Question

Suppose that \(f^{(n+1)}\left(x_{0}\right)\) exists, and let \(T_{n}\) be the \(n\) th Taylor polynomial of \(f\) about \(x_{0}\). Show that the function $$ E_{n}(x)=\left\\{\begin{array}{ll} \frac{f(x)-T_{n}(x)}{\left(x-x_{0}\right)^{n}}, & x \in D_{f}-\left\\{x_{0}\right\\} \\ 0, & x=x_{0} \end{array}\right. $$ is differentiable at \(x_{0}\), and find \(E_{1}^{\prime}\left(x_{0}\right) .\)

Short answer

In conclusion, the error function \(E_n(x)\) is differentiable at \(x_0\), and its derivative at this point is given by \(E_n'(x_0) = \frac{f^{(n+1)}(x_0)}{(n+1)!}\), where \(f^{(n+1)}\) is the (n+1)-th derivative of the function \(f\). When \(n=1\), the derivative of the error function is \(E_1'(x_0) = \frac{f^{(2)}(x_0)}{2!}\).

Step-by-step solution

Define the Taylor polynomial and error function

The given Taylor polynomial is: $$ T_n(x) = \sum_{k=0}^{n}\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k $$ The error function \(E_n(x)\) as given is: $$ E_n(x) = \left\{\begin{array}{ll} \frac{f(x)-T_n(x)}{(x-x_0)^n}, & x \in D_f - \{x_0\} \\ 0, & x=x_0 \end{array}\right. $$

Define the derivative of the error function

To check if \(E_n(x)\) is differentiable at \(x_0\), we need to evaluate: $$ E_n'(x_0) = \lim_{x\to x_0} \frac{E_n(x) - E_n(x_0)}{x - x_0} $$ By substituting the definition of \(E_n(x)\) into the limit, we get: $$ E_n'(x_0) = \lim_{x\to x_0} \frac{\frac{f(x)-T_n(x)}{(x-x_0)^n} - 0}{x - x_0} $$ Simplifying the expression, we get: $$ E_n'(x_0) = \lim_{x\to x_0} \frac{f(x)-T_n(x)}{(x-x_0)^{n+1}} $$

Use Taylor's theorem to find the remainder R_n(x)

According to Taylor's theorem, there exists a number \(c\in(x_0,x)\) such that: $$ f(x) = T_n(x) + R_n(x) $$ Where $$ R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-x_0)^{n+1} $$

Substitute R_n(x) into E_n'(x_0)

Using the expression for \(R_n(x)\) from Taylor's theorem, we can rewrite the limit as: $$ E_n'(x_0) = \lim_{x\to x_0} \frac{R_n(x)}{(x-x_0)^{n+1}} $$ Substituting \(R_n(x)\) in the limit, we get: $$ E_n'(x_0) = \lim_{x\to x_0} \frac{f^{(n+1)}(c)}{(n+1)!} $$

Evaluate the limit

As x approaches \(x_0\), \(c\) also approaches \(x_0\). Therefore: $$ E_n'(x_0) = \frac{f^{(n+1)}(x_0)}{(n+1)!} $$

Find E_1'(x_0)

To find \(E_1'(x_0)\), we can plug \(n=1\) into the expression for \(E_n'(x_0)\): $$ E_1'(x_0) = \frac{f^{(1+1)}(x_0)}{(1+1)!} = \frac{f^{(2)}(x_0)}{2!} $$

Key concepts

Differentiability

Differentiability is a fundamental concept in calculus that describes whether a function has a derivative at a given point. In simpler terms, if a function is differentiable at a point, it means that it is smooth and has no sharp corners or cusps at that point.

To say that a function is differentiable at a point, we must be able to find a unique tangent line to the function's graph at that point. Mathematically, a function \( f \) is differentiable at \( x_0 \) if the limit definition of the derivative \( f'(x_0) \) exists. This means that as we take smaller and smaller intervals around \( x_0 \), the change in \( f \) over the change in \( x \) should approach a single value – the slope of the tangent line. In the exercise provided, we're asked to show that the error function \( E_n(x) \) is differentiable at \( x_0 \), indicating a smooth transition of the function at that point, akin to a ‘smooth curve’ on the graph of \( f \).

A handy tip for students is to visualize differentiability in terms of the graph's appearance. If you can draw a smooth curve without lifting your pencil at the point in question, then the function is likely differentiable there.

Taylor's Theorem

Taylor's theorem is a key principle in calculus that allows us to approximate functions using polynomials, which are much simpler to work with. Specifically, Taylor's theorem states that any smooth function can be approximated by a Taylor polynomial around a point \( x_0 \) up to an error termed as the remainder term \( R_n(x) \).

The theorem provides a formula that represents the function \( f(x) \) as the sum of its Taylor series plus a remainder term. The series is the sum of derivatives of \( f \) at the point \( x_0 \), multiplied by the powers of \( x-x_0 \), and divided by factorial of the number of derivatives. The added remainder term \( R_n(x) \) is what tells us how good our approximation is, representing the difference between the actual value of \( f(x) \) and the value given by the Taylor polynomial.

Understanding the Remainder Term

In the context of the exercise, the remainder term is used to show the differentiability of the error function \( E_n(x) \) at \( x_0 \). The remainder is critical because it’s the piece that tells us how close the Taylor polynomial is to the actual function. The smaller the remainder, the better the approximation.

Limit Definition of Derivative

The limit definition of derivative is a core concept in calculus and underpins the idea of differentiability. The derivative of a function \( f \) at a point \( x_0 \) is defined as the limit of the difference quotient as \( x \) approaches \( x_0 \). In formal mathematical language, it is written as:

\[ f'(x_0) = \text{lim}_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0} \]

This expression calculates the slope of the secant line between points \( (x_0, f(x_0)) \) and \( (x, f(x)) \) on the graph of \( f \), and as \( x \) approaches \( x_0 \), that secant line becomes the tangent line at \( x_0 \). This definition is used directly in the exercise to evaluate the derivative of the error function \( E_n(x) \) at \( x_0 \).

Simplifying the Definition in Practice

During the calculation of \( E_n'(x_0) \), we apply the limit definition of the derivative to the error function. It is crucial for students to recognize that this process mirrors finding the slope of the tangent line to \( E_n(x) \) at \( x_0 \), hence proving differentiability there. The exercise takes advantage of the fact that, by using the limit definition, the complex looking error function becomes more manageable and its behavior at the point \( x_0 \) can be clearly understood.