(x) = \sin x\). With \(a = 0\), we have

\[ \begin{eqnarray} P_1(x) &=& x,\\ P_3(x) &=& x - \frac{1}{6}x^3,\\ P_5(x) &=& x - \frac{1}{6}x^3 + \frac{1}{120}x^5,\\ &\vdots& \end{eqnarray} \]

By plotting the graphs of these polynomials, we can see that they approximate \(\sin x\) with increasingly better accuracies.

Approximation of \(sin x\) by Taylor expansions.

□

Definition (finite Taylor expansion and finite Maclaurin expansion)

The right-hand side of Eq. (eq:taylor) in Taylor's Theorem is called the finite Taylor expansion and the last term \(\frac{1}{n!}f^{(n)}(c_x)(x-a)^n\) is called the remainder or residual. The finite Taylor expansion can be also expressed as

\[f(x) = \sum_{k=0}^{n-1}\frac{1}{k!}f^{(k)}(a)(x-a)^k + \frac{1}{n!}f^{(n)}(a + \theta(x-a))(x-a)^n\]

where \(\theta\in (0,1)\) is a constant dependent on \(x\).

In particular, the finite Taylor expansion with \(a=0\) is called the finite Maclaurin expansion. 

Example. The finite Maclaurin expansion of \(e^x\) is given by

\[e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \cdots + \frac{x^{n-1}}{(n-1)!} + \frac{e^{\theta x}x^n}{n!}.\]

(Exercise!) □

Example. The finite Maclaurin expansion of \(\log(x+1)\) is given by

\[\log(x+1) = x - \frac{1}{2}x^2 + \frac{1}{3}x^3 - \frac{1}{4}x^4 + \cdots + \frac{(-1)^n}{(n-1)}x^{n-1} + \frac{(-1)^{n+1}x^n}{n(\theta x + 1)^n}.\]

(Exercise!) □

Comments

Post a Comment

Popular posts from this blog

Open sets and closed sets in \(\mathbb{R}^n\)

-
Image
Open sets In \(\mathbb{R}\), we have the notion of an open interval such as \((a, b) = \{x \in \mathbb{R} | a < x < b\}\). We want to extend this idea to apply to \(\mathbb{R}^n\). We also introduce the notions of bounded sets and closed sets in \(\mathbb{R}^n\). Recall that the \(\varepsilon\)-neighbor of a point \(x\in\mathbb{R}^n\) is defined as \(N_{\varepsilon}(x) = \{y \in \mathbb{R}^n | d(x, y) < \varepsilon \}\) where \(d(x,y)\) is the distance between \(x\) and \(y\). Definition (Open set) A subset \(U\) of \(\mathbb{R}^n\) is said to be an  open set  if the following holds: \[\forall x \in U ~ \exists \delta > 0 ~ (N_{\delta}(x) \subset U).\tag{Eq:OpenSet}\] That is, for every point in an open set \(U\), we can always find an open ball centered at that point, that is included in \(U\). See the following figure. Perhaps, it is instructive to see what is  not  an open set. Negating (Eq:OpenSet), we have \[\exists x \in U ~ \forall \delta > 0 ~ (N_{\delta}(x) \not
Read more

Euclidean spaces

-
Image
We would like to study multivariate functions (i.e., functions of many variables), continuous multivariate functions in particular. To define continuity, we need a measure of "closeness" between points. One measure of closeness is the Euclidean distance. The set \(\mathbb{R}^n\) (with \(n \in \mathbb{N}\)) with the Euclidean distance function is called a Euclidean space. This is the space where our functions of interest live. The real line is a geometric representation of \(\mathbb{R}\), the set of all real numbers. That is, each \(a \in \mathbb{R}\) is represented as the point \(a\) on the real line. The coordinate plane , or the \(x\)-\(y\) plane , is a geometric representation of \(\mathbb{R}^2\), the set of all pairs of real numbers. Each pair of real numbers \((a, b)\) is visualized as the point \((a, b)\) in the plane. Remark . Recall that \(\mathbb{R}^2 = \mathbb{R}\times\mathbb{R} = \{(x, y) | x, y \in \mathbb{R}\}\) is the Cartesian product of \(\mathbb{R}\) with i
Read more

Applications of multiple integrals

-
Image
We can use multiple integrals to compute areas and volumes of various shapes. Area of a planar region Definition (Area) Let \(D\) be a bounded closed region in \(\mathbb{R}^2\). \(D\) is said to have an area if the multiple integral of the constant function 1 over \(D\), \(\iint_Ddxdy\), exists. Its value is denoted by \(\mu(D)\): \[\mu(D) = \iint_Ddxdy.\] Example . Let us calculate the area of the disk \(D = \{(x,y)\mid x^2 + y^2 \leq a^2\}\). Using the polar coordinates, \(x = r\cos\theta, y = r\sin\theta\), \(dxdy = rdrd\theta\), and the ranges of \(r\) and \(\theta\) are \([0, a]\) and \([0, 2\pi]\), respectively. Thus, \[\begin{eqnarray*} \mu(D) &=& \iint_Ddxdy\\ &=&\int_0^a\left(\int_0^{2\pi}rd\theta\right)dr\\ &=&2\pi\int_0^a rdr\\ &=&2\pi\left[\frac{r^2}{2}\right]_0^a = \pi a^2. \end{eqnarray*}\] □ Volume of a solid figure Definition (Volume) Let \(V\) be a solid figure in the \((x,y,z)\) space \(\mathbb{R}^3\). \(V\) is sai
Read more